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

theorem
  for F being Subset-Family of T holds F is domains-family implies Cl F
  is closed-domains-family
proof
  let F be Subset-Family of T;
  assume
A1: F is domains-family;
  for A being Subset of T holds A in Cl F implies A is closed_condensed
  proof
    let A be Subset of T;
    assume A in Cl F;
    then consider P being Subset of T such that
A2: A = Cl P and
A3: P in F by PCOMPS_1:def 2;
    reconsider P as Subset of T;
    P is condensed by A1,A3;
    hence thesis by A2,TDLAT_1:24;
  end;
  hence thesis;
end;
