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-rw-r--r--megalodon/library/algebra/semigroup.mg0
-rw-r--r--megalodon/library/cardinal.mg941
-rw-r--r--megalodon/library/everything.mg0
-rw-r--r--megalodon/library/relation/equivalence.mg470
-rw-r--r--megalodon/library/set.mg293
-rw-r--r--megalodon/megalodon/scraps.mg5
-rw-r--r--megalodon/ownedSep19202231613
-rw-r--r--megalodon/preamble.mgs434
-rw-r--r--megalodon/scraps.tex17
9 files changed, 33773 insertions, 0 deletions
diff --git a/megalodon/library/algebra/semigroup.mg b/megalodon/library/algebra/semigroup.mg
new file mode 100644
index 0000000..e69de29
--- /dev/null
+++ b/megalodon/library/algebra/semigroup.mg
diff --git a/megalodon/library/cardinal.mg b/megalodon/library/cardinal.mg
new file mode 100644
index 0000000..4d2af41
--- /dev/null
+++ b/megalodon/library/cardinal.mg
@@ -0,0 +1,941 @@
+Let emptyset : set := Empty.
+Let elem : set->set->prop := In.
+Let notelem : set->set->prop := fun a A => ~(In a A).
+Let pow : set->set := Power.
+Let unions : set->set := Union.
+Let union : set->set->set := binunion.
+Let cons : set -> set -> set := fun x X => binunion {x} X.
+Let xor : prop -> prop -> prop := fun p q => (p \/ q) /\ ~(p /\ q).
+Let pair : set -> set -> set := fun a b => {{a}, {a, b}}.
+Let fst : set -> set := fun p => Eps_i (fun a => exists b, p = pair a b).
+Let snd : set -> set := fun p => Eps_i (fun b => exists a, p = pair a b).
+Let nor : prop -> prop -> prop := fun p q => ~(p \/ q) .
+Definition ni := fun x0 x1 : set => (elem x1 x0).
+Fact setext : (forall A B,(((forall a,(((elem a A)<->(elem a B))))->(A = B)))).
+Admitted.
+Theorem neq_witness : (forall A B,(((A <> B)->(exists c,((xor((elem c A)/\~((elem c B))) (~((elem c A))/\(elem c B)))))))).
+Admitted.
+Definition subseteq := fun A B: set => (forall a,(((elem a A)->(elem a B)))).
+Definition is_subset := fun x0 x1 : set => (subseteq x0 x1).
+Definition supseteq := fun x0 x1 : set => (subseteq x1 x0).
+Theorem subseteq_refl : (forall A,((subseteq A A))).
+Admitted.
+Theorem subseteq_antisymmetric : (forall A B,((((subseteq A B)/\(subseteq B A))->(A = B)))).
+Admitted.
+Theorem elem_subseteq : (forall a A B,((((elem a A)/\(subseteq A B))->(elem a B)))).
+Admitted.
+Theorem not_in_subseteq : (forall A B c,((((subseteq A B)/\(notelem c B))->(notelem c A)))).
+Admitted.
+Theorem subseteq_transitive : (forall A B C,((((subseteq A B)/\(subseteq B C))->(subseteq A C)))).
+Admitted.
+Definition subset := fun A B: set => ((subseteq A B)/\(A <> B)).
+Theorem subset_irrefl : (forall A,(~((subset A A)))).
+Admitted.
+Theorem subset_transitive : (forall A B C,((((subseteq A B)/\(subseteq B C))->(subseteq A C)))).
+Admitted.
+Theorem subset_witness : (forall A B,(((subset A B)->(exists b,(((elem b B)/\(notelem b A))))))).
+Admitted.
+Definition family_of_subsets := fun x0 x1 : set => (forall A,(((elem A x0)->(subseteq A x1)))).
+Fact notin_emptyset : (forall a,((notelem a (emptyset)))).
+Admitted.
+Definition inhabited := fun A: set => (exists a,((elem a A))).
+Definition empty := fun x0 : set => ~((inhabited x0)).
+Theorem empty_eq : (forall x y,((((empty x)/\(empty y))->(x = y)))).
+Admitted.
+Theorem emptyset_subseteq : (forall a,((subseteq (emptyset) a))).
+Admitted.
+Theorem subseteq_emptyset_iff : (forall A,(((subseteq A (emptyset))<->(A = (emptyset))))).
+Admitted.
+Definition disjoint := fun A B: set => ~((exists a,(((elem a A)/\(elem a B))))).
+Definition notmeets := fun x0 x1 : set => (disjoint x0 x1).
+Definition meets := fun x0 x1 : set => ~((disjoint x0 x1)).
+Theorem disjoint_symmetric : (forall A B,(((disjoint A B)->(disjoint B A)))).
+Admitted.
+Fact in_cons : (forall x y X,(((elem x (cons y X))<->((x = y)\/(elem x X))))).
+Admitted.
+Theorem in_cons_left : (forall x X,((elem x (cons x X)))).
+Admitted.
+Theorem in_cons_right : (forall y X x,(((elem y X)->(elem y (cons x X))))).
+Admitted.
+Theorem upair_intro_left : (forall a b,((elem a (cons a (cons b (emptyset)))))).
+Admitted.
+Theorem upair_intro_right : (forall b a,((elem b (cons a (cons b (emptyset)))))).
+Admitted.
+Theorem upair_elim : (forall c a b,(((elem c (cons a (cons b (emptyset))))->((a = c)\/(b = c))))).
+Admitted.
+Theorem upair_iff : (forall c a b,(((elem c (cons a (cons b (emptyset))))<->((a = c)\/(b = c))))).
+Admitted.
+Theorem singleton_intro : (forall a,((elem a (cons a (emptyset))))).
+Admitted.
+Theorem singleton_elim : (forall a b,(((elem a (cons b (emptyset)))->(a = b)))).
+Admitted.
+Theorem singleton_iff : (forall a b,(((elem a (cons b (emptyset)))<->(a = b)))).
+Admitted.
+Definition subsingleton := fun x0 : set => (forall a b,((((elem a x0)/\(elem b x0))->(a = b)))).
+Theorem singleton_inhabited : (forall a,((inhabited (cons a (emptyset))))).
+Admitted.
+Theorem singleton_iff_inhabited_subsingleton : (forall A a,((((subsingleton A)/\(elem a A))->(A = (cons a (emptyset)))))).
+Admitted.
+Theorem singleton_subset_intro : (forall a C,(((elem a C)->(subseteq (cons a (emptyset)) C)))).
+Admitted.
+Theorem singleton_subset_elim : (forall a C,(((subseteq (cons a (emptyset)) C)->(elem a C)))).
+Admitted.
+Fact unions_iff_exists : (forall z X,(((elem z (unions X))<->(exists Y,(((elem Y X)/\(elem z Y))))))).
+Admitted.
+Theorem unions_intro : (forall A B C,((((elem A B)/\(elem B C))->(elem A (unions C))))).
+Admitted.
+Theorem unions_emptyset : ((unions (emptyset)) = (emptyset)).
+Admitted.
+Theorem unions_family : (forall F X,(((family_of_subsets F X)->(subseteq (unions F) X)))).
+Admitted.
+Definition closedunderunions := fun x0 : set => (forall M,(((is_subset M x0)->(elem (unions M) x0)))).
+Definition inters := fun A : set => {x :e ((unions A))|(forall a,(((elem a A)->(elem x a))))}.
+Theorem inters_iff_forall : (forall z X,(((elem z (inters X))<->((inhabited X)/\(forall Y,(((elem Y X)->(elem z Y)))))))).
+Admitted.
+Theorem inters_intro : (forall C A,((((inhabited C)/\(forall B,(((elem B C)->(elem A B)))))->(elem A (inters C))))).
+Admitted.
+Theorem inters_destr : (forall A C B,((((elem A (inters C))/\(elem B C))->(elem A B)))).
+Admitted.
+Theorem inters_greatest : (forall A C,((((inhabited A)/\(forall a,(((elem a A)->(subseteq C a)))))->(subseteq C (inters A))))).
+Admitted.
+Theorem subseteq_inters_iff : (forall A C,(((inhabited A)->((subseteq C (inters A))<->(forall a,(((elem a A)->(subseteq C a)))))))).
+Admitted.
+Theorem inters_subseteq_elem : (forall B A,(((elem B A)->(subseteq (inters A) B)))).
+Admitted.
+Theorem inters_singleton : (forall a,(((inters (cons a (emptyset))) = a))).
+Admitted.
+Theorem inters_emptyset : ((inters (cons (emptyset) (emptyset))) = (emptyset)).
+Admitted.
+Fact union_iff : (forall a A B,(((elem a (union A B))<->((elem a A)\/(elem a B))))).
+Admitted.
+Theorem union_intro_left : (forall c A B,(((elem c A)->(elem c (union A B))))).
+Admitted.
+Theorem union_intro_right : (forall c B A,(((elem c B)->(elem c (union A B))))).
+Admitted.
+Theorem union_comm : (forall A B,(((union A B) = (union B A)))).
+Admitted.
+Theorem union_assoc : (forall A B C,(((union (union A B) C) = (union A (union B C))))).
+Admitted.
+Theorem union_idempotent : (forall A,(((union A A) = A))).
+Admitted.
+Theorem subseteq_union_iff : (forall A B C,(((subseteq (union A B) C)<->((subseteq A C)/\(subseteq B C))))).
+Admitted.
+Theorem union_upper_left : (forall A B,((subseteq A (union A B)))).
+Admitted.
+Theorem union_upper_right : (forall B A,((subseteq B (union A B)))).
+Admitted.
+Theorem union_subseteq_union : (forall A C B D,((((subseteq A C)/\(subseteq B D))->(subseteq (union A B) (union C D))))).
+Admitted.
+Theorem union_emptyset : (forall A,(((union A (emptyset)) = A))).
+Admitted.
+Theorem union_emptyset_intro : (forall A B,((((A = (emptyset))/\(B = (emptyset)))->((union A B) = (emptyset))))).
+Admitted.
+Theorem union_emptyset_elim_left : (forall A B,((((union A B) = (emptyset))->(A = (emptyset))))).
+Admitted.
+Theorem union_emptyset_elim_right : (forall A B,((((union A B) = (emptyset))->(B = (emptyset))))).
+Admitted.
+Theorem union_absorb_subseteq_left : (forall A B,(((subseteq A B)->((union A B) = B)))).
+Admitted.
+Theorem union_absorb_subseteq_right : (forall A B,(((subseteq A B)->((union B A) = B)))).
+Admitted.
+Theorem union_eq_self_implies_subseteq : (forall A B,((((union A B) = B)->(subseteq A B)))).
+Admitted.
+Theorem unions_cons : (forall b A,(((unions (cons b A)) = (union b (unions A))))).
+Admitted.
+Theorem union_cons : (forall b A C,(((union (cons b A) C) = (cons b (union A C))))).
+Admitted.
+Theorem union_absorb_left : (forall A B,(((union A (union A B)) = (union A B)))).
+Admitted.
+Theorem union_absorb_right : (forall A B,(((union (union A B) B) = (union A B)))).
+Admitted.
+Theorem union_comm_left : (forall A B C,(((union A (union B C)) = (union B (union A C))))).
+Admitted.
+Definition closedunderunion := fun x0 : set => (forall U V,((((elem U x0)/\(elem V x0))->(elem (union U V) x0)))).
+Definition inter := fun A B : set => {a :e (A)|(elem a B)}.
+Theorem inter_intro : (forall c A B,((((elem c A)/\(elem c B))->(elem c (inter A B))))).
+Admitted.
+Theorem inter_elim_left : (forall c A B,(((elem c (inter A B))->(elem c A)))).
+Admitted.
+Theorem inter_elim_right : (forall c A B,(((elem c (inter A B))->(elem c B)))).
+Admitted.
+Theorem inter_as_inters : (forall A B,(((inters (cons A (cons B (emptyset)))) = (inter A B)))).
+Admitted.
+Theorem inter_comm : (forall A B,(((inter A B) = (inter B A)))).
+Admitted.
+Theorem inter_assoc : (forall A B C,(((inter (inter A B) C) = (inter A (inter B C))))).
+Admitted.
+Theorem inter_idempotent : (forall A,(((inter A A) = A))).
+Admitted.
+Theorem inter_subseteq : (forall A B,((subseteq (inter A B) A))).
+Admitted.
+Theorem inter_emptyset : (forall A,(((inter A (emptyset)) = (emptyset)))).
+Admitted.
+Theorem inter_absorb_supseteq_right : (forall A B,(((subseteq A B)->((inter A B) = A)))).
+Admitted.
+Theorem inter_absorb_supseteq_left : (forall A B,(((subseteq A B)->((inter B A) = A)))).
+Admitted.
+Theorem inter_eq_left_implies_subseteq : (forall A B,((((inter A B) = A)->(subseteq A B)))).
+Admitted.
+Theorem subseteq_inter_iff : (forall C A B,(((subseteq C (inter A B))<->((subseteq C A)/\(subseteq C B))))).
+Admitted.
+Theorem inter_lower_left : (forall A B,((subseteq (inter A B) A))).
+Admitted.
+Theorem inter_lower_right : (forall A B,((subseteq (inter A B) B))).
+Admitted.
+Theorem inter_absorb_left : (forall A B,(((inter A (inter A B)) = (inter A B)))).
+Admitted.
+Theorem inter_absorb_right : (forall A B,(((inter (inter A B) B) = (inter A B)))).
+Admitted.
+Theorem inter_comm_left : (forall A B C,(((inter A (inter B C)) = (inter B (inter A C))))).
+Admitted.
+Definition closedunderinter := fun x0 : set => (forall U V,((((elem U x0)/\(elem V x0))->(elem (inter U V) x0)))).
+Theorem inter_distrib_union : (forall x y z,(((inter x (union y z)) = (union (inter x y) (inter x z))))).
+Admitted.
+Theorem union_distrib_inter : (forall x y z,(((union x (inter y z)) = (inter (union x y) (union x z))))).
+Admitted.
+Theorem union_inter_assoc_intro : (forall C A B,(((subseteq C A)->((union (inter A B) C) = (inter A (union B C)))))).
+Admitted.
+Theorem union_inter_assoc_elim : (forall A B C,((((union (inter A B) C) = (inter A (union B C)))->(subseteq C A)))).
+Admitted.
+Theorem union_inter_crazy : (forall A B C,(((union (union (inter A B) (inter B C)) (inter C A)) = (inter (inter (union A B) (union B C)) (union C A))))).
+Admitted.
+Theorem inters_distrib_union : (forall A B,((((inhabited A)/\(inhabited B))->((inters (union A B)) = (inter (inters A) (inters B)))))).
+Admitted.
+Definition setminus := fun A B : set => {a :e (A)|~((elem a B))}.
+Theorem setminus_intro : (forall a A B,((((elem a A)/\(notelem a B))->(elem a (setminus A B))))).
+Admitted.
+Theorem setminus_elim_left : (forall a A B,(((elem a (setminus A B))->(elem a A)))).
+Admitted.
+Theorem setminus_elim_right : (forall a A B,(((elem a (setminus A B))->(notelem a B)))).
+Admitted.
+Theorem setminus_emptyset : (forall x,(((setminus x (emptyset)) = x))).
+Admitted.
+Theorem emptyset_setminus : (forall x,(((setminus (emptyset) x) = (emptyset)))).
+Admitted.
+Theorem setminus_self : (forall x,(((setminus x x) = (emptyset)))).
+Admitted.
+Theorem setminus_setminus : (forall x y,(((setminus x (setminus x y)) = (inter x y)))).
+Admitted.
+Theorem double_relative_complement : (forall y x,(((subseteq y x)->((setminus x (setminus x y)) = y)))).
+Admitted.
+Theorem setminus_inter : (forall x y z,(((setminus x (inter y z)) = (union (setminus x y) (setminus x z))))).
+Admitted.
+Theorem setminus_union : (forall x y z,(((setminus x (union y z)) = (inter (setminus x y) (setminus x z))))).
+Admitted.
+Theorem inter_setminus : (forall x y z,(((inter x (setminus y z)) = (setminus (inter x y) (inter x z))))).
+Admitted.
+Theorem difference_with_proper_subset_is_inhabited : (forall A B,(((subset A B)->(inhabited (setminus B A))))).
+Admitted.
+Theorem setminus_subseteq : (forall B A,((subseteq (setminus B A) B))).
+Admitted.
+Theorem subseteq_setminus : (forall C A B,((((subseteq C A)/\((inter C B) = (emptyset)))->(subseteq C (setminus A B))))).
+Admitted.
+Theorem subseteq_implies_setminus_supseteq : (forall A B C,(((subseteq A B)->(supseteq (setminus C A) (setminus C B))))).
+Admitted.
+Theorem setminus_absorb_right : (forall A B,((((inter A B) = (emptyset))->((setminus A B) = A)))).
+Admitted.
+Theorem setminus_eq_emptyset_iff_subseteq : (forall A B,((((setminus A B) = (emptyset))<->(subseteq A B)))).
+Admitted.
+Theorem subseteq_setminus_cons_intro : (forall B A C c,((((subseteq B (setminus A C))/\(notelem c B))->(subseteq B (setminus A (cons c C)))))).
+Admitted.
+Theorem subseteq_setminus_cons_elim : (forall B A c C,(((subseteq B (setminus A (cons c C)))->((subseteq B (setminus A C))/\(notelem c B))))).
+Admitted.
+Theorem setminus_cons : (forall A a B,(((setminus A (cons a B)) = (setminus (setminus A (cons a (emptyset))) B)))).
+Admitted.
+Theorem setminus_cons_flip : (forall A a B,(((setminus A (cons a B)) = (setminus (setminus A B) (cons a (emptyset)))))).
+Admitted.
+Theorem setminus_disjoint : (forall A B,(((inter A (setminus B A)) = (emptyset)))).
+Admitted.
+Theorem setminus_partition : (forall A B,(((subseteq A B)->((union A (setminus B A)) = B)))).
+Admitted.
+Theorem subseteq_union_setminus : (forall A B,((subseteq A (union B (setminus A B))))).
+Admitted.
+Theorem double_complement : (forall A B C,((((subseteq A B)/\(subseteq B C))->((setminus B (setminus C A)) = A)))).
+Admitted.
+Theorem double_complement_union : (forall A B,(((setminus (union A B) (setminus B A)) = A))).
+Admitted.
+Theorem setminus_eq_inter_complement : (forall A C B,((((subseteq A C)/\(subseteq B C))->((setminus A B) = (inter A (setminus C B)))))).
+Admitted.
+Fact pair_eq_iff : (forall a b aprime bprime,((((pair a b) = (pair aprime bprime))<->((a = aprime)/\(b = bprime))))).
+Admitted.
+Fact pair_neq_emptyset : (forall a b,(((pair a b) <> (emptyset)))).
+Admitted.
+Fact pair_neq_fst : (forall a b,(((pair a b) <> a))).
+Admitted.
+Fact pair_neq_snd : (forall a b,(((pair a b) <> b))).
+Admitted.
+Theorem triple_eq_iff : (forall a b c aprime bprime cprime,((((pair a (pair b c)) = (pair aprime (pair bprime cprime)))<->(((a = aprime)/\(b = bprime))/\(c = cprime))))).
+Admitted.
+Fact fst_eq : (forall a b,(((fst (pair a b)) = a))).
+Admitted.
+Fact snd_eq : (forall a b,(((snd (pair a b)) = b))).
+Admitted.
+Theorem pair_eq_pair_of_proj : (forall a b,(((pair a b) = (pair (fst (pair a b)) (snd (pair a b)))))).
+Admitted.
+Definition times := fun A B : set => ReplSep2 (A)(fun dummyVar => B)(fun a b => True)(fun a b => (pair a b)).
+Theorem times_tuple_elim : (forall x y X Y,(((elem (pair x y) (times X Y))->((elem x X)/\(elem y Y))))).
+Admitted.
+Theorem times_tuple_intro : (forall x X y Y,((((elem x X)/\(elem y Y))->(elem (pair x y) (times X Y))))).
+Admitted.
+Theorem times_empty_left : (forall Y,(((times (emptyset) Y) = (emptyset)))).
+Admitted.
+Theorem times_empty_right : (forall X,(((times X (emptyset)) = (emptyset)))).
+Admitted.
+Theorem times_empty_iff : (forall X Y,(((empty (times X Y))<->((empty X)\/(empty Y))))).
+Admitted.
