reserve i for object, I for set,
  f for Function,
  x, x1, x2, y, A, B, X, Y, Z for ManySortedSet of I;

theorem     :: ZFMISC_1:124
  A c= X & B c= Y implies [|A,Y|] (/\) [|X,B|] = [|A,B|]
proof
  assume that
A1: A c= X and
A2: B c= Y;
  now
    let i be object;
    assume
A3: i in I;
    then
A4: A.i c= X.i by A1;
A5: B.i c= Y.i by A2,A3;
    thus ([|A,Y|] (/\) [|X,B|]).i
       = [|A,Y|].i /\ [|X,B|].i by A3,PBOOLE:def 5
      .= [:A.i,Y.i:] /\ [|X,B|].i by A3,PBOOLE:def 16
      .= [:A.i,Y.i:] /\ [:X.i,B.i:] by A3,PBOOLE:def 16
      .= [:A.i,B.i:] by A4,A5,ZFMISC_1:101
      .= [|A,B|].i by A3,PBOOLE:def 16;
  end;
  hence thesis;
end;
