From 44d4c1c50ba6e0f12a2f4fdd204b315a15e434db Mon Sep 17 00:00:00 2001
From: Simon-Kor <52245124+Simon-Kor@users.noreply.github.com>
Date: Tue, 25 Jun 2024 00:02:44 +0200
Subject: Improvement for the ATP proof time

---
 library/set.tex            | 3 +++
 library/topology/basis.tex | 2 +-
 2 files changed, 4 insertions(+), 1 deletion(-)

(limited to 'library')

diff --git a/library/set.tex b/library/set.tex
index 2fd18ea..69b9526 100644
--- a/library/set.tex
+++ b/library/set.tex
@@ -654,6 +654,9 @@ The $\operatorname{\textsf{cons}}$ operation is determined by the following axio
     Suppose $(A\inter B)\union C = A\inter (B\union C)$.
     Then $C\subseteq A$.
 \end{proposition}
+\begin{proof}
+    Follows by \cref{union_upper_right,union_upper_left,subseteq_union_iff,subseteq_antisymmetric,subseteq_inter_iff}.
+\end{proof}
 
 % From Isabelle/ZF equalities theory
 \begin{proposition}\label{union_inter_crazy}
diff --git a/library/topology/basis.tex b/library/topology/basis.tex
index bd7d15a..052c551 100644
--- a/library/topology/basis.tex
+++ b/library/topology/basis.tex
@@ -64,7 +64,7 @@
     Then $\unions{F}\in\genOpens{B}{X}$.
 \end{lemma}
 \begin{proof}
-    We have $\unions{F} \in \pow{X}$.
+    We have $\unions{F} \in \pow{X}$ by \cref{genopens,subseteq,pow_iff,unions_family,powerset_elim}.
 
     Show for all $x\in \unions{F}$ there exists $W \in B$
     such that $x\in W$ and $W \subseteq \unions{F}$.
-- 
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