+Theorem fst_type : (forall c A B,(((elem c (times A B))->(elem (fst c) A)))).
+Admitted.
+Theorem snd_type : (forall c A B,(((elem c (times A B))->(elem (snd c) B)))).
+Admitted.
+Theorem times_elem_is_tuple : (forall p X Y,(((elem p (times X Y))->(exists x y,(((elem x X)/\((elem y Y)/\(p = (pair x y))))))))).
+Admitted.
+Theorem times_proj_elim : (forall p X Y,(((elem p (times X Y))->((elem (fst p) X)/\(elem (snd p) Y))))).
+Admitted.
+Theorem cons_subseteq_intro : (forall x X Y,((((elem x X)/\(subseteq Y X))->(subseteq (cons x Y) X)))).
+Admitted.
+Theorem cons_subseteq_elim : (forall x Y X,(((subseteq (cons x Y) X)->((elem x X)/\(subseteq Y X))))).
+Admitted.
+Theorem cons_subseteq_iff : (forall x Y X,(((subseteq (cons x Y) X)<->((elem x X)/\(subseteq Y X))))).
+Admitted.
+Theorem subseteq_cons_right : (forall C B a,(((subseteq C B)->(subseteq C (cons a B))))).
+Admitted.
+Theorem subseteq_cons_self : (forall X y,((subseteq X (cons y X)))).
+Admitted.
+Definition remove_point := fun x0 x1 : set => (setminus x1 (cons x0 (emptyset))).
+Theorem subseteq_cons_intro_left : (forall a C B,((((elem a C)/\(subseteq (remove_point a C) B))->(subseteq C (cons a B))))).
+Admitted.
+Theorem subseteq_cons_intro_right : (forall C B a,(((subseteq C B)->(subseteq C (cons a B))))).
+Admitted.
+Theorem subseteq_cons_elim : (forall C a B,(((subseteq C (cons a B))->((subseteq C B)\/((elem a C)/\(subseteq (remove_point a C) B)))))).
+Admitted.
+Theorem subseteq_cons_iff : (forall C a B,(((subseteq C (cons a B))<->((subseteq C B)\/((elem a C)/\(subseteq (remove_point a C) B)))))).
+Admitted.
+Theorem remove_point_eq_setminus_singletong : (forall a B,(((remove_point a B) = (setminus B (cons a (emptyset)))))).
+Admitted.
+Theorem union_eq_cons : (forall a B,(((union (cons a (emptyset)) B) = (cons a B)))).
+Admitted.
+Theorem cons_comm : (forall a b C,(((cons a (cons b C)) = (cons b (cons a C))))).
+Admitted.
+Theorem cons_absorb : (forall a A,(((elem a A)->((cons a A) = A)))).
+Admitted.
+Theorem cons_remove : (forall a A,(((elem a A)->((cons a (setminus A (cons a (emptyset)))) = A)))).
+Admitted.
+Theorem cons_idempotent : (forall a B,(((cons a (cons a B)) = (cons a B)))).
+Admitted.
+Theorem inters_cons : (forall B a,(((inhabited B)->((inters (cons a B)) = (inter a (inters B)))))).
+Admitted.
+Definition powerset_of := fun x0 : set => (pow x0).
+Fact pow_iff : (forall B A,(((elem B (pow A))<->(subseteq B A)))).
+Admitted.
+Theorem powerset_intro : (forall A B,(((subseteq A B)->(elem A (pow B))))).
+Admitted.
+Theorem powerset_elim : (forall A B,(((elem A (pow B))->(subseteq A B)))).
+Admitted.
+Theorem powerset_bottom : (forall A,((elem (emptyset) (pow A)))).
+Admitted.
+Theorem powerset_top : (forall A,((elem A (pow A)))).
+Admitted.
+Theorem unions_subseteq_of_powerset_is_subseteq : (forall B A,(((is_subset B (pow A))->(subseteq (unions B) A)))).
+Admitted.
+Theorem unions_powerset : (forall A,(((unions (pow A)) = A))).
+Admitted.
+Theorem inters_powerset : (forall A,(((inters (pow A)) = (emptyset)))).
+Admitted.
+Theorem union_powersets_subseteq : (forall A B,((subseteq (union (pow A) (pow B)) (pow (union A B))))).
+Admitted.
+Theorem powerset_emptyset : ((pow (emptyset)) = (cons (emptyset) (emptyset))).
+Admitted.
+Theorem powerset_union_subseteq : (forall A B,((subseteq (union (pow A) (pow B)) (pow (union A B))))).
+Admitted.
+Theorem subseteq_pow_unions : (forall A,((subseteq A (pow (unions A))))).
+Admitted.
+Theorem unions_pow : (forall A,(((unions (pow A)) = A))).
+Admitted.
+Theorem unions_elem_pow_iff : (forall A B,(((elem (unions A) (pow B))<->(elem A (pow (pow B)))))).
+Admitted.
+Theorem pow_inter : (forall A B,(((pow (inter A B)) = (inter (pow A) (pow B))))).
+Admitted.
+Theorem times_subseteq_left : (forall A C B,(((subseteq A C)->(subseteq (times A B) (times C B))))).
+Admitted.
+Theorem times_subseteq_right : (forall B D A,(((subseteq B D)->(subseteq (times A B) (times A D))))).
+Admitted.
+Theorem inter_times_intro : (forall w A B C D,(((elem w (times (inter A B) (inter C D)))->(elem w (inter (times A C) (times B D)))))).
+Admitted.
+Theorem inter_times_elim : (forall w A C B D,(((elem w (inter (times A C) (times B D)))->(elem w (times (inter A B) (inter C D)))))).
+Admitted.
+Theorem inter_times : (forall A B C D,(((times (inter A B) (inter C D)) = (inter (times A C) (times B D))))).
+Admitted.
+Theorem inter_times_right : (forall X Y Z,(((times (inter X Y) Z) = (inter (times X Z) (times Y Z))))).
+Admitted.
+Theorem inter_times_left : (forall X Y Z,(((times X (inter Y Z)) = (inter (times X Y) (times X Z))))).
+Admitted.
+Theorem union_times_intro : (forall w A B C D,(((elem w (times (union A B) (union C D)))->(elem w (union (union (union (times A C) (times B D)) (times A D)) (times B C)))))).
+Admitted.
+Theorem union_times_elim : (forall w A C B D,(((elem w (union (union (union (times A C) (times B D)) (times A D)) (times B C)))->(elem w (times (union A B) (union C D)))))).
+Admitted.
+Theorem union_times : (forall A B C D,(((times (union A B) (union C D)) = (union (union (union (times A C) (times B D)) (times A D)) (times B C))))).
+Admitted.
+Theorem union_times_left : (forall X Y Z,(((times (union X Y) Z) = (union (times X Z) (times Y Z))))).
+Admitted.
+Theorem union_times_right : (forall X Y Z,(((times X (union Y Z)) = (union (times X Y) (times X Z))))).
+Admitted.
+Definition elemminimal := fun x0 x1 : set => ((elem x0 x1)/\(notmeets x0 x1)).
+Theorem regularity_aux : (forall a A,(((elem a A)->(exists b,(((elem b A)/\(notmeets b A))))))).
+Admitted.
+Theorem regularity : (forall A,(((inhabited A)->(exists x1,((elemminimal x1 A)))))).
+Admitted.
+Theorem foundation : (forall A,(((A = (emptyset))\/(exists a,(((elem a A)/\(forall x,(((elem x a)->(notelem x A)))))))))).
+Admitted.
+Theorem in_irrefl : (forall A,(~((elem A A)))).
+Admitted.
+Theorem in_implies_neq : (forall a A,(((elem a A)->(a <> A)))).
+Admitted.
+Theorem in_asymmetric : (forall a b,(((elem a b)->(notelem b a)))).
+Admitted.
+Definition suc := fun x : set => (cons x x).
+Theorem suc_intro_self : (forall x,((elem x (suc x)))).
+Admitted.
+Theorem suc_intro_in : (forall x y,(((elem x y)->(elem x (suc y))))).
+Admitted.
+Theorem suc_elim : (forall x y,(((elem x (suc y))->((x = y)\/(elem x y))))).
+Admitted.
+Theorem suc_iff : (forall x y,(((elem x (suc y))<->((x = y)\/(elem x y))))).
+Admitted.
+Theorem suc_neq_emptyset : (forall x,(((suc x) <> (emptyset)))).
+Admitted.
+Theorem suc_subseteq_implies_in : (forall x y,(((subseteq (suc x) y)->(elem x y)))).
+Admitted.
+Theorem suc_neq_self : (forall x,(((suc x) <> x))).
+Admitted.
+Theorem suc_injective : (forall x y,((((suc x) = (suc y))->(x = y)))).
+Admitted.
+Theorem subseteq_self_suc_intro : (forall x,((subseteq x (suc x)))).
+Admitted.
+Theorem suc_subseteq_intro : (forall x y,((((elem x y)/\(subseteq x y))->(subseteq (suc x) y)))).
+Admitted.
+Theorem suc_subseteq_elim : (forall x y,(((subseteq (suc x) y)->((elem x y)/\(subseteq x y))))).
+Admitted.
+Theorem suc_next_subset : (forall x,(~((exists z,(((subset x z)/\(subset z (suc x)))))))).
+Admitted.
+Definition relation := fun R: set => (forall w,(((elem w R)->(exists x y,((w = (pair x y))))))).
+Definition comparable := fun a b R: set => ((elem (pair a b) R)\/(elem (pair b a) R)).
+Theorem relext : (forall R S,(((((relation R)/\(relation S))/\(forall x y,(((elem (pair x y) R)<->(elem (pair x y) S)))))->(R = S)))).
+Admitted.
+Definition family_of_relations := fun x0 : set => (forall x2,(((elem x2 x0)->(relation x2)))).
+Theorem unions_of_family_of_relations_is_relation : (forall F,(((family_of_relations F)->(relation (unions F))))).
+Admitted.
+Theorem inters_of_family_of_relations_is_relation : (forall F,(((family_of_relations F)->(relation (inters F))))).
+Admitted.
+Theorem union_relations_is_relation : (forall R S,((((relation R)/\(relation S))->(relation (union R S))))).
+Admitted.
+Theorem union_relations_is_relation_type : (forall R A B S C D,((((subseteq R (times A B))/\(subseteq S (times C D)))->(subseteq (union R S) (times (union A C) (union B D)))))).
+Admitted.
+Theorem inter_relations_is_relation : (forall R S,((((relation R)/\(relation S))->(relation (inter R S))))).
+Admitted.
+Theorem setminus_relations_is_relation : (forall R S,((((relation R)/\(relation S))->(relation (setminus R S))))).
+Admitted.
+Definition converse_relation := fun R : set => let MkReplFun := fun w : set => (Eps_i (fun z=>(exists x y,(((w = (pair x y))/\(z = (pair y x))))))) in {MkReplFun w|w :e (R)}.
+Theorem converse_intro : (forall y x R,(((elem (pair y x) R)->(elem (pair x y) (converse_relation R))))).
+Admitted.
+Theorem converse_elim : (forall x y R,(((elem (pair x y) (converse_relation R))->(elem (pair y x) R)))).
+Admitted.
+Theorem converse_iff : (forall x y R,(((elem (pair x y) (converse_relation R))<->(elem (pair y x) R)))).
+Admitted.
+Theorem converse_is_relation : (forall R,((relation (converse_relation R)))).
+Admitted.
+Theorem converse_converse_iff : (forall x y R,(((elem (pair x y) (converse_relation (converse_relation R)))<->(elem (pair x y) R)))).
+Admitted.
+Theorem converse_converse_eq : (forall R,(((relation R)->((converse_relation (converse_relation R)) = R)))).
+Admitted.
+Theorem converse_type : (forall R A B,(((subseteq R (times A B))->(subseteq (converse_relation R) (times B A))))).
+Admitted.
+Theorem converse_times : (forall B A,(((converse_relation (times B A)) = (times A B)))).
+Admitted.
+Theorem converse_emptyset : ((converse_relation (emptyset)) = (emptyset)).
+Admitted.
+Theorem converse_subseteq_intro : (forall R S,(((relation R)->((subseteq R S)->(subseteq (converse_relation R) (converse_relation S)))))).
+Admitted.
+Theorem converse_subseteq_elim : (forall R S,(((relation R)->((subseteq (converse_relation R) (converse_relation S))->(subseteq R S))))).
+Admitted.
+Theorem converse_subseteq_iff : (forall R S,(((relation R)->((subseteq (converse_relation R) (converse_relation S))<->(subseteq R S))))).
+Admitted.
+Theorem converse_union : (forall R S,(((converse_relation (union R S)) = (union (converse_relation R) (converse_relation S))))).
+Admitted.
+Theorem converse_inter : (forall R S,(((converse_relation (inter R S)) = (inter (converse_relation R) (converse_relation S))))).
+Admitted.
+Theorem converse_setminus : (forall R S,(((converse_relation (setminus R S)) = (setminus (converse_relation R) (converse_relation S))))).
+Admitted.
+Definition dom := fun R : set => let MkReplFun := fun w : set => (Eps_i (fun x=>(exists y,((w = (pair x y)))))) in {MkReplFun w|w :e (R)}.
+Theorem dom_iff : (forall a R,(((elem a (dom R))<->(exists b,((elem (pair a b) R)))))).
+Admitted.
+Theorem dom_intro : (forall a b R,(((elem (pair a b) R)->(elem a (dom R))))).
+Admitted.
+Theorem dom_emptyset : ((dom (emptyset)) = (emptyset)).
+Admitted.
+Theorem dom_times : (forall A B,((subseteq (dom (times A B)) A))).
+Admitted.
+Theorem dom_times_inhabited : (forall b B A,(((elem b B)->((dom (times A B)) = A)))).
+Admitted.
+Theorem dom_cons : (forall a b R,(((dom (cons (pair a b) R)) = (cons a (dom R))))).
+Admitted.
+Theorem dom_union : (forall A B,(((dom (union A B)) = (union (dom A) (dom B))))).
+Admitted.
+Theorem dom_inter : (forall A B,((subseteq (dom (inter A B)) (inter (dom A) (dom B))))).
+Admitted.
+Theorem dom_setminus : (forall A B,((supseteq (dom (setminus A B)) (setminus (dom A) (dom B))))).
+Admitted.
+Definition ran := fun R : set => let MkReplFun := fun w : set => (Eps_i (fun y=>(exists x,((w = (pair x y)))))) in {MkReplFun w|w :e (R)}.
+Theorem ran_iff : (forall b R,(((elem b (ran R))<->(exists a,((elem (pair a b) R)))))).
+Admitted.
+Theorem ran_intro : (forall a b R,(((elem (pair a b) R)->(elem b (ran R))))).
+Admitted.
+Theorem ran_emptyset : ((ran (emptyset)) = (emptyset)).
+Admitted.
+Theorem ran_times : (forall A B,((subseteq (ran (times A B)) B))).
+Admitted.
+Theorem ran_times_inhabited : (forall a A B,(((elem a A)->((ran (times A B)) = B)))).
+Admitted.
+Theorem ran_cons : (forall a b R,(((ran (cons (pair a b) R)) = (cons b (ran R))))).
+Admitted.
+Theorem ran_union : (forall A B,(((ran (union A B)) = (union (ran A) (ran B))))).
+Admitted.
+Theorem ran_inter : (forall A B,((subseteq (ran (inter A B)) (inter (ran A) (ran B))))).
+Admitted.
+Theorem ran_setminus : (forall A B,((supseteq (ran (setminus A B)) (setminus (ran A) (ran B))))).
+Admitted.
+Theorem dom_converse : (forall R,(((dom (converse_relation R)) = (ran R)))).
+Admitted.
+Theorem ran_converse : (forall R,(((ran (converse_relation R)) = (dom R)))).
+Admitted.
+Definition fld := fun R : set => (union (dom R) (ran R)).
+Theorem fld_iff : (forall c R,(((elem c (fld R))<->(exists d,(((elem (pair c d) R)\/(elem (pair d c) R))))))).
+Admitted.
+Theorem fld_intro_left : (forall a b R,(((elem (pair a b) R)->(elem a (fld R))))).
+Admitted.
+Theorem fld_intro_right : (forall a b R,(((elem (pair a b) R)->(elem b (fld R))))).
+Admitted.
+Theorem dom_subseteq_fld : (forall R,((subseteq (dom R) (fld R)))).
+Admitted.
+Theorem ran_subseteq_fld : (forall R,((subseteq (ran R) (fld R)))).
+Admitted.
+Theorem fld_times : (forall A B,((subseteq (fld (times A B)) (union A B)))).
+Admitted.
+Theorem relation_elem_times_fld : (forall R w,((((relation R)/\(elem w R))->(elem w (times (fld R) (fld R)))))).
+Admitted.
+Theorem relation_subseteq_times_fld : (forall R,(((relation R)->(subseteq R (times (fld R) (fld R)))))).
+Admitted.
+Theorem fld_universal : (forall A,(((fld (times A A)) = A))).
+Admitted.
+Theorem fld_emptyset : ((fld (emptyset)) = (emptyset)).
+Admitted.
+Theorem fld_cons : (forall a b R,(((fld (cons (pair a b) R)) = (cons a (cons b (fld R)))))).
+Admitted.
+Theorem fld_union : (forall A B,(((fld (union A B)) = (union (fld A) (fld B))))).
+Admitted.
+Theorem fld_inter : (forall A B,((subseteq (fld (inter A B)) (inter (fld A) (fld B))))).
+Admitted.
+Theorem fld_setminus : (forall A B,((supseteq (fld (setminus A B)) (setminus (fld A) (fld B))))).
+Admitted.
+Theorem fld_converse : (forall R,(((fld (converse_relation R)) = (fld R)))).
+Admitted.
+Definition img := fun R A : set => {b :e ((ran R))|(exists a,(((elem a A)/\(elem (pair a b) R))))}.
+Theorem img_elem_intro : (forall a A b R,((((elem a A)/\(elem (pair a b) R))->(elem b (img R A))))).
+Admitted.
+Theorem img_iff : (forall b R A,(((elem b (img R A))<->(exists a,(((elem a A)/\(elem (pair a b) R))))))).
+Admitted.
+Theorem img_subseteq : (forall A B R,(((subseteq A B)->(subseteq (img R A) (img R B))))).
+Admitted.
+Theorem img_subseteq_ran : (forall R A,((subseteq (img R A) (ran R)))).
+Admitted.
+Theorem img_dom : (forall R,(((img R (dom R)) = (ran R)))).
+Admitted.
+Theorem img_union : (forall R A B,(((img R (union A B)) = (union (img R A) (img R B))))).
+Admitted.
+Theorem img_inter : (forall R A B,((subseteq (img R (inter A B)) (inter (img R A) (img R B))))).
+Admitted.
+Theorem img_setminus : (forall R A B,((supseteq (img R (setminus A B)) (setminus (img R A) (img R B))))).
+Admitted.
+Theorem img_singleton_iff : (forall b R a,(((elem b (img R (cons a (emptyset))))<->(elem (pair a b) R)))).
+Admitted.
+Theorem img_singleton_intro : (forall b R a,(((elem b (img R (cons a (emptyset))))->((elem b (ran R))/\(elem (pair a b) R))))).
+Admitted.
+Theorem img_singleton : (forall R a,(((img R (cons a (emptyset))) = {b :e ((ran R))|(elem (pair a b) R)}))).
+Admitted.
+Theorem img_emptyset : (forall R,(((img R (emptyset)) = (emptyset)))).
+Admitted.
+Definition preimg := fun R B : set => {a :e ((dom R))|(exists b,(((elem b B)/\(elem (pair a b) R))))}.
+Theorem preimg_iff : (forall a R B,(((elem a (preimg R B))<->(exists b,(((elem b B)/\(elem (pair a b) R))))))).
+Admitted.
+Theorem preim_eq_img_of_converse : (forall R B,(((preimg R B) = (img (converse_relation R) B)))).
+Admitted.
+Theorem preimg_subseteq : (forall A B R,(((subseteq A B)->(subseteq (preimg R A) (preimg R B))))).
+Admitted.
+Theorem preimg_subseteq_dom : (forall R A,((subseteq (preimg R A) (dom R)))).
+Admitted.
+Theorem preimg_union : (forall R A B,(((preimg R (union A B)) = (union (preimg R A) (preimg R B))))).
+Admitted.
+Theorem preimg_inter : (forall R A B,((subseteq (preimg R (inter A B)) (inter (preimg R A) (preimg R B))))).
