(* egalite' *)
Definition Equal_PA_mor (x y : PA_ob) (f g : PA_mor x y) :=
match f in (PA_mor x' y') return Prop with
| PA_I1 =>
(* PA_I1  *)  True (* PA_I2  *)
| PA_I2 => True
(* PA_f i *)
| PA_f i =>
match g in (PA_mor x' y') return Prop with
| PA_I1 =>
(* PA_I1  *)  False (* PA_I2  *)
| PA_I2 => False
(* PA_f j *)
| PA_f j => k i =_S k j
end
end.

Lemma Equal_PA_mor_Equiv :
forall x y : PA_ob, Equivalence (Equal_PA_mor (x:=x) (y:=y)).