
theorem Th3:
  for T being 1-sorted, F being Subset-Family of T holds
  COMPLEMENT F = {a` where a is Subset of T: a in F}
proof
  let T be 1-sorted, F be Subset-Family of T;
  set X = {a` where a is Subset of T: a in F};
  hereby
    let x be object;
    assume
A1: x in COMPLEMENT F;
    then reconsider P = x as Subset of T;
A2: P` in F by A1,SETFAM_1:def 7;
    P`` = P;
    hence x in X by A2;
  end;
  let x be object;
  assume x in X;
  then ex P being Subset of T st x = P` & P in F;
  hence thesis by YELLOW_8:5;
end;