+Admitted.
+Theorem preimg_setminus : (forall R A B,((supseteq (preimg R (setminus A B)) (setminus (preimg R A) (preimg R B))))).
+Admitted.
+Definition upward_closure := fun R a : set => {b :e ((ran R))|(elem (pair a b) R)}.
+Definition downward_closure := fun R b : set => {a :e ((dom R))|(elem (pair a b) R)}.
+Theorem downward_closure_iff : (forall a R b,(((elem a (downward_closure R b))<->(elem (pair a b) R)))).
+Admitted.
+Definition circ := fun S R : set => ReplSep2 ((dom R))(fun dummyVar => (ran S))(fun x z => (exists y,(((elem (pair x y) R)/\(elem (pair y z) S)))))(fun x z => (pair x z)).
+Theorem circ_is_relation : (forall S R,((relation (circ S R)))).
+Admitted.
+Theorem circ_elem_intro : (forall x y R z S,((((elem (pair x y) R)/\(elem (pair y z) S))->(elem (pair x z) (circ S R))))).
+Admitted.
+Theorem circ_elem_elim : (forall x z S R,(((elem (pair x z) (circ S R))->(exists y,(((elem (pair x y) R)/\(elem (pair y z) S))))))).
+Admitted.
+Theorem circ_iff : (forall x z S R,(((elem (pair x z) (circ S R))<->(exists y,(((elem (pair x y) R)/\(elem (pair y z) S))))))).
+Admitted.
+Theorem circ_assoc : (forall T S R,(((circ (circ T S) R) = (circ T (circ S R))))).
+Admitted.
+Theorem circ_converse_intro_tuple : (forall a c R S,(((elem (pair a c) (circ (converse_relation R) (converse_relation S)))->(elem (pair a c) (converse_relation (circ S R)))))).
+Admitted.
+Theorem circ_converse_elim : (forall a c S R,(((elem (pair a c) (converse_relation (circ S R)))->(elem (pair a c) (circ (converse_relation R) (converse_relation S)))))).
+Admitted.
+Theorem circ_converse : (forall S R,(((converse_relation (circ S R)) = (circ (converse_relation R) (converse_relation S))))).
+Admitted.
+Definition restrl := fun R X : set => {w :e (R)|(exists x y,(((elem x X)/\(w = (pair x y)))))}.
+Theorem restrl_iff : (forall a b R X,(((elem (pair a b) (restrl R X))<->((elem (pair a b) R)/\(elem a X))))).
+Admitted.
+Theorem restrl_subseteq : (forall R X,((subseteq (restrl R X) R))).
+Admitted.
+Theorem elem_dom_of_restrl_implies_elem_dom_and_restr : (forall x R X,(((elem x (dom (restrl R X)))->((elem x (dom R))/\(elem x X))))).
+Admitted.
+Theorem elem_dom_and_restr_implies_elem_of_restr : (forall x R X,((((elem x (dom R))/\(elem x X))->(elem x (dom (restrl R X)))))).
+Admitted.
+Theorem restrl_eq_inter : (forall R X,(((relation R)->((restrl R X) = (inter R (times X (ran R))))))).
+Admitted.
+Theorem dom_of_restrl_eq_inter : (forall R X,(((relation R)->((dom (restrl R X)) = (inter (dom R) X))))).
+Admitted.
+Theorem restrl_restrl : (forall V U R,(((subseteq V U)->((restrl (restrl R U) V) = (restrl R V))))).
+Admitted.
+Theorem restrl_by_dom : (forall R,(((relation R)->((restrl R (dom R)) = R)))).
+Admitted.
+Theorem restrl_dom : (forall R X,((subseteq (dom (restrl R X)) X))).
+Admitted.
+Theorem restrl_ran_elim : (forall X R b,((((subseteq X (dom R))/\(elem b (ran (restrl R X))))->(elem b (img R X))))).
+Admitted.
+Theorem restrl_ran_intro : (forall X R b,((((subseteq X (dom R))/\(elem b (img R X)))->(elem b (ran (restrl R X)))))).
+Admitted.
+Theorem restrl_ran : (forall X R,(((subseteq X (dom R))->((ran (restrl R X)) = (img R X))))).
+Admitted.
+Theorem restrl_img : (forall X R A,(((subseteq X (dom R))->((img (restrl R X) A) = (img R (inter X A)))))).
+Admitted.
+Definition binary_relation_on := fun x0 x1 : set => (subseteq x0 (times x1 x1)).
+Theorem relation_subseteq_intro_elem : (forall R B A w,((((((relation R)/\(subseteq (ran R) B))/\(subseteq (dom R) A))/\(elem w R))->(elem w (times A B))))).
+Admitted.
+Theorem relation_subseteq_intro : (forall R B A,(((((relation R)/\(subseteq (ran R) B))/\(subseteq (dom R) A))->(subseteq R (times A B))))).
+Admitted.
+Theorem relation_subseteq_implies_dom_subseteq_elem : (forall R A B a,((((subseteq R (times A B))/\(elem a (dom R)))->(elem a A)))).
+Admitted.
+Theorem relation_subseteq_implies_dom_subseteq : (forall R A B,(((subseteq R (times A B))->(subseteq (dom R) A)))).
+Admitted.
+Theorem relation_subseteq_implies_ran_subseteq_elem : (forall R A B b,((((subseteq R (times A B))/\(elem b (ran R)))->(elem b B)))).
+Admitted.
+Theorem relation_subseteq_implies_ran_subseteq : (forall R A B,(((subseteq R (times A B))->(subseteq (ran R) B)))).
+Admitted.
+Definition rels := fun A B : set => (pow (times A B)).
+Theorem rels_intro : (forall R A B,(((subseteq R (times A B))->(elem R (rels A B))))).
+Admitted.
+Theorem rels_intro_dom_and_ran : (forall R A B,(((((relation R)/\(subseteq (dom R) A))/\(subseteq (ran R) B))->(elem R (rels A B))))).
+Admitted.
+Theorem rels_elim : (forall R A B,(((elem R (rels A B))->(subseteq R (times A B))))).
+Admitted.
+Theorem rels_weaken_dom : (forall R A B C,((((elem R (rels A B))/\(subseteq A C))->(elem R (rels C B))))).
+Admitted.
+Theorem rels_weaken_codom : (forall R A B D,((((elem R (rels A B))/\(subseteq B D))->(elem R (rels A D))))).
+Admitted.
+Definition id := fun A : set => ReplSep (A)(fun a => True)(fun a => (pair a a)).
+Theorem id_iff : (forall a b A,(((elem (pair a b) (id A))<->((a = b)/\(elem b A))))).
+Admitted.
+Theorem id_elem_intro : (forall a A,(((elem a A)->(elem (pair a a) (id A))))).
+Admitted.
+Theorem id_elem_inspect : (forall w A,(((elem w (id A))->(exists a,(((elem a A)/\(w = (pair a a)))))))).
+Admitted.
+Theorem id_is_relation : (forall A,((relation (id A)))).
+Admitted.
+Theorem id_dom : (forall A,(((dom (id A)) = A))).
+Admitted.
+Theorem id_ran : (forall A,(((ran (id A)) = A))).
+Admitted.
+Theorem id_img : (forall A B,(((img (id A) B) = (inter A B)))).
+Admitted.
+Theorem id_elem_rels : (forall A,((elem (id A) (rels A A)))).
+Admitted.
+Definition memrel := fun A : set => ReplSep2 (A)(fun dummyVar => A)(fun a b => (elem a b))(fun a b => (pair a b)).
+Theorem memrel_elem_intro : (forall a A b,(((((elem a A)/\(elem b A))/\(elem a b))->(elem (pair a b) (memrel A))))).
+Admitted.
+Theorem memrel_elem_inspect : (forall w A,(((elem w (memrel A))->(exists a b,((((elem a A)/\(elem b A))/\((w = (pair a b))/\(elem a b)))))))).
+Admitted.
+Theorem memrel_is_relation : (forall A,((relation (memrel A)))).
+Admitted.
+Definition subseteqrel := fun A : set => ReplSep2 (A)(fun dummyVar => A)(fun a b => (subseteq a b))(fun a b => (pair a b)).
+Theorem subseteqrel_is_relation : (forall A,((relation (subseteqrel A)))).
+Admitted.
+Definition injective := fun R: set => (forall a b aprime,((((elem (pair a b) R)/\(elem (pair aprime b) R))->(a = aprime)))).
+Definition leftunique := fun x0 : set => (injective x0).
+Theorem subseteq_of_injective_is_injective : (forall S R,((((subseteq S R)/\(injective R))->(injective S)))).
+Admitted.
+Theorem restrl_injective : (forall R A,(((injective R)->(injective (restrl R A))))).
+Admitted.
+Theorem circ_injective : (forall R S,((((injective R)/\(injective S))->(injective (circ S R))))).
+Admitted.
+Theorem identity_injective : (forall A,((injective (id A)))).
+Admitted.
+Definition rightunique := fun R: set => (forall a b bprime,((((elem (pair a b) R)/\(elem (pair a bprime) R))->(b = bprime)))).
+Definition onetoone := fun x0 : set => ((rightunique x0)/\(injective x0)).
+Theorem subseteq_of_rightunique_is_rightunique : (forall S R,((((subseteq S R)/\(rightunique R))->(rightunique S)))).
+Admitted.
+Theorem circ_rightunique : (forall R S,((((rightunique R)/\(rightunique S))->(rightunique (circ S R))))).
+Admitted.
+Definition lefttotal := fun R A: set => (forall a,(((elem a A)->(exists b,((elem (pair a b) R)))))).
+Definition righttotal := fun R B: set => (forall b,(((elem b B)->(exists a,((elem (pair a b) R)))))).
+Definition surjective := fun x0 x1 : set => (righttotal x0 x1).
+Definition inductive_set := fun x0 : set => ((elem (emptyset) x0)/\(forall a,(((elem a x0)->(elem (suc a) x0))))).
+Definition transitiveset := fun A: set => (forall x y,((((elem x y)/\(elem y A))->(elem x A)))).
+Theorem transitiveset_iff_subseteq : (forall A,(((transitiveset A)<->(forall a,(((elem a A)->(subseteq a A))))))).
+Admitted.
+Theorem transitiveset_iff_pow : (forall A,(((transitiveset A)<->(subseteq A (pow A))))).
+Admitted.
+Theorem transitiveset_iff_unions_suc : (forall A,(((transitiveset A)<->((unions (suc A)) = A)))).
+Admitted.
+Theorem transitiveset_iff_unions_subseteq : (forall A,(((transitiveset A)<->(subseteq (unions A) A)))).
+Admitted.
+Theorem transitiveset_upair : (forall A a b,((((transitiveset A)/\(elem (cons a (cons b (emptyset))) A))->((elem a A)/\(elem b A))))).
+Admitted.
+Theorem emptyset_transitiveset : (transitiveset (emptyset)).
+Admitted.
+Theorem union_of_transitiveset_is_transitiveset : (forall A B,((((transitiveset A)/\(transitiveset B))->(transitiveset (union A B))))).
+Admitted.
+Theorem inter_of_transitiveset_is_transitiveset : (forall A B,((((transitiveset A)/\(transitiveset B))->(transitiveset (inter A B))))).
+Admitted.
+Theorem suc_of_transitiveset_is_transitiveset : (forall A,(((transitiveset A)->(transitiveset (suc A))))).
+Admitted.
+Theorem unions_of_transitiveset_is_transitiveset : (forall A,(((transitiveset A)->(transitiveset (unions A))))).
+Admitted.
+Theorem unions_family_of_transitiveset_is_transitiveset : (forall A,(((forall x3,(((elem x3 A)->(transitiveset x3))))->(transitiveset (unions A))))).
+Admitted.
+Theorem inters_family_of_transitiveset_is_transitiveset : (forall A,(((forall x4,(((elem x4 A)->(transitiveset x4))))->(transitiveset (inters A))))).
+Admitted.
+Definition ordinal := fun alpha: set => ((transitiveset alpha)/\(forall x5,(((elem x5 alpha)->(transitiveset x5))))).
+Theorem ordinal_intro : (forall alpha,((((transitiveset alpha)/\(forall x6,(((elem x6 alpha)->(transitiveset x6)))))->(ordinal alpha)))).
+Admitted.
+Theorem ordinal_is_transitiveset : (forall alpha,(((ordinal alpha)->(transitiveset alpha)))).
+Admitted.
+Theorem ordinal_elem_is_transitiveset : (forall alpha A,((((ordinal alpha)/\(elem A alpha))->(transitiveset A)))).
+Admitted.
+Theorem elem_of_ordinal_is_ordinal : (forall alpha beta,((((ordinal alpha)/\(elem beta alpha))->(ordinal beta)))).
+Admitted.
+Theorem suc_ordinal_implies_ordinal : (forall alpha,(((ordinal (suc alpha))->(ordinal alpha)))).
+Admitted.
+Theorem transitivesubseteq_of_ordinal_is_ordinal : (forall alpha beta,(((((ordinal alpha)/\(subseteq beta alpha))/\(transitiveset beta))->(ordinal beta)))).
+Admitted.
+Theorem ordinal_elem_implies_subseteq : (forall alpha beta,(((((ordinal alpha)/\(ordinal beta))/\(elem alpha beta))->(subseteq alpha beta)))).
+Admitted.
+Theorem ordinal_transitivity : (forall alpha gamma beta,((((ordinal alpha)/\((elem gamma beta)/\(elem beta alpha)))->(elem gamma alpha)))).
+Admitted.
+Theorem ordinal_suc_subseteq : (forall beta alpha,((((ordinal beta)/\(elem alpha beta))->(subseteq (suc alpha) beta)))).
+Admitted.
+Definition ordinal_prec := fun x0 x1 : set => ((ordinal x1)/\(elem x0 x1)).
+Definition ordinal_preceq := fun x0 x1 : set => ((ordinal x1)/\(subseteq x0 x1)).
+Theorem prec_is_ordinal : (forall alpha beta,(((ordinal_prec alpha beta)->(ordinal alpha)))).
+Admitted.
+Theorem ordinal_elem_connex : (forall alpha beta,((((ordinal alpha)/\(ordinal beta))->(((elem alpha beta)\/(elem beta alpha))\/(alpha = beta))))).
+Admitted.
+Theorem ordinal_proper_subset_implies_elem : (forall alpha beta,(((((ordinal alpha)/\(ordinal beta))/\(subset alpha beta))->(elem alpha beta)))).
+Admitted.
+Theorem ordinal_elem_implies_proper_subset : (forall alpha beta,(((((ordinal alpha)/\(ordinal beta))/\(elem alpha beta))->(subset alpha beta)))).
+Admitted.
+Theorem ordinal_preceq_implies_subseteq : (forall alpha beta,(((((ordinal alpha)/\(ordinal beta))/\(ordinal_preceq alpha beta))->(subseteq alpha beta)))).
+Admitted.
+Theorem ordinal_elem_or_subseteq : (forall alpha beta,((((ordinal alpha)/\(ordinal beta))->((elem alpha beta)\/(subseteq beta alpha))))).
+Admitted.
+Theorem ordinal_subseteq_or_subseteq : (forall alpha beta,((((ordinal alpha)/\(ordinal beta))->((subseteq alpha beta)\/(subseteq beta alpha))))).
+Admitted.
+Theorem ordinal_subseteq_implies_elem_or_eq : (forall alpha beta,(((((ordinal alpha)/\(ordinal beta))/\(subseteq alpha beta))->((elem alpha beta)\/(alpha = beta))))).
+Admitted.
+Theorem ordinal_subset_trichotomy : (forall alpha beta,((((ordinal alpha)/\(ordinal beta))->(((subset alpha beta)\/(subset beta alpha))\/(alpha = beta))))).
+Admitted.
+Theorem ordinal_nor_elem_implies_eq : (forall alpha beta,(((((ordinal alpha)/\(ordinal beta))/\(nor(elem alpha beta) (elem beta alpha)))->(alpha = beta)))).
+Admitted.
+Theorem ordinal_in_trichotomy : (forall alpha beta,((((ordinal alpha)/\(ordinal beta))->(((elem alpha beta)\/(elem beta alpha))\/(alpha = beta))))).
+Admitted.
+Theorem ordinal_prec_trichotomy : (forall alpha beta,(((((ordinal alpha)/\(ordinal beta))/\(nor(ordinal_prec alpha beta) (ordinal_prec beta alpha)))->(alpha = beta)))).
+Admitted.
+Theorem ordinal_elem_or_superset : (forall alpha beta,((((ordinal alpha)/\(ordinal beta))->((elem alpha beta)\/(subseteq beta alpha))))).
+Admitted.
+Theorem emptyset_is_ordinal : (ordinal (emptyset)).
+Admitted.
+Theorem suc_ordinal : (forall alpha,(((ordinal alpha)->(ordinal (suc alpha))))).
+Admitted.
+Theorem ordinal_iff_suc_ordinal : (forall alpha,(((ordinal alpha)<->(ordinal (suc alpha))))).
+Admitted.
+Theorem ordinal_in_suc : (forall alpha,(((ordinal alpha)->(elem alpha (suc alpha))))).
+Admitted.
+Theorem ordinal_precedes_suc : (forall alpha,(((ordinal alpha)->(ordinal_prec alpha (suc alpha))))).
+Admitted.
+Theorem ordinal_elem_implies_subset_of_suc : (forall alpha beta,(((((ordinal alpha)/\(ordinal beta))/\(elem alpha beta))->(subseteq alpha (suc beta))))).
+Admitted.
+Theorem unions_of_ordinal_is_ordinal : (forall alpha,(((ordinal alpha)->(ordinal (unions alpha))))).
+Admitted.
+Theorem ordinal_subseteq_unions : (forall alpha,(((ordinal alpha)->(subseteq (unions alpha) alpha)))).
+Admitted.
+Theorem union_of_two_ordinals_is_ordinal : (forall alpha beta,((((ordinal alpha)/\(ordinal beta))->(ordinal (union alpha beta))))).
+Admitted.
+Theorem ordinal_empty_or_emptyset_elem : (forall alpha,(((ordinal alpha)->((alpha = (emptyset))\/(elem (emptyset) alpha))))).
+Admitted.
+Theorem transitive_set_of_ordinals_is_ordinal : (forall A,((((forall alpha,(((elem alpha A)->(ordinal alpha))))/\(transitiveset A))->(ordinal A)))).
+Admitted.
+Theorem buraliforti_antinomy : ~((exists Omega,((forall alpha,(((elem alpha Omega)<->(ordinal alpha))))))).
+Admitted.
+Theorem inters_of_ordinals_is_ordinal : (forall A,((((inhabited A)/\(forall alpha,(((elem alpha A)->(ordinal alpha)))))->(ordinal (inters A))))).
+Admitted.
+Theorem inters_of_ordinals_subseteq : (forall A,((((inhabited A)/\(forall alpha,(((elem alpha A)->(ordinal alpha)))))->(forall alpha,(((elem alpha A)->(subseteq (inters A) alpha))))))).
+Admitted.
+Theorem inters_of_ordinals_elem : (forall A,((((inhabited A)/\(forall alpha,(((elem alpha A)->(ordinal alpha)))))->(elem (inters A) A)))).
+Admitted.
+Theorem inters_of_ordinals_is_minimal : (forall A,((((inhabited A)/\(forall alpha,(((elem alpha A)->(ordinal alpha)))))->(elemminimal (inters A) A)))).
+Admitted.
+Theorem inters_of_ordinals_is_minimal_alternate : (forall A,((((inhabited A)/\(forall alpha,(((elem alpha A)->(ordinal alpha)))))->(forall alpha,(((elem alpha A)->(((inters A) = alpha)\/(elem (inters A) alpha)))))))).
+Admitted.
+Theorem inter_of_two_ordinals_is_ordinal : (forall alpha beta,((((ordinal alpha)/\(ordinal beta))->(ordinal (inter alpha beta))))).
+Admitted.
+Definition limit_ordinal := fun lambda: set => ((ordinal_prec (emptyset) lambda)/\(forall alpha,(((elem alpha lambda)->(elem (suc alpha) lambda))))).
+Definition successor_ordinal := fun alpha: set => (exists beta,(((ordinal beta)/\(alpha = (suc beta))))).
+Theorem positive_ordinal_is_limit_or_successor : (forall alpha,((((ordinal_prec (emptyset) alpha)/\(ordinal alpha))->((limit_ordinal alpha)\/(successor_ordinal alpha))))).
+Admitted.
