Library GeoCoq.Elements.OriginalProofs.lemma_parallelsymmetric

Require Export GeoCoq.Elements.OriginalProofs.euclidean_tactics.

Section Euclid.

Context `{Ax1:euclidean_neutral}.

Lemma lemma_parallelsymmetric :
   ∀ A B C D,
   Par A B C D →
   Par C D A B.
Proof.
intros.
let Tf:=fresh in
assert (Tf:∃ a b c d m, (neq A B ∧ neq C D ∧ Col A B a ∧ Col A B b ∧ neq a b ∧ Col C D c ∧ Col C D d ∧ neq c d ∧ ¬ Meet A B C D ∧ BetS a m d ∧ BetS c m b)) by (conclude_def Par );destruct Tf as [a[b[c[d[m]]]]];spliter.
assert (¬ Meet C D A B).
 {
 intro.
 let Tf:=fresh in
 assert (Tf:∃ P, (neq C D ∧ neq A B ∧ Col C D P ∧ Col A B P)) by (conclude_def Meet );destruct Tf as [P];spliter.
 assert (Meet A B C D) by (conclude_def Meet ).
 contradict.
 }
assert (Par C D A B) by (conclude_def Par ).
close.
Qed.

End Euclid.