
theorem Th6:
  for X being 1-sorted, F being Subset-Family of X holds Intersect
  COMPLEMENT F = (union F)`
proof
  let X be 1-sorted, F be Subset-Family of X;
  per cases;
  suppose
A1: F <> {};
    then COMPLEMENT F <> {} by SETFAM_1:32;
    hence Intersect COMPLEMENT F = meet COMPLEMENT F by SETFAM_1:def 9
      .= ([#] the carrier of X) \ union F by A1,SETFAM_1:33
      .= (union F)`;
  end;
  suppose
    F = {};
    then reconsider G = F as empty Subset-Family of X;
    COMPLEMENT G = {};
    hence thesis by SETFAM_1:def 9,ZFMISC_1:2;
  end;
end;
