Library GeoCoq.Elements.OriginalProofs.lemma_PGsymmetric

Require Export GeoCoq.Elements.OriginalProofs.lemma_parallelsymmetric.
Require Export GeoCoq.Elements.OriginalProofs.lemma_parallelflip.

Section Euclid.

Context `{Ax1:euclidean_neutral}.

Lemma lemma_PGsymmetric :
   ∀ A B C D,
   PG A B C D →
   PG C D A B.
Proof.
intros.
assert ((Par A B C D ∧ Par A D B C)) by (conclude_def PG ).
assert (Par C D A B) by (conclude lemma_parallelsymmetric).
assert (Par B C A D) by (conclude lemma_parallelsymmetric).
assert (Par C B D A) by (forward_using lemma_parallelflip).
assert (PG C D A B) by (conclude_def PG ).
close.
Qed.

End Euclid.