Definition combine A B C f g₁ g₂ dom₁ dom₂ (m₁ : t A dom₁) (m₂ : t B dom₂) : t C (Dom.union dom₁ dom₂) :=
  exist (OK (Dom.union dom₁ dom₂)) (Scombine f g₁ g₂ (proj1_sig m₁) (proj1_sig m₂)) (combine_OK f g₁ g₂ m₁ m₂).

(* Specifications: *)
Lemma cast_OK : forall A dom₁ dom₂ (Heq : Dom.eq dom₁ dom₂) (m : S.t A), OK dom₁ m -> OK dom₂ m.