+Theorem zero_not_successorordinal : ~((successor_ordinal (emptyset))).
+Admitted.
+Theorem zero_not_limitordinal : ~((limit_ordinal (emptyset))).
+Admitted.
+Theorem suc_elem_limitordinal : (forall lambda alpha,((((limit_ordinal lambda)/\(elem alpha lambda))->(elem (suc alpha) lambda)))).
+Admitted.
+Theorem limitordinal_eq_unions : (forall lambda,(((limit_ordinal lambda)->((unions lambda) = lambda)))).
+Admitted.
+Definition function := fun x0 : set => ((rightunique x0)/\(relation x0)).
+Definition appl := fun f x : set => (unions (img f (cons x (emptyset)))).
+Theorem function_rightunique : (forall f a b bprime,((((function f)/\((elem (pair a b) f)/\(elem (pair a bprime) f)))->(b = bprime)))).
+Admitted.
+Theorem function_appl_intro : (forall f a b,((((function f)/\(elem (pair a b) f))->((appl f a) = b)))).
+Admitted.
+Theorem function_member_elim : (forall f w,((((function f)/\(elem w f))->(exists x,(((elem x (dom f))/\(w = (pair x (appl f x))))))))).
+Admitted.
+Theorem function_appl_elim : (forall f x,((((function f)/\(elem x (dom f)))->(elem (pair x (appl f x)) f)))).
+Admitted.
+Theorem function_appl_iff : (forall f a b,(((function f)->((elem (pair a b) f)<->((elem a (dom f))/\((appl f a) = b)))))).
+Admitted.
+Theorem fun_subseteq : (forall f g,((((((function f)/\(function g))/\(subseteq (dom f) (dom g)))/\(forall x,(((elem x (dom f))->((appl f x) = (appl g x))))))->(subseteq f g)))).
+Admitted.
+Theorem funext : (forall f g,((((((function f)/\(function g))/\((dom f) = (dom g)))/\(forall x,(((appl f x) = (appl g x)))))->(f = g)))).
+Admitted.
+Definition function_on := fun x0 x1 : set => ((function x0)/\(x1 = (dom x0))).
+Definition function_to := fun x0 x1 : set => ((function x0)/\(forall x,(((elem x (dom x0))->(elem (appl x0 x) x1))))).
+Definition function_from_to := fun x0 x1 x2 : set => ((function_to x0 x2)/\((dom x0) = x1)).
+Theorem function_on_weaken_codom : (forall f B C,((((function_to f B)/\(subseteq B C))->(function_to f C)))).
+Admitted.
+Theorem function_to_ran : (forall f B,(((function_to f B)->(subseteq (ran f) B)))).
+Admitted.
+Theorem function_from_to_weaken_codom : (forall f A B C,((((function_from_to f A B)/\(subseteq B C))->(function_from_to f A C)))).
+Admitted.
+Definition funs := fun A B : set => {f :e ((rels A B))|((subseteq A (dom f))/\(rightunique f))}.
+Theorem funs_subseteq_rels : (forall A B,((subseteq (funs A B) (rels A B)))).
+Admitted.
+Theorem funs_intro : (forall f A B,(((function_from_to f A B)->(elem f (funs A B))))).
+Admitted.
+Theorem funs_weaken_codom : (forall f A B D,((((elem f (funs A B))/\(subseteq B D))->(elem f (funs A D))))).
+Admitted.
+Theorem img_of_function_intro : (forall f x X,((((function f)/\(elem x (inter (dom f) X)))->(elem (appl f x) (img f X))))).
+Admitted.
+Theorem img_of_function_elim : (forall f y X,((((function f)/\(elem y (img f X)))->(exists x,(((elem x (inter (dom f) X))/\(y = (appl f x)))))))).
+Admitted.
+Theorem img_of_function : (forall f X,(((function f)->((img f X) = ReplSep ((inter (dom f) X))(fun x => True)(fun x => (appl f x)))))).
+Admitted.
+Definition family_of_functions := fun x0 : set => (forall x11,(((elem x11 x0)->(function x11)))).
+Theorem unions_of_compatible_family_of_function_is_function : (forall F,((((family_of_functions F)/\(forall f g,((((elem f F)/\(elem g F))->((subseteq f g)\/(subseteq g f))))))->(function (unions F))))).
+Admitted.
+Theorem emptyset_is_function : (function (emptyset)).
+Admitted.
+Theorem emptyset_is_function_on_emptyset : (function_on (emptyset) (emptyset)).
+Admitted.
+Theorem codom_of_emptyset_can_be_anything : (forall X,((function_to (emptyset) X))).
+Admitted.
+Theorem emptyset_is_injective : (injective (emptyset)).
+Admitted.
+Definition composable := fun x0 x1 : set => (subseteq (ran x1) (dom x0)).
+Theorem function_circ : (forall f g,((((rightunique f)/\(rightunique g))->(function (circ g f))))).
+Admitted.
+Theorem circ_appl : (forall f g x,((((((function f)/\(function g))/\(composable g f))/\(elem x (dom f)))->((appl (circ g f) x) = (appl g (appl f x)))))).
+Admitted.
+Theorem dom_of_circ : (forall f g,(((((function f)/\(function g))/\(composable g f))->((dom (circ g f)) = (preimg f (dom g)))))).
+Admitted.
+Theorem dom_circ_exact : (forall f g,(((((function f)/\(function g))/\((ran f) = (dom g)))->((dom (circ g f)) = (dom f))))).
+Admitted.
+Theorem ran_of_circ_intro : (forall f g y,((((((function f)/\(function g))/\(composable g f))/\(elem y (img g (ran f))))->(elem y (ran (circ g f)))))).
+Admitted.
+Theorem ran_of_circ_elim : (forall f g y,((((((function f)/\(function g))/\(composable g f))/\(elem y (ran (circ g f))))->(elem y (img g (ran f)))))).
+Admitted.
+Theorem ran_of_circ : (forall f g,(((((function f)/\(function g))/\(composable g f))->((ran (circ g f)) = (img g (ran f)))))).
+Admitted.
+Theorem ran_circ_exact : (forall f g,(((((function f)/\(function g))/\((ran f) = (dom g)))->((ran (circ g f)) = (ran g))))).
+Admitted.
+Theorem img_of_circ_elim : (forall f g c A,((((((function f)/\(function g))/\(subseteq (ran f) (dom g)))/\(elem c (img (circ g f) A)))->(elem c (img g (img f A)))))).
+Admitted.
+Theorem img_of_circ : (forall f g A,(((((function f)/\(function g))/\(subseteq (ran f) (dom g)))->((img (circ g f) A) = (img g (img f A)))))).
+Admitted.
+Theorem restrl_of_function_is_function : (forall f A,(((function f)->(function (restrl f A))))).
+Admitted.
+Theorem restrl_of_function_appl : (forall f A a,(((((function f)/\(subseteq A (dom f)))/\(elem a A))->((appl (restrl f A) a) = (appl f a))))).
+Admitted.
+Theorem function_appl_default : (forall x f,(((notelem x (dom f))->((appl f x) = (emptyset))))).
+Admitted.
+Theorem injective_function : (forall f,(((function f)->((injective f)<->(forall x y,((((elem x (dom f))/\(elem y (dom f)))->(((appl f x) = (appl f y))->(x = y))))))))).
+Admitted.
+Definition injection := fun x0 : set => ((function x0)/\(injective x0)).
+Definition surjects := fun f Y: set => (Y = ReplSep ((dom f))(fun x => True)(fun x => (appl f x))).
+Theorem surjects_img : (forall f,((surjects f (img f (dom f))))).
+Admitted.
+Theorem surjects_implies_img : (forall f Y,(((surjects f Y)->(Y = (img f (dom f)))))).
+Admitted.
+Theorem surjects_implies_ran_eq : (forall f Y,((((function f)/\(surjects f Y))->((ran f) = Y)))).
+Admitted.
+Theorem ran_eq_implies_surjects : (forall f Y,((((function f)/\((ran f) = Y))->(surjects f Y)))).
+Admitted.
+Theorem surjects_iff_ran_eq : (forall f Y,(((function f)->((surjects f Y)<->((ran f) = Y))))).
+Admitted.
+Definition bijection := fun f X Y: set => (((dom f) = X)/\((surjects f Y)/\(injection f))).
+Theorem bijection_circ : (forall f A B g C,((((bijection f A B)/\(bijection g B C))->(bijection (circ g f) A C)))).
+Admitted.
+Theorem converse_of_function_is_injective : (forall f,(((function f)->(injective (converse_relation f))))).
+Admitted.
+Theorem injective_converse_is_function : (forall f,(((injective f)->(function (converse_relation f))))).
+Admitted.
+Theorem bijective_converse_is_function : (forall f A B,(((bijection f A B)->(function (converse_relation f))))).
+Admitted.
+Theorem bijection_converse_is_bijection : (forall f A B,(((bijection f A B)->(bijection (converse_relation f) B A)))).
+Admitted.
+Definition leftinverse := fun x0 x1 : set => (forall x,(((elem x (dom x1))->((appl x0 (appl x1 x)) = x)))).
+Definition rightinverse := fun x0 x1 : set => ((circ x1 x0) = (id (dom x0))).
+Definition rightinverseon := fun x0 x1 x2 : set => ((circ x1 x0) = (id x2)).
+Theorem injective_converse_is_leftinverse : (forall f,(((injection f)->(leftinverse (converse_relation f) f)))).
+Admitted.
+Theorem id_rightunique : (forall A,((rightunique (id A)))).
+Admitted.
+Theorem id_is_function : (forall A,((function (id A)))).
+Admitted.
+Theorem id_is_function_on : (forall A,((function_on (id A) A))).
+Admitted.
+Theorem id_is_function_to : (forall A,((function_to (id A) A))).
+Admitted.
+Theorem id_is_function_to_form : (forall A,((function_from_to (id A) A A))).
+Admitted.
+Theorem id_elem_funs : (forall A,((elem (id A) (funs A A)))).
+Admitted.
+Theorem id_appl : (forall a A f,((((elem a A)/\(f = (id A)))->((appl f a) = a)))).
+Admitted.
+Theorem id_is_bijection : (forall A,((bijection (id A) A A))).
+Admitted.
diff --git a/megalodon/library/everything.mg b/megalodon/library/everything.mg
new file mode 100644
index 0000000..e69de29
--- /dev/null
+++ b/megalodon/library/everything.mg
diff --git a/megalodon/library/relation/equivalence.mg b/megalodon/library/relation/equivalence.mg
new file mode 100644
index 0000000..e9c8211
--- /dev/null
+++ b/megalodon/library/relation/equivalence.mg
@@ -0,0 +1,470 @@
+Let emptyset : set := Empty.
+Let elem : set->set->prop := In.
+Let notelem : set->set->prop := fun a A => ~(In a A).
+Let pow : set->set := Power.
+Let unions : set->set := Union.
+Let union : set->set->set := binunion.
+Let cons : set -> set -> set := fun x X => binunion {x} X.
+Let xor : prop -> prop -> prop := fun p q => (p \/ q) /\ ~(p /\ q).
+Let pair : set -> set -> set := fun a b => {{a}, {a, b}}.
+Let fst : set -> set := fun p => Eps_i (fun a => exists b, p = pair a b).
+Let snd : set -> set := fun p => Eps_i (fun b => exists a, p = pair a b).
+Let nor : prop -> prop -> prop := fun p q => ~(p \/ q) .
+Definition ni := fun x0 x1 : set => (elem x1 x0).
+Fact setext : (forall A B,(((forall a,(((elem a A)<->(elem a B))))->(A = B)))).
+Admitted.
+Theorem neq_witness : (forall A B,(((A <> B)->(exists c,((xor((elem c A)/\~((elem c B))) (~((elem c A))/\(elem c B)))))))).
+Admitted.
+Definition subseteq := fun A B: set => (forall a,(((elem a A)->(elem a B)))).
+Definition is_subset := fun x0 x1 : set => (subseteq x0 x1).
+Theorem subseteq_refl : (forall A,((subseteq A A))).
+Admitted.
+Theorem subseteq_antisymmetric : (forall A B,((((subseteq A B)/\(subseteq B A))->(A = B)))).
+Admitted.
+Theorem elem_subseteq : (forall a A B,((((elem a A)/\(subseteq A B))->(elem a B)))).
+Admitted.
+Theorem not_in_subseteq : (forall A B c,((((subseteq A B)/\(notelem c B))->(notelem c A)))).
+Admitted.
+Theorem subseteq_transitive : (forall A B C,((((subseteq A B)/\(subseteq B C))->(subseteq A C)))).
+Admitted.
+Definition subset := fun A B: set => ((subseteq A B)/\(A <> B)).
+Theorem subset_irrefl : (forall A,(~((subset A A)))).
+Admitted.
+Theorem subset_transitive : (forall A B C,((((subseteq A B)/\(subseteq B C))->(subseteq A C)))).
+Admitted.
+Theorem subset_witness : (forall A B,(((subset A B)->(exists b,(((elem b B)/\(notelem b A))))))).
+Admitted.
+Definition family_of_subsets := fun x0 x1 : set => (forall A,(((elem A x0)->(subseteq A x1)))).
+Fact notin_emptyset : (forall a,((notelem a (emptyset)))).
+Admitted.
+Definition inhabited := fun A: set => (exists a,((elem a A))).
+Definition empty := fun x0 : set => ~((inhabited x0)).
+Theorem empty_eq : (forall x y,((((empty x)/\(empty y))->(x = y)))).
+Admitted.
+Theorem emptyset_subseteq : (forall a,((subseteq (emptyset) a))).
+Admitted.
+Theorem subseteq_emptyset_iff : (forall A,(((subseteq A (emptyset))<->(A = (emptyset))))).
+Admitted.
+Definition disjoint := fun A B: set => ~((exists a,(((elem a A)/\(elem a B))))).
+Definition notmeets := fun x0 x1 : set => (disjoint x0 x1).
+Definition meets := fun x0 x1 : set => ~((disjoint x0 x1)).
+Theorem disjoint_symmetric : (forall A B,(((disjoint A B)->(disjoint B A)))).
+Admitted.
+Fact in_cons : (forall x y X,(((elem x (cons y X))<->((x = y)\/(elem x X))))).
+Admitted.
+Theorem in_cons_left : (forall x X,((elem x (cons x X)))).
+Admitted.
+Theorem in_cons_right : (forall y X x,(((elem y X)->(elem y (cons x X))))).
+Admitted.
+Theorem upair_intro_left : (forall a b,((elem a (cons a (cons b (emptyset)))))).
+Admitted.
+Theorem upair_intro_right : (forall b a,((elem b (cons a (cons b (emptyset)))))).
+Admitted.
+Theorem upair_elim : (forall c a b,(((elem c (cons a (cons b (emptyset))))->((a = c)\/(b = c))))).
+Admitted.
+Theorem singleton_intro : (forall a,((elem a (cons a (emptyset))))).
+Admitted.
+Theorem singleton_elim : (forall a b,(((elem a (cons b (emptyset)))->(a = b)))).
+Admitted.
+Theorem singleton_iff : (forall a b,(((elem a (cons b (emptyset)))<->(a = b)))).
+Admitted.
+Theorem singleton_subset_intro : (forall a C,(((elem a C)->(subseteq (cons a (emptyset)) C)))).
+Admitted.
+Theorem singleton_subset_elim : (forall a C,(((subseteq (cons a (emptyset)) C)->(elem a C)))).
+Admitted.
+Fact unions_iff_exists : (forall z X,(((elem z (unions X))<->(exists Y,(((elem Y X)/\(elem z Y))))))).
+Admitted.
+Theorem unions_intro : (forall A B C,((((elem A B)/\(elem B C))->(elem A (unions C))))).
+Admitted.
+Theorem unions_emptyset : ((unions (emptyset)) = (emptyset)).
+Admitted.
+Definition inters := fun A : set => {x :e ((unions A))|(forall a,(((elem a A)->(elem x a))))}.
+Theorem inters_iff_forall : (forall z X,(((elem z (inters X))<->((inhabited X)/\(forall Y,(((elem Y X)->(elem z Y)))))))).
+Admitted.
+Theorem inters_intro : (forall C A,((((inhabited C)/\(forall B,(((elem B C)->(elem A B)))))->(elem A (inters C))))).
+Admitted.
+Theorem inters_destr : (forall A C B,((((elem A (inters C))/\(elem B C))->(elem A B)))).
+Admitted.
+Theorem inters_singleton : (forall a,(((inters (cons a (emptyset))) = a))).
+Admitted.
+Theorem inters_emptyset : ((inters (cons (emptyset) (emptyset))) = (emptyset)).
+Admitted.
+Fact union_iff : (forall a A B,(((elem a (union A B))<->((elem a A)\/(elem a B))))).
+Admitted.
+Theorem union_intro_left : (forall c A B,(((elem c A)->(elem c (union A B))))).
+Admitted.
+Theorem union_intro_right : (forall c B A,(((elem c B)->(elem c (union A B))))).
+Admitted.
+Theorem union_comm : (forall A B,(((union A B) = (union B A)))).
+Admitted.
+Theorem union_assoc : (forall A B C,(((union (union A B) C) = (union A (union B C))))).
+Admitted.
+Theorem union_idempotent : (forall A,(((union A A) = A))).
+Admitted.
+Theorem union_subseteq_iff : (forall A B C,(((subseteq (union A B) C)<->((subseteq A C)/\(subseteq B C))))).
+Admitted.
+Theorem union_upper_left : (forall A B,((subseteq A (union A B)))).
+Admitted.
+Theorem union_upper_right : (forall B A,((subseteq B (union A B)))).
+Admitted.
+Theorem union_emptyset : (forall A,(((union A (emptyset)) = A))).
+Admitted.
+Theorem union_emptyset_intro : (forall A B,((((A = (emptyset))/\(B = (emptyset)))->((union A B) = (emptyset))))).
+Admitted.
+Theorem union_emptyset_elim_left : (forall A B,((((union A B) = (emptyset))->(A = (emptyset))))).
+Admitted.
+Theorem union_emptyset_elim_right : (forall A B,((((union A B) = (emptyset))->(B = (emptyset))))).
+Admitted.
+Theorem union_absorb_subseteq_left : (forall A B,(((subseteq A B)->((union A B) = B)))).
+Admitted.
+Theorem union_absorb_subseteq_right : (forall A B,(((subseteq A B)->((union B A) = B)))).
+Admitted.
+Theorem union_eq_self_implies_subseteq : (forall A B,((((union A B) = B)->(subseteq A B)))).
+Admitted.
+Theorem unions_cons : (forall b A,(((unions (cons b A)) = (union b (unions A))))).
+Admitted.
+Theorem union_cons : (forall b A C,(((union (cons b A) C) = (cons b (union A C))))).
+Admitted.
+Theorem union_absorb_left : (forall A B,(((union A (union A B)) = (union A B)))).
+Admitted.
+Theorem union_absorb_right : (forall A B,(((union (union A B) B) = (union A B)))).
+Admitted.
+Theorem union_comm_left : (forall A B C,(((union A (union B C)) = (union B (union A C))))).
+Admitted.
+Definition inter := fun A B : set => {a :e (A)|(elem a B)}.
+Theorem inter_intro : (forall c A B,((((elem c A)/\(elem c B))->(elem c (inter A B))))).
+Admitted.
+Theorem inter_elim_left : (forall c A B,(((elem c (inter A B))->(elem c A)))).
+Admitted.
+Theorem inter_elim_right : (forall c A B,(((elem c (inter A B))->(elem c B)))).
+Admitted.
+Theorem inter_comm : (forall A B,(((inter A B) = (inter B A)))).
+Admitted.
+Theorem inter_assoc : (forall A B C,(((inter (inter A B) C) = (inter A (inter B C))))).
+Admitted.
+Theorem inter_idempotent : (forall A,(((inter A A) = A))).
+Admitted.
+Theorem inter_subseteq : (forall A B,((subseteq (inter A B) A))).
+Admitted.
+Theorem inter_emptyset : (forall A,(((inter A (emptyset)) = (emptyset)))).
+Admitted.
+Theorem inter_absorb_superseteq_right : (forall A B,(((subseteq A B)->((inter A B) = A)))).
+Admitted.
+Theorem inter_absorb_superseteq_left : (forall A B,(((subseteq A B)->((inter B A) = A)))).
+Admitted.
+Theorem inter_eq_left_implies_subseteq : (forall A B,((((inter A B) = A)->(subseteq A B)))).
