split; [ try assumption | idtac ].
red in |- *.
intros x; red in |- *.
elim x.
unfold cart_eq in |- *; simpl in |- *; auto with algebra.
red in |- *.
split; [ idtac | try assumption ].
red in |- *.
unfold app_rel, cart_eq in |- *.
intros x y z H' H'0; split; [ try assumption | idtac ].
apply Trans with (cart_l y); intuition.
apply Trans with (cart_r y); intuition.
red in |- *.
unfold app_rel, cart_eq in |- *.
intuition.
Qed.
Definition cart : Setoid := Build_Setoid cart_eq_equiv.
Comments "We will denote the cartesian product of" E "and" F "with"
(cart E F).
End Def.
Section Projections.
Variable E F : Setoid.
Definition proj1 (x : cart E F) : E := cart_l x.
Definition proj2 (x : cart E F) : F := cart_r x.
Comments "We note" (proj1 x) "and" (proj2 x) "the components of a couple" x
"in " (cart E F).

Lemma proj1_comp :
forall x y : cart E F, Equal x y -> Equal (proj1 x) (proj1 y).