Lemma struct_Prop_ext: forall m {A0 A1} (P0: forall it, A0 it -> Prop) (P1: forall it, A1 it -> Prop) v0 v1, members_no_replicate m = true -> (forall i d0 d1, in_members i m -> P0 _ (proj_struct i m v0 d0) = P1 _ (proj_struct i m v1 d1)) -> struct_Prop m P0 v0 = struct_Prop m P1 v1.