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

theorem :: ZFMISC_1:98
  for B be ManySortedSet of I for A be non-empty ManySortedSet of I
   st for X be ManySortedSet of I st X in A holds X (/\) B = EmptyMS I
 holds union(A) (/\) B = EmptyMS I
proof
  let B be ManySortedSet of I, A be non-empty ManySortedSet of I;
  assume
A1: for X be ManySortedSet of I st X in A holds X (/\) B = EmptyMS I;
  now
    let i be object such that
A2: i in I;
    for X9 be set st X9 in A.i holds X9 misses (B.i)
    proof
      consider M be ManySortedSet of I such that
A3:   M in A by PBOOLE:134;
      let X9 be set such that
A4:   X9 in A.i;
      dom (M +* (i .--> X9)) = I by A2,Lm1;
      then reconsider K = M +* (i .--> X9) as ManySortedSet of I by
PARTFUN1:def 2,RELAT_1:def 18;
A5:   dom (i .--> X9) = {i};
      i in {i} by TARSKI:def 1;
      then
A6:   K.i = (i .--> X9).i by A5,FUNCT_4:13
        .= X9 by FUNCOP_1:72;
      K in A
      proof
        let j be object such that
A7:     j in I;
        now
          per cases;
          case
            j = i;
            hence thesis by A4,A6;
          end;
          case
            j <> i;
            then not j in dom (i .--> X9) by TARSKI:def 1;
            then K.j = M.j by FUNCT_4:11;
            hence thesis by A3,A7;
          end;
        end;
        hence thesis;
      end;
      then K (/\) B = EmptyMS I by A1;
      then K.i /\ B.i = EmptyMS I.i by A2,PBOOLE:def 5;
      then X9 /\ B.i = {} by A6,PBOOLE:5;
      hence thesis by XBOOLE_0:def 7;
    end;
    then union (A.i) misses (B.i) by ZFMISC_1:80;
    then union (A.i) /\ (B.i) = {} by XBOOLE_0:def 7;
    then (union A).i /\ B.i = {} by A2,Def2;
    then (union A (/\) B).i = {} by A2,PBOOLE:def 5;
    hence (union(A) (/\) B).i = EmptyMS I.i by PBOOLE:5;
  end;
  hence thesis;
end;
