Definition closed_mem m_a := forall x y, e x y -> in_mem x m_a = in_mem y m_a.Definition closure_mem m_a : pred T :=
  fun x => ~~ disjoint (mem (connect x)) m_a.Notation n_comp e a := (n_comp_mem e (mem a)).Notation closed e a := (closed_mem e (mem a)).Notation closure e a := (closure_mem e (mem a)).Notation fconnect f := (connect (coerced_frel f)).Notation froot f := (root (coerced_frel f)).Notation froots f := (roots (coerced_frel f)).Notation fcard_mem f := (n_comp_mem (coerced_frel f)).Notation fcard f a := (fcard_mem f (mem a)).Notation fclosed f a := (closed (coerced_frel f) a).Notation fclosure f a := (closure (coerced_frel f) a).Implicit Types (e : rel T) (a : pred T).

Lemma relU_sym e e' : connect_sym e -> connect_sym e' -> connect_sym (relU e e').