Definition compl : part_set E -> part_set E.

(* Specifications: *)
Lemma compl_in : forall (A : part_set E) (x : E), ~ in_part x A -> in_part x (compl A).
Lemma in_compl : forall (A : part_set E) (x : E), in_part x (compl A) -> ~ in_part x A.
Lemma compl_comp : forall A B : part_set E, Equal A B -> Equal (compl A) (compl B).
Lemma compl_comp_rev : forall A B : part_set E, Equal (compl A) (compl B) -> Equal A B.
Lemma compl_compl : forall A : part_set E, Equal (compl (compl A)) A.
Lemma compl_not_in : forall (A : part_set E) (x : E), in_part x A -> ~ in_part x (compl A).
Lemma not_in_compl : forall (A : part_set E) (x : E), in_part x (compl A) -> ~ in_part x A.
Lemma compl_included : forall A B : part_set E, included A B -> included (compl B) (compl A).
Lemma compl_not_compl : forall (A : part_set E) (x : E), in_part x A \/ in_part x (compl A).