proposition: Lemma existence_similitude_Ika : forall (I M : PO) (k a : R), k > 0 -> exists M' : PO, M' = similitude I k a M.
proof: Proof. intros. elim existence_rotation_Ia with (I := I) (M := M) (a := a); intros M1; intros. elim existence_homothetique with (k := k) (I := I) (A := M1); intros M' H1; try clear existence_homothetique; try exact H1. exists M'. apply composee_rotation_homothetie_pos with M1; auto. Qed.