reserve x, y for object, I for set,
  A, B, X, Y for ManySortedSet of I;

theorem
  X c= A (\) B iff X c= A & X misses B
proof
  thus X c= A (\) B implies X c= A & X misses B
  proof
    assume X c= A (\) B;
    then
A1: X in bool (A (\) B) by Th1;
    thus X c= A
    proof
      let i be object;
      assume
A2:   i in I;
      then X.i in (bool (A (\) B)).i by A1;
      then X.i in bool (A.i \ B.i) by A2,Lm4;
      hence thesis by XBOOLE_1:106;
    end;
    let i be object;
    assume
A3: i in I;
    then X.i in (bool (A (\) B)).i by A1;
    then X.i in bool (A.i \ B.i) by A3,Lm4;
    hence thesis by XBOOLE_1:106;
  end;
  assume
A4: X c= A & X misses B;
  let i be object;
  assume
A5: i in I;
  then X.i c= A.i & X.i misses B.i by A4;
  then X.i c= A.i \ B.i by XBOOLE_1:86;
  hence thesis by A5,PBOOLE:def 6;
end;
