reserve T for non empty TopSpace,
  A, B for Subset of T,
  F, G for Subset-Family of T;

theorem
  for T being set, F being Subset-Family of T st F is compl-closed holds
  F = COMPLEMENT F
proof
  let T be set, F be Subset-Family of T;
  assume
A1: F is compl-closed;
  thus F c= COMPLEMENT F
  proof
    let x be object;
    assume
A2: x in F;
    then reconsider x9 = x as Subset of T;
    x9` in F by A1,A2;
    hence thesis by SETFAM_1:def 7;
  end;
  let x be object;
  assume
A3: x in COMPLEMENT F;
  then reconsider x9 = x as Subset of T;
  x9` in F by A3,SETFAM_1:def 7;
  then x9`` in F by A1;
  hence thesis;
end;