+Admitted.
+Theorem subseteq_inter_iff : (forall C A B,(((subseteq C (inter A B))<->((subseteq C A)/\(subseteq C B))))).
+Admitted.
+Theorem inter_lower_left : (forall A B,((subseteq (inter A B) A))).
+Admitted.
+Theorem inter_lower_right : (forall A B,((subseteq (inter A B) B))).
+Admitted.
+Theorem inter_absorb_left : (forall A B,(((inter A (inter A B)) = (inter A B)))).
+Admitted.
+Theorem inter_absorb_right : (forall A B,(((inter (inter A B) B) = (inter A B)))).
+Admitted.
+Theorem inter_comm_left : (forall A B C,(((inter A (inter B C)) = (inter B (inter A C))))).
+Admitted.
+Theorem inter_distrib_union : (forall x y z,(((inter x (union y z)) = (union (inter x y) (inter x z))))).
+Admitted.
+Theorem union_distrib_inter : (forall x y z,(((union x (inter y z)) = (inter (union x y) (union x z))))).
+Admitted.
+Theorem union_inter_assoc_intro : (forall C A B,(((subseteq C A)->((union (inter A B) C) = (inter A (union B C)))))).
+Admitted.
+Theorem union_inter_assoc_elim : (forall A B C,((((union (inter A B) C) = (inter A (union B C)))->(subseteq C A)))).
+Admitted.
+Theorem union_inter_crazy : (forall A B C,(((union (union (inter A B) (inter B C)) (inter C A)) = (inter (inter (union A B) (union B C)) (union C A))))).
+Admitted.
+Definition setminus := fun A B : set => {a :e (A)|~((elem a B))}.
+Theorem setminus_intro : (forall a A B,((((elem a A)/\(notelem a B))->(elem a (setminus A B))))).
+Admitted.
+Theorem setminus_elim_left : (forall a A B,(((elem a (setminus A B))->(elem a A)))).
+Admitted.
+Theorem setminus_elim_right : (forall a A B,(((elem a (setminus A B))->(notelem a B)))).
+Admitted.
+Theorem setminus_emptyset : (forall x,(((setminus x (emptyset)) = x))).
+Admitted.
+Theorem emptyset_setminus : (forall x,(((setminus (emptyset) x) = (emptyset)))).
+Admitted.
+Theorem setminus_self : (forall x,(((setminus x x) = (emptyset)))).
+Admitted.
+Theorem setminus_setminus : (forall x y,(((setminus x (setminus x y)) = (inter x y)))).
+Admitted.
+Theorem setminus_inter : (forall x y z,(((setminus x (inter y z)) = (union (setminus x y) (setminus x z))))).
+Admitted.
+Theorem setminus_union : (forall x y z,(((setminus x (union y z)) = (inter (setminus x y) (setminus x z))))).
+Admitted.
+Theorem inter_setminus : (forall x y z,(((inter x (setminus y z)) = (setminus (inter x y) (inter x z))))).
+Admitted.
+Theorem difference_with_proper_subset_is_inhabited : (forall A B,(((subset A B)->(inhabited (setminus B A))))).
+Admitted.
+Theorem setminus_subseteq : (forall B A,((subseteq (setminus B A) B))).
+Admitted.
+Theorem subseteq_setminus : (forall C A B,((((subseteq C A)/\((inter C B) = (emptyset)))->(subseteq C (setminus A B))))).
+Admitted.
+Theorem setminus_absorb_right : (forall A B,((((inter A B) = (emptyset))->((setminus A B) = A)))).
+Admitted.
+Theorem setminus_eq_emptyset_iff_subseteq : (forall A B,((((setminus A B) = (emptyset))<->(subseteq A B)))).
+Admitted.
+Theorem subseteq_setminus_cons_intro : (forall B A C c,((((subseteq B (setminus A C))/\(notelem c B))->(subseteq B (setminus A (cons c C)))))).
+Admitted.
+Theorem subseteq_setminus_cons_elim : (forall B A c C,(((subseteq B (setminus A (cons c C)))->((subseteq B (setminus A C))/\(notelem c B))))).
+Admitted.
+Theorem setminus_cons : (forall A a B,(((setminus A (cons a B)) = (setminus (setminus A (cons a (emptyset))) B)))).
+Admitted.
+Theorem setminus_cons_flip : (forall A a B,(((setminus A (cons a B)) = (setminus (setminus A B) (cons a (emptyset)))))).
+Admitted.
+Theorem setminus_disjoint : (forall A B,(((inter A (setminus B A)) = (emptyset)))).
+Admitted.
+Theorem setminus_partition : (forall A B,(((subseteq A B)->((union A (setminus B A)) = B)))).
+Admitted.
+Theorem subseteq_union_setminus : (forall A B,((subseteq A (union B (setminus A B))))).
+Admitted.
+Theorem double_complement : (forall A B C,((((subseteq A B)/\(subseteq B C))->((setminus B (setminus C A)) = A)))).
+Admitted.
+Theorem double_complement_union : (forall A B,(((setminus (union A B) (setminus B A)) = A))).
+Admitted.
+Theorem setminus_eq_inter_complement : (forall A C B,((((subseteq A C)/\(subseteq B C))->((setminus A B) = (inter A (setminus C B)))))).
+Admitted.
+Fact pair_eq_iff : (forall a b aprime bprime,((((pair a b) = (pair aprime bprime))<->((a = aprime)/\(b = bprime))))).
+Admitted.
+Fact pair_neq_emptyset : (forall a b,(((pair a b) <> (emptyset)))).
+Admitted.
+Fact pair_neq_fst : (forall a b,(((pair a b) <> a))).
+Admitted.
+Fact pair_neq_snd : (forall a b,(((pair a b) <> b))).
+Admitted.
+Theorem triple_eq_iff : (forall a b c aprime bprime cprime,((((pair a (pair b c)) = (pair aprime (pair bprime cprime)))<->(((a = aprime)/\(b = bprime))/\(c = cprime))))).
+Admitted.
+Fact fst_eq : (forall a b,(((fst (pair a b)) = a))).
+Admitted.
+Fact snd_eq : (forall a b,(((snd (pair a b)) = b))).
+Admitted.
+Theorem pair_eq_pair_of_proj : (forall a b,(((pair a b) = (pair (fst (pair a b)) (snd (pair a b)))))).
+Admitted.
+Definition times := fun A B : set => Eps_i (fun frs : set => (forall frv,(((elem frv frs)<->(exists a b,((((elem a A)/\(elem b B))/\((pair a b) = frv)))))))).
+Theorem times_tuple_elim : (forall x y X Y,(((elem (pair x y) (times X Y))->((elem x X)/\(elem y Y))))).
+Admitted.
+Theorem times_tuple_intro : (forall x X y Y,((((elem x X)/\(elem y Y))->(elem (pair x y) (times X Y))))).
+Admitted.
+Theorem times_empty_left : (forall Y,(((times (emptyset) Y) = (emptyset)))).
+Admitted.
+Theorem times_empty_right : (forall X,(((times X (emptyset)) = (emptyset)))).
+Admitted.
+Theorem times_empty_iff : (forall X Y,(((empty (times X Y))<->((empty X)\/(empty Y))))).
+Admitted.
+Theorem fst_type : (forall c A B,(((elem c (times A B))->(elem (fst c) A)))).
+Admitted.
+Theorem snd_type : (forall c A B,(((elem c (times A B))->(elem (snd c) B)))).
+Admitted.
+Theorem times_elem_is_tuple : (forall p X Y,(((elem p (times X Y))->(exists x y,(((elem x X)/\((elem y Y)/\(p = (pair x y))))))))).
+Admitted.
+Theorem times_proj_elim : (forall p X Y,(((elem p (times X Y))->((elem (fst p) X)/\(elem (snd p) Y))))).
+Admitted.
+Definition partition := fun P: set => ((notelem (emptyset) P)/\(forall B C,(((((elem B P)/\(elem C P))/\(B <> C))->(disjoint B C))))).
+Definition partition_of := fun x0 x1 : set => ((partition x0)/\((unions x0) = x1)).
+Theorem partition_emptyset : (partition_of (emptyset) (emptyset)).
+Admitted.
+Definition partition_refinement := fun Pprime P: set => (forall Aprime,(((elem Aprime Pprime)->(exists A,(((elem A P)/\(subseteq Aprime A))))))).
+Definition partition_refines := fun x0 x1 : set => (partition_refinement x0 x1).
+Theorem partition_refinement_transitive : (forall Pprimeprime Pprime P,((((partition_refines Pprimeprime Pprime)/\(partition_refines Pprime P))->(partition_refines Pprimeprime P)))).
+Admitted.
+Definition powerset_of := fun x0 : set => (pow x0).
+Fact pow_iff : (forall B A,(((elem B (pow A))<->(subseteq B A)))).
+Admitted.
+Theorem powerset_intro : (forall A B,(((subseteq A B)->(elem A (pow B))))).
+Admitted.
+Theorem powerset_elim : (forall A B,(((elem A (pow B))->(subseteq A B)))).
+Admitted.
+Theorem powerset_bottom : (forall A,((elem (emptyset) (pow A)))).
+Admitted.
+Theorem powerset_top : (forall A,((elem A (pow A)))).
+Admitted.
+Theorem unions_subseteq_of_powerset_is_subseteq : (forall B A,(((is_subset B (pow A))->(subseteq (unions B) A)))).
+Admitted.
+Theorem unions_powerset : (forall A,(((unions (pow A)) = A))).
+Admitted.
+Theorem inters_powerset : (forall A,(((inters (pow A)) = (emptyset)))).
+Admitted.
+Theorem union_powersets_subseteq : (forall A B,((subseteq (union (pow A) (pow B)) (pow (union A B))))).
+Admitted.
+Theorem times_subseteq_left : (forall A C B,(((subseteq A C)->(subseteq (times A B) (times C B))))).
+Admitted.
+Theorem times_subseteq_right : (forall B D A,(((subseteq B D)->(subseteq (times A B) (times A D))))).
+Admitted.
+Theorem inter_times_intro : (forall w A B C D,(((elem w (times (inter A B) (inter C D)))->(elem w (inter (times A C) (times B D)))))).
+Admitted.
+Theorem inter_times_elim : (forall w A C B D,(((elem w (inter (times A C) (times B D)))->(elem w (times (inter A B) (inter C D)))))).
+Admitted.
+Theorem inter_times : (forall A B C D,(((times (inter A B) (inter C D)) = (inter (times A C) (times B D))))).
+Admitted.
+Theorem inter_times_right : (forall X Y Z,(((times (inter X Y) Z) = (inter (times X Z) (times Y Z))))).
+Admitted.
+Theorem inter_times_left : (forall X Y Z,(((times X (inter Y Z)) = (inter (times X Y) (times X Z))))).
+Admitted.
+Theorem union_times_intro : (forall w A B C D,(((elem w (times (union A B) (union C D)))->(elem w (union (union (union (times A C) (times B D)) (times A D)) (times B C)))))).
+Admitted.
+Theorem union_times_elim : (forall w A C B D,(((elem w (union (union (union (times A C) (times B D)) (times A D)) (times B C)))->(elem w (times (union A B) (union C D)))))).
+Admitted.
+Theorem union_times : (forall A B C D,(((times (union A B) (union C D)) = (union (union (union (times A C) (times B D)) (times A D)) (times B C))))).
+Admitted.
+Theorem union_times_left : (forall X Y Z,(((times (union X Y) Z) = (union (times X Z) (times Y Z))))).
+Admitted.
+Theorem union_times_right : (forall X Y Z,(((times X (union Y Z)) = (union (times X Y) (times X Z))))).
+Admitted.
+Definition relation := fun R: set => (forall w,(((elem w R)->(exists x y,((w = (pair x y))))))).
+Theorem relext : (forall R S,(((((relation R)/\(relation S))/\(forall x y,(((elem (pair x y) R)<->(elem (pair x y) S)))))->(R = S)))).
+Admitted.
+Definition family_of_relations := fun x0 : set => (forall x9,(((elem x9 x0)->(relation x9)))).
+Theorem unions_of_family_of_relations_is_relation : (forall F,(((family_of_relations F)->(relation (unions F))))).
+Admitted.
+Definition converse_relation := fun R : set => let MkReplFun := fun w : set => (Eps_i (fun z=>(exists x y,(((w = (pair x y))/\(z = (pair y x))))))) in {MkReplFun w|w :e (R)}.
+Theorem converse_intro : (forall y x R,(((elem (pair y x) R)->(elem (pair x y) (converse_relation R))))).
+Admitted.
+Theorem converse_elim : (forall x y R,(((elem (pair x y) (converse_relation R))->(elem (pair y x) R)))).
+Admitted.
+Theorem converse_iff : (forall x y R,(((elem (pair x y) (converse_relation R))<->(elem (pair y x) R)))).
+Admitted.
+Theorem converse_converse_iff : (forall x y R,(((elem (pair x y) (converse_relation (converse_relation R)))<->(elem (pair x y) R)))).
+Admitted.
+Theorem converse_converse_eq : (forall R,(((relation R)->((converse_relation (converse_relation R)) = R)))).
+Admitted.
+Theorem converse_type : (forall R A B,(((subseteq R (times A B))->(subseteq (converse_relation R) (times B A))))).
+Admitted.
+Theorem converse_times : (forall B A,(((converse_relation (times B A)) = (times A B)))).
+Admitted.
+Theorem converse_emptyset : ((converse_relation (emptyset)) = (emptyset)).
+Admitted.
+Theorem converse_subseteq_intro : (forall R S,(((relation R)->((subseteq R S)->(subseteq (converse_relation R) (converse_relation S)))))).
+Admitted.
+Theorem converse_subseteq_elim : (forall R S,(((relation R)->((subseteq (converse_relation R) (converse_relation S))->(subseteq R S))))).
+Admitted.
+Theorem converse_subseteq_iff : (forall R S,(((relation R)->((subseteq (converse_relation R) (converse_relation S))<->(subseteq R S))))).
+Admitted.
+Definition dom := fun R : set => let MkReplFun := fun w : set => (Eps_i (fun x=>(exists y,((w = (pair x y)))))) in {MkReplFun w|w :e (R)}.
+Definition ran := fun R : set => let MkReplFun := fun w : set => (Eps_i (fun y=>(exists x,((w = (pair x y)))))) in {MkReplFun w|w :e (R)}.
+Definition fld := fun R : set => (union (dom R) (ran R)).
+Definition img := fun R A : set => {b :e ((ran R))|(exists a,(((elem a A)/\(elem (pair a b) R))))}.
+Definition preimg := fun R B : set => {a :e ((dom R))|(exists b,(((elem b B)/\(elem (pair a b) R))))}.
+Theorem preim_eq_img_of_converse : (forall R B,(((preimg R B) = (img (converse_relation R) B)))).
+Admitted.
+Theorem dom_emptyset : ((dom (emptyset)) = (emptyset)).
+Admitted.
+Definition upward_closure := fun R a : set => {b :e ((ran R))|(elem (pair a b) R)}.
+Definition downward_closure := fun R b : set => {a :e ((dom R))|(elem (pair a b) R)}.
+Definition circ := fun S R : set => {p :e ((times (dom R) (ran S)))|(exists x z y,(((p = (pair x z))/\((elem (pair x y) R)/\(elem (pair y z) S)))))}.
+Theorem circ_is_relation : (forall S R,((relation (circ S R)))).
+Admitted.
+Theorem circ_elem_intro : (forall x y R z S,((((elem (pair x y) R)/\(elem (pair y z) S))->(elem (pair x z) (circ S R))))).
+Admitted.
+Theorem circ_elem_elim : (forall x z S R,(((elem (pair x z) (circ S R))->(exists y,(((elem (pair x y) R)/\(elem (pair y z) S))))))).
+Admitted.
+Theorem circ_iff : (forall x z S R,(((elem (pair x z) (circ S R))<->(exists y,(((elem (pair x y) R)/\(elem (pair y z) S))))))).
+Admitted.
+Theorem circ_assoc : (forall T S R,(((circ (circ T S) R) = (circ T (circ S R))))).
+Admitted.
+Theorem circ_converse_intro : (forall a c R S,(((elem (pair a c) (circ (converse_relation R) (converse_relation S)))->(elem (pair a c) (converse_relation (circ S R)))))).
+Admitted.
+Theorem circ_converse_elim : (forall x y S R,(((elem (pair x y) (converse_relation (circ S R)))->(elem (pair x y) (circ (converse_relation R) (converse_relation S)))))).
+Admitted.
+Theorem circ_converse : (forall S R,(((converse_relation (circ S R)) = (circ (converse_relation R) (converse_relation S))))).
+Admitted.
+Definition restrl := fun R X : set => {w :e (R)|(exists x y,(((elem x X)/\(w = (pair x y)))))}.
+Theorem restrl_subseteq : (forall R X,((subseteq (restrl R X) R))).
+Admitted.
+Theorem elem_dom_of_restrl_implies_elem_dom_and_restr : (forall x R X,(((elem x (dom (restrl R X)))->((elem x (dom R))/\(elem x X))))).
+Admitted.
+Theorem elem_dom_and_restr_implies_elem_of_restr : (forall x R X,((((elem x (dom R))/\(elem x X))->(elem x (dom (restrl R X)))))).
+Admitted.
+Theorem restrl_eq_inter : (forall R X,(((relation R)->((restrl R X) = (inter R (times X (ran R))))))).
+Admitted.
+Theorem dom_of_restrl_eq_inter : (forall R X,(((relation R)->((dom (restrl R X)) = (inter (dom R) X))))).
+Admitted.
+Theorem restrl_restrl : (forall V U R,(((subseteq V U)->((restrl (restrl R U) V) = (restrl R V))))).
+Admitted.
+Theorem restrl_by_dom : (forall R,(((relation R)->((restrl R (dom R)) = R)))).
+Admitted.
+Theorem restrl_dom : (forall R X,((subseteq (dom (restrl R X)) X))).
+Admitted.
+Theorem restrl_ran : (forall X R,(((subseteq X (dom R))->((ran (restrl R X)) = (img R X))))).
+Admitted.
+Theorem restrl_img : (forall X R A,(((subseteq X (dom R))->((img (restrl R X) A) = (img R (inter X A)))))).
+Admitted.
+Definition binary_relation_on := fun x0 x1 : set => (subseteq x0 (times x1 x1)).
+Theorem relation_subseteq_intro_elem : (forall R B A w,((((((relation R)/\(subseteq (ran R) B))/\(subseteq (dom R) A))/\(elem w R))->(elem w (times A B))))).
+Admitted.
+Theorem relation_subseteq_intro : (forall R B A,(((((relation R)/\(subseteq (ran R) B))/\(subseteq (dom R) A))->(subseteq R (times A B))))).
+Admitted.
+Theorem relation_subseteq_implies_dom_subseteq_elem : (forall R A B a,((((subseteq R (times A B))/\(elem a (dom R)))->(elem a A)))).
+Admitted.
+Theorem relation_subseteq_implies_dom_subseteq : (forall R A B,(((subseteq R (times A B))->(subseteq (dom R) A)))).
+Admitted.
+Theorem relation_subseteq_implies_ran_subseteq_elem : (forall R A B b,((((subseteq R (times A B))/\(elem b (ran R)))->(elem b B)))).
+Admitted.
+Theorem relation_subseteq_implies_ran_subseteq : (forall R A B,(((subseteq R (times A B))->(subseteq (ran R) B)))).
+Admitted.
+Definition rels := fun A B : set => (pow (times A B)).
+Theorem rels_intro : (forall R A B,(((subseteq R (times A B))->(elem R (rels A B))))).
+Admitted.
+Theorem rels_intro_dom_and_ran : (forall R A B,(((((relation R)/\(subseteq (dom R) A))/\(subseteq (ran R) B))->(elem R (rels A B))))).
+Admitted.
+Theorem rels_elim : (forall R A B,(((elem R (rels A B))->(subseteq R (times A B))))).
+Admitted.
+Theorem rels_weaken_dom : (forall R A B C,((((elem R (rels A B))/\(subseteq A C))->(elem R (rels C B))))).
+Admitted.
+Theorem rels_weaken_codom : (forall R A B D,((((elem R (rels A B))/\(subseteq B D))->(elem R (rels A D))))).
+Admitted.
+Definition id := fun A : set => Eps_i (fun frs : set => (forall frv,(((elem frv frs)<->(exists a,(((elem a A)/\((pair a a) = frv)))))))).
