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

theorem Th69:
  for F being Subset-Family of T holds F is domains-family implies
  (meet F) /\ (Cl Int(meet F)) is condensed
proof
  let F be Subset-Family of T;
A1: Int Int (meet F) c= Int Cl Int(meet F) by PRE_TOPC:18,TOPS_1:19;
  assume F is domains-family;
  then Int Cl(meet F) c= meet F by Th66;
  then
A2: Int Cl((meet F) /\ (Cl Int(meet F))) c= (meet F) /\ (Cl Int(meet F)) by Th6
;
  Cl Int((meet F) /\ (Cl Int(meet F))) = Cl(Int(meet F) /\ Int(Cl Int(meet
  F))) by TOPS_1:17
    .= Cl(Int(meet F)) by A1,XBOOLE_1:28;
  then
  (meet F) /\ (Cl Int(meet F)) c= Cl Int((meet F) /\ (Cl Int(meet F))) by
XBOOLE_1:17;
  hence thesis by A2,TOPS_1:def 6;
end;
