Library GeoCoq.Elements.OriginalProofs.lemma_layoff

Require Export GeoCoq.Elements.OriginalProofs.lemma_equalitysymmetric.
Require Export GeoCoq.Elements.OriginalProofs.lemma_betweennotequal.

Section Euclid.

Context `{Ax:euclidean_neutral}.

Lemma lemma_layoff :
   ∀ A B C D,
   neq A B → neq C D →
   ∃ X, Out A B X ∧ Cong A X C D.
Proof.
intros.
assert (¬ eq B A).
 {
 intro.
 assert (eq A B) by (conclude lemma_equalitysymmetric).
 contradict.
 }
let Tf:=fresh in
assert (Tf:∃ E, (BetS B A E ∧ Cong A E C D)) by (conclude postulate_extension);destruct Tf as [E];spliter.
assert (BetS E A B) by (conclude axiom_betweennesssymmetry).
assert (neq E A) by (forward_using lemma_betweennotequal).
assert (BetS E A B) by (conclude axiom_betweennesssymmetry).
let Tf:=fresh in
assert (Tf:∃ P, (BetS E A P ∧ Cong A P C D)) by (conclude postulate_extension);destruct Tf as [P];spliter.
assert (Out A B P) by (conclude_def Out ).
close.
Qed.

End Euclid.