Section fun2_to_map2.
Variable A B C : Setoid.
Definition Map2 := Map A (B ==> C).
Definition Map2_alt := Map (SPROD A B) C.
Variable f : A -> B -> C.
Definition Map2_congl_law :=
forall (b1 b2 : B) (a : A), b1 =_S b2 -> f a b1 =_S f a b2.
Definition Map2_congr_law :=
forall (a1 a2 : A) (b : B), a1 =_S a2 -> f a1 b =_S f a2 b.
Definition Map2_cong_law :=
forall (a1 a2 : A) (b1 b2 : B),
a1 =_S a2 -> b1 =_S b2 -> f a1 b1 =_S f a2 b2.
Hypothesis pgcl : Map2_congl_law.
Hypothesis pgcr : Map2_congr_law.

Lemma Map2_map_law1 : forall a : A, Map_law (f a).