+Theorem id_elem_intro : (forall a A,(((elem a A)->(elem (pair a a) (id A))))).
+Admitted.
+Theorem id_elem_inspect : (forall w A,(((elem w (id A))->(exists a,(((elem a A)/\(w = (pair a a)))))))).
+Admitted.
+Theorem id_is_relation : (forall A,((relation (id A)))).
+Admitted.
+Theorem id_dom : (forall A,(((dom (id A)) = A))).
+Admitted.
+Theorem id_ran : (forall A,(((ran (id A)) = A))).
+Admitted.
+Theorem id_elem_rels : (forall A,((elem (id A) (rels A A)))).
+Admitted.
+Definition injective := fun R: set => (forall a b aprime,((((elem (pair a b) R)/\(elem (pair aprime b) R))->(a = aprime)))).
+Theorem subseteq_of_injective_is_injective : (forall S R,((((subseteq S R)/\(injective R))->(injective S)))).
+Admitted.
+Theorem restrl_injective : (forall R A,(((injective R)->(injective (restrl R A))))).
+Admitted.
+Theorem circ_injective : (forall R S,((((injective R)/\(injective S))->(injective (circ S R))))).
+Admitted.
+Definition left_quasireflexive_relation := fun R: set => (forall x y,(((elem (pair x y) R)->(elem (pair x x) R)))).
+Definition right_quasireflexive_relation := fun R: set => (forall x y,(((elem (pair x y) R)->(elem (pair y y) R)))).
+Definition quasireflexive_relation := fun R: set => (forall x y,(((elem (pair x y) R)->((elem (pair x x) R)/\(elem (pair y y) R))))).
+Definition coreflexive_relation := fun R: set => (forall x y,(((elem (pair x y) R)->(x = y)))).
+Definition reflexive_on := fun x0 x1 : set => (forall x,(((elem x x1)->(elem (pair x x) x0)))).
+Theorem quasireflexive_iff_reflexive_on_fld : (forall R,(((quasireflexive_relation R)<->(reflexive_on R (fld R))))).
+Admitted.
+Definition antisymmetric_relation := fun R: set => (forall x y,((((elem (pair x y) R)/\(elem (pair y x) R))->(x = y)))).
+Definition symmetric_relation := fun R: set => (forall x y,(((elem (pair x y) R)<->(elem (pair y x) R)))).
+Definition transitive_relation := fun R: set => (forall x y z,((((elem (pair x y) R)/\(elem (pair y z) R))->(elem (pair x z) R)))).
+Theorem transitive_downward_elem : (forall R a b c,(((((transitive_relation R)/\(elem a (downward_closure R b)))/\(elem c (downward_closure R a)))->(elem c (downward_closure R b))))).
+Admitted.
+Theorem transitive_downward_subseteq : (forall R a b,((((transitive_relation R)/\(elem a (downward_closure R b)))->(subseteq (downward_closure R a) (downward_closure R b))))).
+Admitted.
+Definition quasiorder := fun x0 : set => ((quasireflexive_relation x0)/\(transitive_relation x0)).
+Definition equivalence_relation := fun x0 : set => ((quasiorder x0)/\(symmetric_relation x0)).
+Definition equiv_class := fun x0 x1 : set => (downward_closure x0 x1).
+Definition equiv_class_abbr := fun x0 x1 : set => (equiv_class x0 x1).
+Theorem equiv_classes_inhabited : (forall E a,((((equivalence_relation E)/\(elem a (dom E)))->(elem a (equiv_class E a))))).
+Admitted.
+Theorem equiv_classes_diseq_implies_disjoint : (forall E a b,((((equivalence_relation E)/\((equiv_class E a) <> (equiv_class E b)))->(disjoint (equiv_class E a) (equiv_class E b))))).
+Admitted.
+Definition quotient_set := fun E : set => Eps_i (fun frs : set => (forall frv,(((elem frv frs)<->(exists a,(((elem a (dom E))/\((equiv_class E a) = frv)))))))).
+Theorem quotient_emptyset : ((quotient_set (emptyset)) = (emptyset)).
+Admitted.
+Theorem quotient_elems_disjoint : (forall E B C,((((equivalence_relation E)/\(((elem B (quotient_set E))/\(elem C (quotient_set E)))/\(B <> C)))->(disjoint B C)))).
+Admitted.
+Theorem quotient_elems_inhabited : (forall E A,((((equivalence_relation E)/\(elem A (quotient_set E)))->(inhabited A)))).
+Admitted.
+Theorem quotient_notni_emptyset : (forall E,(((equivalence_relation E)->(notelem (emptyset) (quotient_set E))))).
+Admitted.
+Theorem quotient_partition : (forall E,(((equivalence_relation E)->(partition (quotient_set E))))).
+Admitted.
diff --git a/megalodon/library/set.mg b/megalodon/library/set.mg
new file mode 100644
index 0000000..3606b4e
--- /dev/null
+++ b/megalodon/library/set.mg
@@ -0,0 +1,293 @@
+Let emptyset : set := Empty.
+Let elem : set->set->prop := In.
+Let notelem : set->set->prop := fun a A => ~(In a A).
+Let pow : set->set := Power.
+Let unions : set->set := Union.
+Let union : set->set->set := binunion.
+Let cons : set -> set -> set := fun x X => binunion {x} X.
+Let xor : prop -> prop -> prop := fun p q => (p \/ q) /\ ~(p /\ q).
+Let pair : set -> set -> set := fun a b => {{a}, {a, b}}.
+Let fst : set -> set := fun p => Eps_i (fun a => exists b, p = pair a b).
+Let snd : set -> set := fun p => Eps_i (fun b => exists a, p = pair a b).
+Let nor : prop -> prop -> prop := fun p q => ~(p \/ q) .
+Definition ni := fun x0 x1 : set => (elem x1 x0).
+Fact setext : (forall A B,(((forall a,(((elem a A)<->(elem a B))))->(A = B)))).
+Admitted.
+Theorem neq_witness : (forall A B,(((A <> B)->(exists c,((xor((elem c A)/\~((elem c B))) (~((elem c A))/\(elem c B)))))))).
+Admitted.
+Definition subseteq := fun A B: set => (forall a,(((elem a A)->(elem a B)))).
+Definition is_subset := fun x0 x1 : set => (subseteq x0 x1).
+Definition supseteq := fun x0 x1 : set => (subseteq x1 x0).
+Theorem subseteq_refl : (forall A,((subseteq A A))).
+Admitted.
+Theorem subseteq_antisymmetric : (forall A B,((((subseteq A B)/\(subseteq B A))->(A = B)))).
+Admitted.
+Theorem elem_subseteq : (forall a A B,((((elem a A)/\(subseteq A B))->(elem a B)))).
+Admitted.
+Theorem not_in_subseteq : (forall A B c,((((subseteq A B)/\(notelem c B))->(notelem c A)))).
+Admitted.
+Theorem subseteq_transitive : (forall A B C,((((subseteq A B)/\(subseteq B C))->(subseteq A C)))).
+Admitted.
+Definition subset := fun A B: set => ((subseteq A B)/\(A <> B)).
+Theorem subset_irrefl : (forall A,(~((subset A A)))).
+Admitted.
+Theorem subset_transitive : (forall A B C,((((subseteq A B)/\(subseteq B C))->(subseteq A C)))).
+Admitted.
+Theorem subset_witness : (forall A B,(((subset A B)->(exists b,(((elem b B)/\(notelem b A))))))).
+Admitted.
+Definition family_of_subsets := fun x0 x1 : set => (forall A,(((elem A x0)->(subseteq A x1)))).
+Fact notin_emptyset : (forall a,((notelem a (emptyset)))).
+Admitted.
+Definition inhabited := fun A: set => (exists a,((elem a A))).
+Definition empty := fun x0 : set => ~((inhabited x0)).
+Theorem empty_eq : (forall x y,((((empty x)/\(empty y))->(x = y)))).
+Admitted.
+Theorem emptyset_subseteq : (forall a,((subseteq (emptyset) a))).
+Admitted.
+Theorem subseteq_emptyset_iff : (forall A,(((subseteq A (emptyset))<->(A = (emptyset))))).
+Admitted.
+Definition disjoint := fun A B: set => ~((exists a,(((elem a A)/\(elem a B))))).
+Definition notmeets := fun x0 x1 : set => (disjoint x0 x1).
+Definition meets := fun x0 x1 : set => ~((disjoint x0 x1)).
+Theorem disjoint_symmetric : (forall A B,(((disjoint A B)->(disjoint B A)))).
+Admitted.
+Fact in_cons : (forall x y X,(((elem x (cons y X))<->((x = y)\/(elem x X))))).
+Admitted.
+Theorem in_cons_left : (forall x X,((elem x (cons x X)))).
+Admitted.
+Theorem in_cons_right : (forall y X x,(((elem y X)->(elem y (cons x X))))).
+Admitted.
+Theorem upair_intro_left : (forall a b,((elem a (cons a (cons b (emptyset)))))).
+Admitted.
+Theorem upair_intro_right : (forall b a,((elem b (cons a (cons b (emptyset)))))).
+Admitted.
+Theorem upair_elim : (forall c a b,(((elem c (cons a (cons b (emptyset))))->((a = c)\/(b = c))))).
+Admitted.
+Theorem upair_iff : (forall c a b,(((elem c (cons a (cons b (emptyset))))<->((a = c)\/(b = c))))).
+Admitted.
+Theorem singleton_intro : (forall a,((elem a (cons a (emptyset))))).
+Admitted.
+Theorem singleton_elim : (forall a b,(((elem a (cons b (emptyset)))->(a = b)))).
+Admitted.
+Theorem singleton_iff : (forall a b,(((elem a (cons b (emptyset)))<->(a = b)))).
+Admitted.
+Definition subsingleton := fun x0 : set => (forall a b,((((elem a x0)/\(elem b x0))->(a = b)))).
+Theorem singleton_inhabited : (forall a,((inhabited (cons a (emptyset))))).
+Admitted.
+Theorem singleton_iff_inhabited_subsingleton : (forall A a,((((subsingleton A)/\(elem a A))->(A = (cons a (emptyset)))))).
+Admitted.
+Theorem singleton_subset_intro : (forall a C,(((elem a C)->(subseteq (cons a (emptyset)) C)))).
+Admitted.
+Theorem singleton_subset_elim : (forall a C,(((subseteq (cons a (emptyset)) C)->(elem a C)))).
+Admitted.
+Fact unions_iff_exists : (forall z X,(((elem z (unions X))<->(exists Y,(((elem Y X)/\(elem z Y))))))).
+Admitted.
+Theorem unions_intro : (forall A B C,((((elem A B)/\(elem B C))->(elem A (unions C))))).
+Admitted.
+Theorem unions_emptyset : ((unions (emptyset)) = (emptyset)).
+Admitted.
+Theorem unions_family : (forall F X,(((family_of_subsets F X)->(subseteq (unions F) X)))).
+Admitted.
+Definition closedunderunions := fun x0 : set => (forall M,(((is_subset M x0)->(elem (unions M) x0)))).
+Definition inters := fun A : set => {x :e ((unions A))|(forall a,(((elem a A)->(elem x a))))}.
+Theorem inters_iff_forall : (forall z X,(((elem z (inters X))<->((inhabited X)/\(forall Y,(((elem Y X)->(elem z Y)))))))).
+Admitted.
+Theorem inters_intro : (forall C A,((((inhabited C)/\(forall B,(((elem B C)->(elem A B)))))->(elem A (inters C))))).
+Admitted.
+Theorem inters_destr : (forall A C B,((((elem A (inters C))/\(elem B C))->(elem A B)))).
+Admitted.
+Theorem inters_greatest : (forall A C,((((inhabited A)/\(forall a,(((elem a A)->(subseteq C a)))))->(subseteq C (inters A))))).
+Admitted.
+Theorem subseteq_inters_iff : (forall A C,(((inhabited A)->((subseteq C (inters A))<->(forall a,(((elem a A)->(subseteq C a)))))))).
+Admitted.
+Theorem inters_subseteq_elem : (forall B A,(((elem B A)->(subseteq (inters A) B)))).
+Admitted.
+Theorem inters_singleton : (forall a,(((inters (cons a (emptyset))) = a))).
+Admitted.
+Theorem inters_emptyset : ((inters (cons (emptyset) (emptyset))) = (emptyset)).
+Admitted.
+Fact union_iff : (forall a A B,(((elem a (union A B))<->((elem a A)\/(elem a B))))).
+Admitted.
+Theorem union_intro_left : (forall c A B,(((elem c A)->(elem c (union A B))))).
+Admitted.
+Theorem union_intro_right : (forall c B A,(((elem c B)->(elem c (union A B))))).
+Admitted.
+Theorem union_comm : (forall A B,(((union A B) = (union B A)))).
+Admitted.
+Theorem union_assoc : (forall A B C,(((union (union A B) C) = (union A (union B C))))).
+Admitted.
+Theorem union_idempotent : (forall A,(((union A A) = A))).
+Admitted.
+Theorem subseteq_union_iff : (forall A B C,(((subseteq (union A B) C)<->((subseteq A C)/\(subseteq B C))))).
+Admitted.
+Theorem union_upper_left : (forall A B,((subseteq A (union A B)))).
+Admitted.
+Theorem union_upper_right : (forall B A,((subseteq B (union A B)))).
+Admitted.
+Theorem union_subseteq_union : (forall A C B D,((((subseteq A C)/\(subseteq B D))->(subseteq (union A B) (union C D))))).
+Admitted.
+Theorem union_emptyset : (forall A,(((union A (emptyset)) = A))).
+Admitted.
+Theorem union_emptyset_intro : (forall A B,((((A = (emptyset))/\(B = (emptyset)))->((union A B) = (emptyset))))).
+Admitted.
+Theorem union_emptyset_elim_left : (forall A B,((((union A B) = (emptyset))->(A = (emptyset))))).
+Admitted.
+Theorem union_emptyset_elim_right : (forall A B,((((union A B) = (emptyset))->(B = (emptyset))))).
+Admitted.
+Theorem union_absorb_subseteq_left : (forall A B,(((subseteq A B)->((union A B) = B)))).
+Admitted.
+Theorem union_absorb_subseteq_right : (forall A B,(((subseteq A B)->((union B A) = B)))).
+Admitted.
+Theorem union_eq_self_implies_subseteq : (forall A B,((((union A B) = B)->(subseteq A B)))).
+Admitted.
+Theorem unions_cons : (forall b A,(((unions (cons b A)) = (union b (unions A))))).
+Admitted.
+Theorem union_cons : (forall b A C,(((union (cons b A) C) = (cons b (union A C))))).
+Admitted.
+Theorem union_absorb_left : (forall A B,(((union A (union A B)) = (union A B)))).
+Admitted.
+Theorem union_absorb_right : (forall A B,(((union (union A B) B) = (union A B)))).
+Admitted.
+Theorem union_comm_left : (forall A B C,(((union A (union B C)) = (union B (union A C))))).
+Admitted.
+Definition closedunderunion := fun x0 : set => (forall U V,((((elem U x0)/\(elem V x0))->(elem (union U V) x0)))).
+Definition inter := fun A B : set => {a :e (A)|(elem a B)}.
+Theorem inter_intro : (forall c A B,((((elem c A)/\(elem c B))->(elem c (inter A B))))).
+Admitted.
+Theorem inter_elim_left : (forall c A B,(((elem c (inter A B))->(elem c A)))).
+Admitted.
+Theorem inter_elim_right : (forall c A B,(((elem c (inter A B))->(elem c B)))).
+Admitted.
+Theorem inter_as_inters : (forall A B,(((inters (cons A (cons B (emptyset)))) = (inter A B)))).
+Admitted.
+Theorem inter_comm : (forall A B,(((inter A B) = (inter B A)))).
+Admitted.
+Theorem inter_assoc : (forall A B C,(((inter (inter A B) C) = (inter A (inter B C))))).
+Admitted.
+Theorem inter_idempotent : (forall A,(((inter A A) = A))).
+Admitted.
+Theorem inter_subseteq : (forall A B,((subseteq (inter A B) A))).
+Admitted.
+Theorem inter_emptyset : (forall A,(((inter A (emptyset)) = (emptyset)))).
+Admitted.
+Theorem inter_absorb_supseteq_right : (forall A B,(((subseteq A B)->((inter A B) = A)))).
+Admitted.
+Theorem inter_absorb_supseteq_left : (forall A B,(((subseteq A B)->((inter B A) = A)))).
+Admitted.
+Theorem inter_eq_left_implies_subseteq : (forall A B,((((inter A B) = A)->(subseteq A B)))).
+Admitted.
+Theorem subseteq_inter_iff : (forall C A B,(((subseteq C (inter A B))<->((subseteq C A)/\(subseteq C B))))).
+Admitted.
+Theorem inter_lower_left : (forall A B,((subseteq (inter A B) A))).
+Admitted.
+Theorem inter_lower_right : (forall A B,((subseteq (inter A B) B))).
+Admitted.
+Theorem inter_absorb_left : (forall A B,(((inter A (inter A B)) = (inter A B)))).
+Admitted.
+Theorem inter_absorb_right : (forall A B,(((inter (inter A B) B) = (inter A B)))).
+Admitted.
+Theorem inter_comm_left : (forall A B C,(((inter A (inter B C)) = (inter B (inter A C))))).
+Admitted.
+Definition closedunderinter := fun x0 : set => (forall U V,((((elem U x0)/\(elem V x0))->(elem (inter U V) x0)))).
+Theorem inter_distrib_union : (forall x y z,(((inter x (union y z)) = (union (inter x y) (inter x z))))).
+Admitted.
+Theorem union_distrib_inter : (forall x y z,(((union x (inter y z)) = (inter (union x y) (union x z))))).
+Admitted.
+Theorem union_inter_assoc_intro : (forall C A B,(((subseteq C A)->((union (inter A B) C) = (inter A (union B C)))))).
+Admitted.
+Theorem union_inter_assoc_elim : (forall A B C,((((union (inter A B) C) = (inter A (union B C)))->(subseteq C A)))).
+Admitted.
+Theorem union_inter_crazy : (forall A B C,(((union (union (inter A B) (inter B C)) (inter C A)) = (inter (inter (union A B) (union B C)) (union C A))))).
+Admitted.
+Theorem inters_distrib_union : (forall A B,((((inhabited A)/\(inhabited B))->((inters (union A B)) = (inter (inters A) (inters B)))))).
+Admitted.
+Definition setminus := fun A B : set => {a :e (A)|~((elem a B))}.
+Theorem setminus_intro : (forall a A B,((((elem a A)/\(notelem a B))->(elem a (setminus A B))))).
+Admitted.
+Theorem setminus_elim_left : (forall a A B,(((elem a (setminus A B))->(elem a A)))).
+Admitted.
+Theorem setminus_elim_right : (forall a A B,(((elem a (setminus A B))->(notelem a B)))).
+Admitted.
+Theorem setminus_emptyset : (forall x,(((setminus x (emptyset)) = x))).
+Admitted.
+Theorem emptyset_setminus : (forall x,(((setminus (emptyset) x) = (emptyset)))).
+Admitted.
+Theorem setminus_self : (forall x,(((setminus x x) = (emptyset)))).
+Admitted.
+Theorem setminus_setminus : (forall x y,(((setminus x (setminus x y)) = (inter x y)))).
+Admitted.
+Theorem double_relative_complement : (forall y x,(((subseteq y x)->((setminus x (setminus x y)) = y)))).
+Admitted.
+Theorem setminus_inter : (forall x y z,(((setminus x (inter y z)) = (union (setminus x y) (setminus x z))))).
+Admitted.
+Theorem setminus_union : (forall x y z,(((setminus x (union y z)) = (inter (setminus x y) (setminus x z))))).
+Admitted.
+Theorem inter_setminus : (forall x y z,(((inter x (setminus y z)) = (setminus (inter x y) (inter x z))))).
+Admitted.
+Theorem difference_with_proper_subset_is_inhabited : (forall A B,(((subset A B)->(inhabited (setminus B A))))).
+Admitted.
+Theorem setminus_subseteq : (forall B A,((subseteq (setminus B A) B))).
+Admitted.
+Theorem subseteq_setminus : (forall C A B,((((subseteq C A)/\((inter C B) = (emptyset)))->(subseteq C (setminus A B))))).
+Admitted.
+Theorem subseteq_implies_setminus_supseteq : (forall A B C,(((subseteq A B)->(supseteq (setminus C A) (setminus C B))))).
