Library GeoCoq.Elements.OriginalProofs.lemma_3_7a

Require Export GeoCoq.Elements.OriginalProofs.lemma_extensionunique.

Section Euclid.

Context `{Ax:euclidean_neutral}.

Lemma lemma_3_7a :
   ∀ A B C D,
   BetS A B C → BetS B C D →
   BetS A C D.
Proof.
intros.
assert (neq A C) by (forward_using lemma_betweennotequal).
assert (neq C D) by (forward_using lemma_betweennotequal).
let Tf:=fresh in
assert (Tf:∃ E, (BetS A C E ∧ Cong C E C D)) by (conclude postulate_extension);destruct Tf as [E];spliter.
assert (Cong C E C E) by (conclude cn_congruencereflexive).
assert (Cong C D C E) by (conclude lemma_congruencesymmetric).
assert (BetS B C E) by (conclude lemma_3_6a).
assert (neq B C) by (forward_using lemma_betweennotequal).
assert (eq D E) by (conclude lemma_extensionunique).
assert (BetS A C D) by (conclude cn_equalitysub).
close.
Qed.

End Euclid.