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

theorem Th67:
  for F being Subset-Family of T holds F is domains-family implies
  (union F) \/ (Int Cl(union F)) is condensed
proof
  let F be Subset-Family of T;
A1: Cl Int Cl(union F) c= Cl Cl(union F) by PRE_TOPC:19,TOPS_1:16;
  assume F is domains-family;
  then union F c= Cl Int(union F) by Th65;
  then
A2: (union F) \/ (Int Cl(union F)) c= Cl Int((union F) \/ (Int Cl(union F))
  ) by Th5;
  Int Cl((union F) \/ (Int Cl(union F))) = Int(Cl(union F) \/ Cl(Int Cl(
  union F))) by PRE_TOPC:20
    .= Int(Cl(union F)) by A1,XBOOLE_1:12;
  then
  Int Cl((union F) \/ (Int Cl(union F))) c= (union F) \/ (Int Cl(union F))
  by XBOOLE_1:7;
  hence thesis by A2,TOPS_1:def 6;
end;