+Admitted.
+Theorem setminus_absorb_right : (forall A B,((((inter A B) = (emptyset))->((setminus A B) = A)))).
+Admitted.
+Theorem setminus_eq_emptyset_iff_subseteq : (forall A B,((((setminus A B) = (emptyset))<->(subseteq A B)))).
+Admitted.
+Theorem subseteq_setminus_cons_intro : (forall B A C c,((((subseteq B (setminus A C))/\(notelem c B))->(subseteq B (setminus A (cons c C)))))).
+Admitted.
+Theorem subseteq_setminus_cons_elim : (forall B A c C,(((subseteq B (setminus A (cons c C)))->((subseteq B (setminus A C))/\(notelem c B))))).
+Admitted.
+Theorem setminus_cons : (forall A a B,(((setminus A (cons a B)) = (setminus (setminus A (cons a (emptyset))) B)))).
+Admitted.
+Theorem setminus_cons_flip : (forall A a B,(((setminus A (cons a B)) = (setminus (setminus A B) (cons a (emptyset)))))).
+Admitted.
+Theorem setminus_disjoint : (forall A B,(((inter A (setminus B A)) = (emptyset)))).
+Admitted.
+Theorem setminus_partition : (forall A B,(((subseteq A B)->((union A (setminus B A)) = B)))).
+Admitted.
+Theorem subseteq_union_setminus : (forall A B,((subseteq A (union B (setminus A B))))).
+Admitted.
+Theorem double_complement : (forall A B C,((((subseteq A B)/\(subseteq B C))->((setminus B (setminus C A)) = A)))).
+Admitted.
+Theorem double_complement_union : (forall A B,(((setminus (union A B) (setminus B A)) = A))).
+Admitted.
+Theorem setminus_eq_inter_complement : (forall A C B,((((subseteq A C)/\(subseteq B C))->((setminus A B) = (inter A (setminus C B)))))).
+Admitted.
+Fact pair_eq_iff : (forall a b aprime bprime,((((pair a b) = (pair aprime bprime))<->((a = aprime)/\(b = bprime))))).
+Admitted.
+Fact pair_neq_emptyset : (forall a b,(((pair a b) <> (emptyset)))).
+Admitted.
+Fact pair_neq_fst : (forall a b,(((pair a b) <> a))).
+Admitted.
+Fact pair_neq_snd : (forall a b,(((pair a b) <> b))).
+Admitted.
+Theorem triple_eq_iff : (forall a b c aprime bprime cprime,((((pair a (pair b c)) = (pair aprime (pair bprime cprime)))<->(((a = aprime)/\(b = bprime))/\(c = cprime))))).
+Admitted.
+Fact fst_eq : (forall a b,(((fst (pair a b)) = a))).
+Admitted.
+Fact snd_eq : (forall a b,(((snd (pair a b)) = b))).
+Admitted.
+Theorem pair_eq_pair_of_proj : (forall a b,(((pair a b) = (pair (fst (pair a b)) (snd (pair a b)))))).
+Admitted.
+Definition times := fun A B : set => ReplSep2 (A)(fun dummyVar => B)(fun a b => True)(fun a b => (pair a b)).
+Theorem times_tuple_elim : (forall x y X Y,(((elem (pair x y) (times X Y))->((elem x X)/\(elem y Y))))).
+Admitted.
+Theorem times_tuple_intro : (forall x X y Y,((((elem x X)/\(elem y Y))->(elem (pair x y) (times X Y))))).
+Admitted.
+Theorem times_empty_left : (forall Y,(((times (emptyset) Y) = (emptyset)))).
+Admitted.
+Theorem times_empty_right : (forall X,(((times X (emptyset)) = (emptyset)))).
+Admitted.
+Theorem times_empty_iff : (forall X Y,(((empty (times X Y))<->((empty X)\/(empty Y))))).
+Admitted.
+Theorem fst_type : (forall c A B,(((elem c (times A B))->(elem (fst c) A)))).
+Admitted.
+Theorem snd_type : (forall c A B,(((elem c (times A B))->(elem (snd c) B)))).
+Admitted.
+Theorem times_elem_is_tuple : (forall p X Y,(((elem p (times X Y))->(exists x y,(((elem x X)/\((elem y Y)/\(p = (pair x y))))))))).
+Admitted.
+Theorem times_proj_elim : (forall p X Y,(((elem p (times X Y))->((elem (fst p) X)/\(elem (snd p) Y))))).
+Admitted.
diff --git a/megalodon/megalodon/scraps.mg b/megalodon/megalodon/scraps.mg
new file mode 100644
index 0000000..ec6d190
--- /dev/null
+++ b/megalodon/megalodon/scraps.mg
@@ -0,0 +1,5 @@
+Axiom trivial : (forall x,(x = x)).
+Axiom empty_axiom : (~(exists x,(elem x (emptyset)))).
+Axiom union_eq : (forall x X,((elem x (unions X))<->(exists Y,((elem x Y)/\(elem Y X))))).
+Theorem eq_refl : (forall x,(x = x)).
+Admitted.
diff --git a/megalodon/ownedSep192022 b/megalodon/ownedSep192022
new file mode 100644
index 0000000..4b9e0d2
--- /dev/null
+++ b/megalodon/ownedSep192022
@@ -0,0 +1,31613 @@
+Obj d37f2a956f8825aaa56873f4ee343d8c083825c53252d63e7bcba4515cb620aa 2277 09d0b23a0e5bda2fb8ebfe741b1b45e1baf5f0709214f6604aae7714ae2c7642 TMEg2RE3xgWpRbKa5zDuUPhECTsfNrwyn5Z TMEg2RE3xgWpRbKa5zDuUPhECTsfNrwyn5Z PrGxvhtV11eKSAxtzEcT1Ku9MkDCTq1NrUp 0 PrGxvhtV11eKSAxtzEcT1Ku9MkDCTq1NrUp 0
+Prop d5ae307b481651b2a6c91ae9380e2cc17182d7637cd48b7440019f101db5a0f1 6761 5f187260e033281111655ae0676b5d5325c934ef62451c58a1e391a90687ee5c TMEg3K9LDCCnd2cuJbgeRFbZKTQCSr7xdJo TMEg3K9LDCCnd2cuJbgeRFbZKTQCSr7xdJo Pr6PcLD1ZFK7NER7E2CoDmtTrD5GEF7R1wq 0 Pr6PcLD1ZFK7NER7E2CoDmtTrD5GEF7R1wq 0
+Obj 6fd89bddcd9c74a75655ecf8a684f3851ab32ffcec122d43794c6f87c87f0a2a 4958 30acc532eaa669658d7b9166abf687ea3e2b7c588c03b36ba41be23d1c82e448 TMEg3b8dv5i7oFXmXe8scTrFUcM8PzJ4zpW TMEg3b8dv5i7oFXmXe8scTrFUcM8PzJ4zpW Pr6PcLD1ZFK7NER7E2CoDmtTrD5GEF7R1wq 0 Pr6PcLD1ZFK7NER7E2CoDmtTrD5GEF7R1wq 0
+Prop dc6aa4947e518943ba67b1da3f25ad325c28b3fd8e8ab28d5679b58df4cd0069 15602 518f8766bcd89cc387e477ffe0d14cc0533a08b5c94d7cc56b6af58c2e4e9b47 TMEg4ZNDX4xuWPSQKZc9BsNnt4JbkWfvynB TMEg4ZNDX4xuWPSQKZc9BsNnt4JbkWfvynB Pr4zBe5RPZVASWz9JcEocYoVA1L3QqJ9Pk2 0 Pr4zBe5RPZVASWz9JcEocYoVA1L3QqJ9Pk2 0
+Prop a7cf8ef2688d65d9fafd0271e0ba7c54890a40655c28bb9f60f486c7cd96340a 4898 7758f10f23cd5bde5f62e6e19176c07070d03250e7ee2f5445ddd169a94ac9cc TMEg6jVocmXvkpGecshFe76iHzxRPb9JEDU TMEg6jVocmXvkpGecshFe76iHzxRPb9JEDU Pr6PcLD1ZFK7NER7E2CoDmtTrD5GEF7R1wq 0 Pr6PcLD1ZFK7NER7E2CoDmtTrD5GEF7R1wq 0
+NegProp f2929efb756ca4982e50b81a130b3bf6e36ea327a2f342ee2c747df0473a0a93 14958 TMEgAFxgJ2qLwCGZ1nN1jwv5aTfyQLJNZzR Pr4zBe5RPZVASWz9JcEocYoVA1L3QqJ9Pk2 0
+Prop ce9cd7601d86521201b84755adf45f2c05a59f2698a3d7d593a1d1f90c9408cd 11848 8b0cef70f636e6ee67ab0c14c07c5ea4b80c291b0e6ff90630f439d9d03ef5ba TMEgGQcC6KMJoHGy3h6GCWLDJ1KTmBWCcKq TMEgGQcC6KMJoHGy3h6GCWLDJ1KTmBWCcKq PrGVSsTu3sRqvuwGfGwdtAB4V6dBozJHLki 0 PrGVSsTu3sRqvuwGfGwdtAB4V6dBozJHLki 0
+Prop d9c88bcbd3ca30e18ff0262b322d071ec6edb19561f1b5fc540379f5ec606d36 4903 11f51b14cc9177b5bc57ce79c943b2dcdfe3d83f93b1a6336d2174ba4e9df0b7 TMEgLJHQtw8TKqiFHfyA8RQk6wHyjMKHyrr TMEgLJHQtw8TKqiFHfyA8RQk6wHyjMKHyrr Pr6PcLD1ZFK7NER7E2CoDmtTrD5GEF7R1wq 0 Pr6PcLD1ZFK7NER7E2CoDmtTrD5GEF7R1wq 0
+Obj 1de15ad196986a1868df28b54762d2c03ffefd0b4dd417b89092bcbe67d20013 4972 a1fba96d2d6ac5b62c8785d2dcc33aea17914aba665f308fd346d23d626d3e0c TMEgZeoteeGJu859kpBbmJ6NbLCpAp3Jyzi TMEgZeoteeGJu859kpBbmJ6NbLCpAp3Jyzi Pr6PcLD1ZFK7NER7E2CoDmtTrD5GEF7R1wq 0 Pr6PcLD1ZFK7NER7E2CoDmtTrD5GEF7R1wq 0
+Prop ab31560c6c6643b0d158ed4d8f3d85a10b930e9268cdd27e0e7610e9782e5aad 19222 616c991f45f4fe9d3efb0220d6bfc2b3a9576d02c827cfd1aa156159e180bc78 TMEgbCN37qkgxnFg2qhU7k4qRwjFDnn5YbV TMEgbCN37qkgxnFg2qhU7k4qRwjFDnn5YbV Pr4zBe5RPZVASWz9JcEocYoVA1L3QqJ9Pk2 0 Pr4zBe5RPZVASWz9JcEocYoVA1L3QqJ9Pk2 0
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diff --git a/megalodon/preamble.mgs b/megalodon/preamble.mgs
new file mode 100644
index 0000000..931f1c8
--- /dev/null
+++ b/megalodon/preamble.mgs
@@ -0,0 +1,434 @@
+(* Parameter Eps_i "174b78e53fc239e8c2aab4ab5a996a27e3e5741e88070dad186e05fb13f275e5" *)
+Parameter Eps_i : (set->prop)->set.
+
+Axiom Eps_i_ax : forall P:set->prop, forall x:set, P x -> P (Eps_i P).
+
+Definition True : prop := forall p:prop, p -> p.
+Definition False : prop := forall p:prop, p.
+
+Definition not : prop -> prop := fun A:prop => A -> False.
+
+(* Unicode ~ "00ac" *)
+Prefix ~ 700 := not.
+
+Definition and : prop -> prop -> prop := fun A B:prop => forall p:prop, (A -> B -> p) -> p.
+
+(* Unicode /\ "2227" *)
+Infix /\ 780 left := and.
+
+Definition or : prop -> prop -> prop := fun A B:prop => forall p:prop, (A -> p) -> (B -> p) -> p.
+
+(* Unicode \/ "2228" *)
+Infix \/ 785 left := or.
+
+Definition iff : prop -> prop -> prop := fun A B:prop => and (A -> B) (B -> A).
+
+(* Unicode <-> "2194" *)
+Infix <-> 805 := iff.
+
+Section Eq.
+Variable A:SType.
+Definition eq : A->A->prop := fun x y:A => forall Q:A->A->prop, Q x y -> Q y x.
+Definition neq : A->A->prop := fun x y:A => ~ eq x y.
+End Eq.
+
+Infix = 502 := eq.
+(* Unicode <> "2260" *)
+Infix <> 502 := neq.
+
+Section FE.
+Variable A B : SType.
+Axiom func_ext : forall f g : A -> B , (forall x : A , f x = g x) -> f = g.
+End FE.
+
+Section Ex.
+Variable A:SType.
+Definition ex : (A->prop)->prop := fun Q:A->prop => forall P:prop, (forall x:A, Q x -> P) -> P.
+End Ex.
+
+(* Unicode exists "2203" *)
+Binder+ exists , := ex.
+
+Axiom prop_ext : forall p q:prop, iff p q -> p = q.
+
+Parameter In:set->set->prop.
+
+Definition Subq : set -> set -> prop := fun A B => forall x :e A, x :e B.
+
+Axiom set_ext : forall X Y:set, X c= Y -> Y c= X -> X = Y.
+
+Axiom In_ind : forall P:set->prop, (forall X:set, (forall x :e X, P x) -> P X) -> forall X:set, P X.
+
+Binder+ exists , := ex; and.
+
+Parameter Empty : set.
+Axiom EmptyAx : ~exists x:set, x :e Empty.
+
+(* Unicode Union "22C3" *)
+Parameter Union : set->set.
+
+Axiom UnionEq : forall X x, x :e Union X <-> exists Y, x :e Y /\ Y :e X.
+
+(* Unicode Power "1D4AB" *)
+Parameter Power : set->set.
+
+Axiom PowerEq : forall X Y:set, Y :e Power X <-> Y c= X.
+
+Parameter Repl : set -> (set -> set) -> set.
+Notation Repl Repl.
+
+Axiom ReplEq : forall A:set, forall F:set->set, forall y:set, y :e {F x|x :e A} <-> exists x :e A, y = F x.
+
+Definition TransSet : set->prop := fun U:set => forall x :e U, x c= U.
+
+Definition Union_closed : set->prop := fun U:set => forall X:set, X :e U -> Union X :e U.
+Definition Power_closed : set->prop := fun U:set => forall X:set, X :e U -> Power X :e U.
+Definition Repl_closed : set->prop := fun U:set => forall X:set, X :e U -> forall F:set->set,
+ (forall x:set, x :e X -> F x :e U) -> {F x|x :e X} :e U.
+Definition ZF_closed : set->prop := fun U:set =>
+ Union_closed U
+/\ Power_closed U
+/\ Repl_closed U.
+
+Parameter UnivOf : set->set.
+
+Axiom UnivOf_In : forall N:set, N :e UnivOf N.
+
+Axiom UnivOf_TransSet : forall N:set, TransSet (UnivOf N).
+
+Axiom UnivOf_ZF_closed : forall N:set, ZF_closed (UnivOf N).
+
+Axiom UnivOf_Min : forall N U:set, N :e U
+ -> TransSet U
+ -> ZF_closed U
+ -> UnivOf N c= U.
+
+Axiom FalseE : False -> forall p:prop, p.
+
+Axiom TrueI : True.
+
+Axiom notI : forall A:prop, (A -> False) -> ~A.
+
+Axiom notE : forall A:prop, ~A -> A -> False.
+
+Axiom andI : forall (A B : prop), A -> B -> A /\ B.
+
+Axiom andEL : forall (A B : prop), A /\ B -> A.
+
+Axiom andER : forall (A B : prop), A /\ B -> B.
+
+Axiom orIL : forall (A B : prop), A -> A \/ B.
+
+Axiom orIR : forall (A B : prop), B -> A \/ B.
+
+Axiom orE : forall (A B C:prop), (A -> C) -> (B -> C) -> A \/ B -> C.
+
+Section PropN.
+
+Variable P1 P2 P3:prop.
+
+Axiom and3I : P1 -> P2 -> P3 -> P1 /\ P2 /\ P3.
+Axiom and3E : P1 /\ P2 /\ P3 -> (forall p:prop, (P1 -> P2 -> P3 -> p) -> p).
+Axiom or3I1 : P1 -> P1 \/ P2 \/ P3.
+Axiom or3I2 : P2 -> P1 \/ P2 \/ P3.
+Axiom or3I3 : P3 -> P1 \/ P2 \/ P3.
+Axiom or3E : P1 \/ P2 \/ P3 -> (forall p:prop, (P1 -> p) -> (P2 -> p) -> (P3 -> p) -> p).
+
+Variable P4:prop.
+
+Axiom and4I : P1 -> P2 -> P3 -> P4 -> P1 /\ P2 /\ P3 /\ P4.
+Axiom and4E : P1 /\ P2 /\ P3 /\ P4 -> (forall p:prop, (P1 -> P2 -> P3 -> P4 -> p) -> p).
+Axiom or4I1 : P1 -> P1 \/ P2 \/ P3 \/ P4.
+Axiom or4I2 : P2 -> P1 \/ P2 \/ P3 \/ P4.
+Axiom or4I3 : P3 -> P1 \/ P2 \/ P3 \/ P4.
+Axiom or4I4 : P4 -> P1 \/ P2 \/ P3 \/ P4.
+Axiom or4E : P1 \/ P2 \/ P3 \/ P4 -> (forall p:prop, (P1 -> p) -> (P2 -> p) -> (P3 -> p) -> (P4 -> p) -> p).
+
+Variable P5:prop.
+
+Axiom and5I : P1 -> P2 -> P3 -> P4 -> P5 -> P1 /\ P2 /\ P3 /\ P4 /\ P5.
+Axiom and5E : P1 /\ P2 /\ P3 /\ P4 /\ P5 -> (forall p:prop, (P1 -> P2 -> P3 -> P4 -> P5 -> p) -> p).
+Axiom or5I1 : P1 -> P1 \/ P2 \/ P3 \/ P4 \/ P5.
+Axiom or5I2 : P2 -> P1 \/ P2 \/ P3 \/ P4 \/ P5.
+Axiom or5I3 : P3 -> P1 \/ P2 \/ P3 \/ P4 \/ P5.
+Axiom or5I4 : P4 -> P1 \/ P2 \/ P3 \/ P4 \/ P5.
+Axiom or5I5 : P5 -> P1 \/ P2 \/ P3 \/ P4 \/ P5.
+Axiom or5E : P1 \/ P2 \/ P3 \/ P4 \/ P5 -> (forall p:prop, (P1 -> p) -> (P2 -> p) -> (P3 -> p) -> (P4 -> p) -> (P5 -> p) -> p).
+
+Variable P6:prop.
+
+Axiom and6I: P1 -> P2 -> P3 -> P4 -> P5 -> P6 -> P1 /\ P2 /\ P3 /\ P4 /\ P5 /\ P6.
+Axiom and6E : P1 /\ P2 /\ P3 /\ P4 /\ P5 /\ P6 -> (forall p:prop, (P1 -> P2 -> P3 -> P4 -> P5 -> P6 -> p) -> p).
+
+Variable P7:prop.
+
+Axiom and7I: P1 -> P2 -> P3 -> P4 -> P5 -> P6 -> P7 -> P1 /\ P2 /\ P3 /\ P4 /\ P5 /\ P6 /\ P7.
+Axiom and7E : P1 /\ P2 /\ P3 /\ P4 /\ P5 /\ P6 /\ P7 -> (forall p:prop, (P1 -> P2 -> P3 -> P4 -> P5 -> P6 -> P7 -> p) -> p).
+
+End PropN.
+
+Axiom iffI : forall A B:prop, (A -> B) -> (B -> A) -> (A <-> B).
+Axiom iffEL : forall A B:prop, (A <-> B) -> A -> B.
+Axiom iffER : forall A B:prop, (A <-> B) -> B -> A.
+Axiom iff_ref : forall A:prop, A <-> A.
+
+Axiom neq_i_sym: forall x y, x <> y -> y <> x.
+
+Definition nIn : set->set->prop :=
+fun x X => ~In x X.
+
+(* Unicode /:e "2209" *)
+Infix /:e 502 := nIn.
+
+Axiom Eps_i_ex : forall P:set -> prop, (exists x, P x) -> P (Eps_i P).
