Library GeoCoq.Elements.OriginalProofs.lemma_rightreverse

Require Export GeoCoq.Elements.OriginalProofs.lemma_congruenceflip.
Require Export GeoCoq.Elements.OriginalProofs.lemma_extensionunique.

Section Euclid.

Context `{Ax:euclidean_neutral}.

Lemma lemma_rightreverse :
   ∀ A B C D,
   Per A B C → BetS A B D → Cong A B B D →
   Cong A C D C.
Proof.
intros.
let Tf:=fresh in
assert (Tf:∃ E, (BetS A B E ∧ Cong A B E B ∧ Cong A C E C ∧ neq B C)) by (conclude_def Per );destruct Tf as [E];spliter.
assert (neq A B) by (forward_using lemma_betweennotequal).
assert (Cong B D A B) by (conclude lemma_congruencesymmetric).
assert (Cong B D E B) by (conclude lemma_congruencetransitive).
assert (Cong B D B E) by (forward_using lemma_congruenceflip).
assert (eq D E) by (conclude lemma_extensionunique).
assert (Cong A C D C) by (conclude cn_equalitysub).
close.
Qed.

End Euclid.