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

theorem Th14:
  for F being Subset-Family of T holds F is all-closed-containing
  compl-closed iff F is all-open-containing compl-closed
proof
  let F be Subset-Family of T;
  hereby
    assume
A1: F is all-closed-containing compl-closed;
    for A being Subset of T st A is open holds A in F
    proof
      let A be Subset of T;
      assume A is open;
      then A` in F by A1;
      then A`` in F by A1;
      hence thesis;
    end;
    hence F is all-open-containing compl-closed by A1;
  end;
  assume
A2: F is all-open-containing compl-closed;
  for A being Subset of T st A is closed holds A in F
  proof
    let A be Subset of T;
    assume A is closed;
    then A` in F by A2;
    then A`` in F by A2;
    hence thesis;
  end;
  hence thesis by A2;
end;
