# Library GeoCoq.Elements.OriginalProofs.lemma_raystrict

Require Export GeoCoq.Elements.OriginalProofs.lemma_betweennotequal.

Section Euclid.

Context `{Ax1:euclidean_neutral}.

Lemma lemma_raystrict :

∀ A B C,

Out A B C →

neq A C.

Proof.

intros.

let Tf:=fresh in

assert (Tf:∃ J, (BetS J A C ∧ BetS J A B)) by (conclude_def Out );destruct Tf as [J];spliter.

assert (neq A C) by (forward_using lemma_betweennotequal).

close.

Qed.

End Euclid.

Section Euclid.

Context `{Ax1:euclidean_neutral}.

Lemma lemma_raystrict :

∀ A B C,

Out A B C →

neq A C.

Proof.

intros.

let Tf:=fresh in

assert (Tf:∃ J, (BetS J A C ∧ BetS J A B)) by (conclude_def Out );destruct Tf as [J];spliter.

assert (neq A C) by (forward_using lemma_betweennotequal).

close.

Qed.

End Euclid.