Set Implicit Arguments.
Unset Strict Implicit.
Parameter V : Type.
Parameter AV : Type.
Parameter cons : V -> V -> AV.
Parameter R : AV -> AV -> Prop.
Parameter opp : V -> V.
Parameter zero : AV.
Parameter pi : AV.
Parameter plus : AV -> AV -> AV.
intros u v; try assumption.
apply (transitive (a:=plus (cons u v) (cons v u)) (b:=cons u u) (c:=zero));
auto.
exact (Chasles u v u).
apply angle_nul.
Qed.
Axiom
compatible : forall a b c d : AV, R a b -> R c d -> R (plus a c) (plus b d).
Parameter vR : V -> V -> Prop.
Axiom
vR_R_compatible :
forall u u' v v' : V, vR u u' -> vR v v' -> R (cons u v) (cons u' v').
Parameter PO : Type.
Parameter vec : PO -> PO -> V.
intros u; try assumption.
elim non_vide_P; intros A H; clear H.
elim v_vec with (u := u) (A := A); intros B H; try exact H.
apply v_trans with (opp (opp (vec A B))).
apply opp_compatible; apply opp_compatible.
try exact H.
apply v_trans with (opp (vec B A)).
apply opp_compatible.
apply v_trans with (vec B A).
apply opp_vect.
apply v_refl.
apply v_trans with (vec A B).
apply opp_vect.
apply v_sym; auto.
Qed.
Hint Resolve opp_opp opp_compatible v_refl v_sym.
intros u; try assumption.
apply transitive with (cons (opp u) (opp (opp u))).
apply vR_R_compatible.
apply v_refl.
apply v_sym; apply opp_opp.
apply angle_plat.
Qed.
elim non_vide_V.
intros u H; try assumption.
apply transitive with (plus (cons u (opp u)) (cons (opp u) u)).
apply compatible.
apply symetrique; auto.
apply angle_plat.
apply symetrique; auto.
apply oppu_u.
apply transitive with (cons u u).
apply Chasles.
auto.
apply angle_nul.
Qed.
intros u v; try assumption.
apply transitive with (plus (cons u v) (cons v (opp v))).
apply symetrique; auto.
apply Chasles.
apply compatible.
apply reflexive.
apply angle_plat.
Qed.
intros u v; try assumption.
apply transitive with (plus (cons (opp u) u) (cons u v)).
apply symetrique; auto.
apply Chasles.
apply compatible.
apply oppu_u.
apply reflexive.
Qed.

Lemma Chasles_2 :
forall u v w t : V,
R (cons u t) (plus (cons u v) (plus (cons v w) (cons w t))).