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

theorem :: ZFMISC_1:87
  X c= A (\+\) B iff X c= A (\/) B & X misses A (/\) B
proof
  thus X c= A (\+\) B implies X c= A (\/) B & X misses A (/\) B
  proof
    assume X c= A (\+\) B;
    then
A1: X in bool (A (\+\) B) by Th1;
    thus X c= A (\/) B
    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,Lm5;
      then X.i c= A.i \/ B.i by XBOOLE_1:107;
      hence thesis by A2,PBOOLE:def 4;
    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,Lm5;
    then X.i misses A.i /\ B.i by XBOOLE_1:107;
    hence thesis by A3,PBOOLE:def 5;
  end;
  assume that
A4: X c= A (\/) B and
A5: X misses A (/\) B;
  let i be object;
  assume
A6: i in I;
  then X.i misses (A (/\) B).i by A5;
  then
A7: X.i misses A.i /\ B.i by A6,PBOOLE:def 5;
  X.i c= (A (\/) B).i by A4,A6;
  then X.i c= A.i \/ B.i by A6,PBOOLE:def 4;
  then X.i c= A.i \+\ B.i by A7,XBOOLE_1:107;
  hence thesis by A6,PBOOLE:4;
end;
