intros H' x y H'0 H'1 H'2; try assumption.
cut (Same_set U (Approximants U D x) (Approximants U D y)).
intro H'3; elim H'3.
generalize (Rel_of_antisymmetric U D); auto with sets.
rewrite H'2; auto with sets.
Qed.
Hint Resolve Approximants_define_element.
Hint Unfold not.

(* Specifications: *)
Theorem Corollary1_1 :
Domain U D ->
forall x y : U,
Compact U D x ->
In U (Carrier_of U D) y ->
Strict_Rel_of U D x y ->
exists z : U, In U (Approximants U D y) z /\ Strict_Rel_of U D x z.
Theorem Corollary1_2 :
Domain U D ->
forall x : U,
In U (Carrier_of U D) x ->
~ Compact U D x -> ~ Finite U (Approximants U D x).