reserve T for TopSpace;
reserve T for non empty TopSpace;
reserve F for Subset-Family of T;
reserve T for non empty TopSpace;

theorem Th65:
  for F being Subset-Family of T holds F is domains-family implies
  union F c= Cl Int(union F) & Cl(union F) = Cl Int Cl(union F)
proof
  let F be Subset-Family of T;
  assume
A1: F is domains-family;
  now
    let A be Subset of T;
    reconsider B = A as Subset of T;
    assume A in F;
    then B is condensed by A1;
    hence A c= Cl Int A by TOPS_1:def 6;
  end;
  hence thesis by Th56;
end;
