fof(pair_emptyset_in_times_unit,conjecture,elem(pair(emptyset,emptyset),times(unit,unit))).
fof(unit,axiom,![Xany]:(elem(Xany,unit)<=>(elem(Xany,emptyset)|Xany=emptyset))).
fof(times,axiom,![XA,XB]:![Xfrv]:(elem(Xfrv,times(XA,XB))<=>?[Xa,Xb]:(elem(Xa,XA)&elem(Xb,XB)&pair(Xa,Xb)=Xfrv))).
fof(cons,axiom,![Xx,Xy,XX]:(elem(Xx,cons(Xy,XX))<=>(Xx=Xy|elem(Xx,XX)))).
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fof(suc,conjecture,![X0]:(elem(X0,fa)=>![X1,X2]:((X1=cons(X0,emptyset)&X2=cons(X0,emptyset))=>X1=X2))).
fof(pair_emptyset_in_times_unit,axiom,elem(pair(emptyset,emptyset),times(unit,unit))).
fof(unit,axiom,![Xany]:(elem(Xany,unit)<=>(elem(Xany,emptyset)|Xany=emptyset))).
fof(times,axiom,![XA,XB]:![Xfrv]:(elem(Xfrv,times(XA,XB))<=>?[Xa,Xb]:(elem(Xa,XA)&elem(Xb,XB)&pair(Xa,Xb)=Xfrv))).
fof(cons,axiom,![Xx,Xy,XX]:(elem(Xx,cons(Xy,XX))<=>(Xx=Xy|elem(Xx,XX)))).
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fof(times_replacement_test,conjecture,![Xfrv]:(elem(Xfrv,times(fA,fB))<=>?[Xa,Xb]:(elem(Xa,fA)&elem(Xb,fB)&pair(Xa,Xb)=Xfrv))).
fof(suc,axiom,![X1,Xa]:(elem(X1,suc(Xa))<=>?[X0]:(elem(X0,Xa)&X1=cons(X0,emptyset)))).
fof(pair_emptyset_in_times_unit,axiom,elem(pair(emptyset,emptyset),times(unit,unit))).
fof(unit,axiom,![Xany]:(elem(Xany,unit)<=>(elem(Xany,emptyset)|Xany=emptyset))).
fof(times,axiom,![XA,XB]:![Xfrv]:(elem(Xfrv,times(XA,XB))<=>?[Xa,Xb]:(elem(Xa,XA)&elem(Xb,XB)&pair(Xa,Xb)=Xfrv))).
fof(cons,axiom,![Xx,Xy,XX]:(elem(Xx,cons(Xy,XX))<=>(Xx=Xy|elem(Xx,XX)))).