+
+Axiom pred_ext : forall P Q:set -> prop, (forall x, P x <-> Q x) -> P = Q.
+Axiom prop_ext_2 : forall p q:prop, (p -> q) -> (q -> p) -> p = q.
+Axiom pred_ext_2 : forall P Q:set -> prop, P c= Q -> Q c= P -> P = Q.
+
+Axiom Subq_ref : forall X:set, X c= X.
+Axiom Subq_tra : forall X Y Z:set, X c= Y -> Y c= Z -> X c= Z.
+Axiom Subq_contra : forall X Y z:set, X c= Y -> z /:e Y -> z /:e X.
+
+Axiom EmptyE : forall x:set, x /:e Empty.
+Axiom Subq_Empty : forall X:set, Empty c= X.
+Axiom Empty_Subq_eq : forall X:set, X c= Empty -> X = Empty.
+Axiom Empty_eq : forall X:set, (forall x, x /:e X) -> X = Empty.
+
+Axiom UnionI : forall X x Y:set, x :e Y -> Y :e X -> x :e Union X.
+Axiom UnionE : forall X x:set, x :e Union X -> exists Y:set, x :e Y /\ Y :e X.
+Axiom UnionE_impred : forall X x:set, x :e Union X -> forall p:prop, (forall Y:set, x :e Y -> Y :e X -> p) -> p.
+
+Axiom Union_Empty : Union Empty = Empty.
+
+Axiom PowerI : forall X Y:set, Y c= X -> Y :e Power X.
+Axiom PowerE : forall X Y:set, Y :e Power X -> Y c= X.
+Axiom Power_Subq : forall X Y:set, X c= Y -> Power X c= Power Y.
+Axiom Empty_In_Power : forall X:set, Empty :e Power X.
+Axiom Self_In_Power : forall X:set, X :e Power X.
+
+Axiom Union_Power_Subq : forall X:set, Union (Power X) c= X.
+
+Axiom xm : forall P:prop, P \/ ~P.
+Axiom dneg : forall P:prop, ~~P -> P.
+Axiom imp_not_or : forall p q:prop, (p -> q) -> ~p \/ q.
+Axiom not_and_or_demorgan : forall p q:prop, ~(p /\ q) -> ~p \/ ~q.
+
+(* Parameter exactly1of2 "3578b0d6a7b318714bc5ea889c6a38cf27f08eaccfab7edbde3cb7a350cf2d9b" "163602f90de012a7426ee39176523ca58bc964ccde619b652cb448bd678f7e21" *)
+Parameter exactly1of2 : prop->prop->prop.
+
+Axiom exactly1of2_I1 : forall A B:prop, A -> ~B -> exactly1of2 A B.
+Axiom exactly1of2_I2 : forall A B:prop, ~A -> B -> exactly1of2 A B.
+Axiom exactly1of2_impI1 : forall A B:prop, (A -> ~B) -> (~A -> B) -> exactly1of2 A B.
+Axiom exactly1of2_impI2 : forall A B:prop, (B -> ~A) -> (~B -> A) -> exactly1of2 A B.
+
+Axiom exactly1of2_E : forall A B:prop, exactly1of2 A B ->
+forall p:prop,
+(A -> ~B -> p) ->
+(~A -> B -> p) ->
+p.
+
+Axiom exactly1of2_or : forall A B:prop, exactly1of2 A B -> A \/ B.
+Axiom exactly1of2_impn12 : forall A B:prop, exactly1of2 A B -> A -> ~B.
+Axiom exactly1of2_impn21 : forall A B:prop, exactly1of2 A B -> B -> ~A.
+Axiom exactly1of2_nimp12 : forall A B:prop, exactly1of2 A B -> ~A -> B.
+Axiom exactly1of2_nimp21 : forall A B:prop, exactly1of2 A B -> ~B -> A.
+
+(* Parameter exactly1of3 "d2a0e4530f6e4a8ef3d5fadfbb12229fa580c2add302f925c85ede027bb4b175" "aa4bcd059b9a4c99635877362627f7d5998ee755c58679934cc62913f8ef06e0" *)
+Parameter exactly1of3 : prop->prop->prop->prop.
+
+Axiom exactly1of3_I1 : forall A B C:prop, A -> ~B -> ~C -> exactly1of3 A B C.
+Axiom exactly1of3_I2 : forall A B C:prop, ~A -> B -> ~C -> exactly1of3 A B C.
+Axiom exactly1of3_I3 : forall A B C:prop, ~A -> ~B -> C -> exactly1of3 A B C.
+Axiom exactly1of3_impI1 : forall A B C:prop, (A -> ~B) -> (A -> ~C) -> (B -> ~C) -> (~A -> B \/ C) -> exactly1of3 A B C.
+Axiom exactly1of3_impI2 : forall A B C:prop, (B -> ~A) -> (B -> ~C) -> (A -> ~C) -> (~B -> A \/ C) -> exactly1of3 A B C.
+Axiom exactly1of3_impI3 : forall A B C:prop, (C -> ~A) -> (C -> ~B) -> (A -> ~B) -> (~A -> B) -> exactly1of3 A B C.
+
+Axiom exactly1of3_E : forall A B C:prop, exactly1of3 A B C ->
+forall p:prop,
+(A -> ~B -> ~C -> p) ->
+(~A -> B -> ~C -> p) ->
+(~A -> ~B -> C -> p) ->
+p.
+
+Axiom exactly1of3_or : forall A B C:prop, exactly1of3 A B C -> A \/ B \/ C.
+Axiom exactly1of3_impn12 : forall A B C:prop, exactly1of3 A B C -> A -> ~B.
+Axiom exactly1of3_impn13 : forall A B C:prop, exactly1of3 A B C -> A -> ~C.
+Axiom exactly1of3_impn21 : forall A B C:prop, exactly1of3 A B C -> B -> ~A.
+Axiom exactly1of3_impn23 : forall A B C:prop, exactly1of3 A B C -> B -> ~C.
+Axiom exactly1of3_impn31 : forall A B C:prop, exactly1of3 A B C -> C -> ~A.
+Axiom exactly1of3_impn32 : forall A B C:prop, exactly1of3 A B C -> C -> ~B.
+Axiom exactly1of3_nimp1 : forall A B C:prop, exactly1of3 A B C -> ~A -> B \/ C.
+Axiom exactly1of3_nimp2 : forall A B C:prop, exactly1of3 A B C -> ~B -> A \/ C.
+Axiom exactly1of3_nimp3 : forall A B C:prop, exactly1of3 A B C -> ~C -> A \/ B.
+
+Axiom ReplI : forall A:set, forall F:set->set, forall x:set, x :e A -> F x :e {F x|x :e A}.
+
+Axiom ReplE : forall A:set, forall F:set->set, forall y:set, y :e {F x|x :e A} -> exists x :e A, y = F x.
+Axiom ReplE_impred : forall A:set, forall F:set->set, forall y:set, y :e {F x|x :e A} -> forall p:prop, (forall x:set, x :e A -> y = F x -> p) -> p.
+Axiom ReplE' : forall X, forall f:set -> set, forall p:set -> prop, (forall x :e X, p (f x)) -> forall y :e {f x|x :e X}, p y.
+
+Axiom Repl_Empty : forall F:set -> set, {F x|x :e Empty} = Empty.
+
+Axiom ReplEq_ext_sub : forall X, forall F G:set -> set, (forall x :e X, F x = G x) -> {F x|x :e X} c= {G x|x :e X}.
+
+Axiom ReplEq_ext : forall X, forall F G:set -> set, (forall x :e X, F x = G x) -> {F x|x :e X} = {G x|x :e X}.
+
+(* Parameter If_i "8c8f550868df4fdc93407b979afa60092db4b1bb96087bc3c2f17fadf3f35cbf" "b8ff52f838d0ff97beb955ee0b26fad79602e1529f8a2854bda0ecd4193a8a3c" *)
+Parameter If_i : prop->set->set->set.
+
+Notation IfThenElse If_i.
+
+Axiom If_i_correct : forall p:prop, forall x y:set,
+p /\ (if p then x else y) = x \/ ~p /\ (if p then x else y) = y.
+
+Axiom If_i_0 : forall p:prop, forall x y:set,
+~ p -> (if p then x else y) = y.
+
+Axiom If_i_1 : forall p:prop, forall x y:set,
+p -> (if p then x else y) = x.
+
+Axiom If_i_or : forall p:prop, forall x y:set, (if p then x else y) = x \/ (if p then x else y) = y.
+
+Axiom If_i_eta : forall p:prop, forall x:set, (if p then x else x) = x.
+
+(* Parameter UPair "80aea0a41bb8a47c7340fe8af33487887119c29240a470e920d3f6642b91990d" "74243828e4e6c9c0b467551f19c2ddaebf843f72e2437cc2dea41d079a31107f" *)
+Parameter UPair : set->set->set.
+
+Notation SetEnum2 UPair.
+
+Axiom UPairE :
+forall x y z:set, x :e {y,z} -> x = y \/ x = z.
+
+Axiom UPairI1 : forall y z:set, y :e {y,z}.
+
+Axiom UPairI2 : forall y z:set, z :e {y,z}.
+
+Axiom UPair_com : forall x y:set, {x,y} = {y,x}.
+
+(* Parameter Sing "158bae29452f8cbf276df6f8db2be0a5d20290e15eca88ffe1e7b41d211d41d7" "bd01a809e97149be7e091bf7cbb44e0c2084c018911c24e159f585455d8e6bd0" *)
+Parameter Sing : set -> set.
+Notation SetEnum1 Sing.
+
+Axiom SingI : forall x:set, x :e {x}.
+Axiom SingE : forall x y:set, y :e {x} -> y = x.
+
+(* Parameter binunion "0a445311c45f0eb3ba2217c35ecb47f122b2301b2b80124922fbf03a5c4d223e" "5e1ac4ac93257583d0e9e17d6d048ff7c0d6ccc1a69875b2a505a2d4da305784" *)
+Parameter binunion : set -> set -> set.
+
+(* Unicode :\/: "222a" *)
+Infix :\/: 345 left := binunion.
+
+Axiom binunionI1 : forall X Y z:set, z :e X -> z :e X :\/: Y.
+
+Axiom binunionI2 : forall X Y z:set, z :e Y -> z :e X :\/: Y.
+
+Axiom binunionE : forall X Y z:set, z :e X :\/: Y -> z :e X \/ z :e Y.
+Axiom binunionE' : forall X Y z, forall p:prop, (z :e X -> p) -> (z :e Y -> p) -> (z :e X :\/: Y -> p).
+Axiom binunion_Repl_E' : forall X Y, forall f g:set -> set, forall p:set -> prop,
+ (forall x :e X, p (f x))
+ -> (forall y :e Y, p (g y))
+ -> (forall w :e {f x|x :e X} :\/: {g y|y :e Y}, p w).
+
+Definition SetAdjoin : set->set->set := fun X y => X :\/: {y}.
+
+Notation SetEnum Empty Sing UPair SetAdjoin.
+
+Axiom Power_0_Sing_0 : Power Empty = {Empty}.
+
+Axiom Repl_UPair : forall F:set->set, forall x y:set, {F z|z :e {x,y}} = {F x,F y}.
+
+Axiom Repl_Sing : forall F:set->set, forall x:set, {F z|z :e {x}} = {F x}.
+
+Axiom Repl_restr : forall X:set, forall F G:set -> set, (forall x:set, x :e X -> F x = G x) -> {F x|x :e X} = {G x|x :e X}.
+
+(* Parameter famunion "d772b0f5d472e1ef525c5f8bd11cf6a4faed2e76d4eacfa455f4d65cc24ec792" "b3e3bf86a58af5d468d398d3acad61ccc50261f43c856a68f8594967a06ec07a" *)
+Parameter famunion:set->(set->set)->set.
+
+(* Unicode \/_ "22C3" *)
+(* Subscript \/_ *)
+Binder \/_ , := famunion.
+
+Axiom famunionI:forall X:set, forall F:(set->set), forall x y:set, x :e X -> y :e F x -> y :e \/_ x :e X, F x.
+
+Axiom famunionE:forall X:set, forall F:(set->set), forall y:set, y :e (\/_ x :e X, F x) -> exists x :e X, y :e F x.
+
+Axiom famunionE_impred : forall X : set , forall F : (set -> set) , forall y : set , y :e (\/_ x :e X , F x) -> forall p:prop, (forall x, x :e X -> y :e F x -> p) -> p.
+
+Axiom UnionEq_famunionId:forall X:set, Union X = \/_ x :e X, x.
+
+Axiom ReplEq_famunion_Sing:forall X:set, forall F:(set->set), {F x|x :e X} = \/_ x :e X, {F x}.
+
+Axiom Power_Sing : forall x:set, Power {x} = {Empty,{x}}.
+Axiom Power_Sing_0 : Power {Empty} = {Empty,{Empty}}.
+
+(* Parameter Sep "f7e63d81e8f98ac9bc7864e0b01f93952ef3b0cbf9777abab27bcbd743b6b079" "f336a4ec8d55185095e45a638507748bac5384e04e0c48d008e4f6a9653e9c44" *)
+Parameter Sep: set -> (set -> prop) -> set.
+
+Notation Sep Sep.
+
+Axiom SepI:forall X:set, forall P:(set->prop), forall x:set, x :e X -> P x -> x :e {x :e X|P x}.
+Axiom SepE:forall X:set, forall P:(set->prop), forall x:set, x :e {x :e X|P x} -> x :e X /\ P x.
+Axiom SepE1:forall X:set, forall P:(set->prop), forall x:set, x :e {x :e X|P x} -> x :e X.
+Axiom SepE2:forall X:set, forall P:(set->prop), forall x:set, x :e {x :e X|P x} -> P x.
+
+Axiom Sep_Subq : forall X:set, forall P:set->prop, {x :e X|P x} c= X.
+
+Axiom Sep_In_Power : forall X:set, forall P:set->prop, {x :e X|P x} :e Power X.
+
+(* Parameter ReplSep "f627d20f1b21063483a5b96e4e2704bac09415a75fed6806a2587ce257f1f2fd" "ec807b205da3293041239ff9552e2912636525180ddecb3a2b285b91b53f70d8" *)
+Parameter ReplSep : set->(set->prop)->(set->set)->set.
+Notation ReplSep ReplSep.
+
+Axiom ReplSepI: forall X:set, forall P:set->prop, forall F:set->set, forall x:set, x :e X -> P x -> F x :e {F x|x :e X, P x}.
+
+Axiom ReplSepE:forall X:set, forall P:set->prop, forall F:set->set, forall y:set, y :e {F x|x :e X, P x} -> exists x:set, x :e X /\ P x /\ y = F x.
+
+Axiom ReplSepE_impred: forall X:set, forall P:set->prop, forall F:set->set, forall y:set, y :e {F x|x :e X, P x} -> forall p:prop, (forall x :e X, P x -> y = F x -> p) -> p.
+
+(* Parameter ReplSep2 "816cc62796568c2de8e31e57b826d72c2e70ee3394c00fbc921f2e41e996e83a" "da098a2dd3a59275101fdd49b6d2258642997171eac15c6b60570c638743e785" *)
+Parameter ReplSep2 : set -> (set -> set) -> (set -> set -> prop) -> (set -> set -> set) -> set.
+
+Axiom ReplSep2I : forall A, forall B:set -> set, forall P:set -> set -> prop, forall F:set -> set -> set, forall x :e A, forall y :e B x, P x y -> F x y :e ReplSep2 A B P F.
+
+Axiom ReplSep2E_impred : forall A, forall B:set -> set, forall P:set -> set -> prop, forall F:set -> set -> set, forall r :e ReplSep2 A B P F, forall p:prop, (forall x :e A, forall y :e B x, P x y -> r = F x y -> p) -> p.
+
+Axiom ReplSep2E : forall A, forall B:set -> set, forall P:set -> set -> prop, forall F:set -> set -> set, forall r :e ReplSep2 A B P F, exists x :e A, exists y :e B x, P x y /\ r = F x y.
+
+Axiom binunion_asso: forall X Y Z:set, X :\/: (Y :\/: Z) = (X :\/: Y) :\/: Z.
+Axiom binunion_com: forall X Y:set, X :\/: Y = Y :\/: X.
+Axiom binunion_idl: forall X:set, Empty :\/: X = X.
+Axiom binunion_idr: forall X:set, X :\/: Empty = X.
+Axiom binunion_idem: forall X:set, X :\/: X = X.
+Axiom binunion_Subq_1: forall X Y:set, X c= X :\/: Y.
+Axiom binunion_Subq_2: forall X Y:set, Y c= X :\/: Y.
+Axiom binunion_Subq_min: forall X Y Z:set, X c= Z -> Y c= Z -> X :\/: Y c= Z.
+Axiom Subq_binunion_eq:forall X Y, (X c= Y) = (X :\/: Y = Y).
+Axiom binunion_nIn_I : forall X Y z:set, z /:e X -> z /:e Y -> z /:e X :\/: Y.
+Axiom binunion_nIn_E : forall X Y z:set, z /:e X :\/: Y -> z /:e X /\ z /:e Y.
+
+(* Parameter binintersect "8cf6b1f490ef8eb37db39c526ab9d7c756e98b0eb12143156198f1956deb5036" "b2abd2e5215c0170efe42d2fa0fb8a62cdafe2c8fbd0d37ca14e3497e54ba729" *)
+Parameter binintersect:set->set->set.
+
+(* Unicode :/\: "2229" *)
+Infix :/\: 340 left := binintersect.
+
+Axiom binintersectI:forall X Y z, z :e X -> z :e Y -> z :e X :/\: Y.
+Axiom binintersectE:forall X Y z, z :e X :/\: Y -> z :e X /\ z :e Y.
+Axiom binintersectE1:forall X Y z, z :e X :/\: Y -> z :e X.
+Axiom binintersectE2:forall X Y z, z :e X :/\: Y -> z :e Y.
+Axiom binintersect_Subq_1:forall X Y:set, X :/\: Y c= X.
+Axiom binintersect_Subq_2:forall X Y:set, X :/\: Y c= Y.
+Axiom binintersect_Subq_eq_1 : forall X Y, X c= Y -> X :/\: Y = X.
+Axiom binintersect_Subq_max:forall X Y Z:set, Z c= X -> Z c= Y -> Z c= X :/\: Y.
+Axiom binintersect_asso:forall X Y Z:set, X :/\: (Y :/\: Z) = (X :/\: Y) :/\: Z.
+Axiom binintersect_com: forall X Y:set, X :/\: Y = Y :/\: X.
+Axiom binintersect_annil:forall X:set, Empty :/\: X = Empty.
+Axiom binintersect_annir:forall X:set, X :/\: Empty = Empty.
+Axiom binintersect_idem:forall X:set, X :/\: X = X.
+Axiom binintersect_binunion_distr:forall X Y Z:set, X :/\: (Y :\/: Z) = X :/\: Y :\/: X :/\: Z.
+Axiom binunion_binintersect_distr:forall X Y Z:set, X :\/: Y :/\: Z = (X :\/: Y) :/\: (X :\/: Z).
+Axiom Subq_binintersection_eq:forall X Y:set, (X c= Y) = (X :/\: Y = X).
+Axiom binintersect_nIn_I1 : forall X Y z:set, z /:e X -> z /:e X :/\: Y.
+Axiom binintersect_nIn_I2 : forall X Y z:set, z /:e Y -> z /:e X :/\: Y.
+Axiom binintersect_nIn_E : forall X Y z:set, z /:e X :/\: Y -> z /:e X \/ z /:e Y.
diff --git a/megalodon/scraps.tex b/megalodon/scraps.tex
new file mode 100644
index 0000000..f79a630
--- /dev/null
+++ b/megalodon/scraps.tex
@@ -0,0 +1,17 @@
+\begin{axiom}\label{trivial}
+ For all $x$ we have $x = x$.
+\end{axiom}
+
+% Axiom EmptyAx : ~exists x:set, x :e Empty.
+\begin{axiom}\label{empty_axiom}
+ There exists no set $x$ such that $x\in\emptyset$.
+\end{axiom}
+
+% UnionEq : forall X x, x :e Union X <-> exists Y, x :e Y /\ Y :e X.
+\begin{axiom}\label{union_eq}
+ For all $x, X$ we have $x\in\unions{X}$ iff there exists $Y$ such that $x\in Y\in X$.
+\end{axiom}
+
+\begin{theorem}\label{eq_refl}
+ For all $x$ we have $x=x$.
+\end{theorem}