theorem
  A1` = { x1 : not x1 in A1 }
proof
  thus A1` c= { x1 : not x1 in A1 }
  proof
    let a be object;
    assume
A1: a in A1`;
    then not a in A1 by XBOOLE_0:def 5;
    hence thesis by A1;
  end;
  let a be object;
  assume a in { x1 : not x1 in A1 };
  then ex x1 st a = x1 & not x1 in A1;
  hence thesis by SUBSET_1:29;
end;
