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_1:120
  [|X (\/) Y, Z|] = [|X, Z|] (\/) [|Y, Z|] &
  [|Z, X (\/) Y|] = [|Z, X|] (\/) [|Z, Y|]
proof
  now
    let i be object;
    assume
A1: i in I;
    hence [|X (\/) Y, Z|].i = [:(X (\/) Y).i, Z.i:] by PBOOLE:def 16
      .= [:X.i \/ Y.i, Z.i:] by A1,PBOOLE:def 4
      .= [:X.i, Z.i:] \/ [:Y.i, Z.i:] by ZFMISC_1:97
      .= [|X,Z|].i \/ [:Y.i, Z.i:] by A1,PBOOLE:def 16
      .= [|X,Z|].i \/ [|Y,Z|].i by A1,PBOOLE:def 16
      .= ([|X, Z|] (\/) [|Y, Z|]).i by A1,PBOOLE:def 4;
  end;
  hence [|X (\/) Y, Z|] = [|X, Z|] (\/) [|Y, Z|];
  now
    let i be object;
    assume
A2: i in I;
    hence [|Z, X (\/) Y|].i = [:Z.i, (X (\/) Y).i:] by PBOOLE:def 16
      .= [:Z.i, X.i \/ Y.i:] by A2,PBOOLE:def 4
      .= [:Z.i,X.i:] \/ [:Z.i,Y.i:] by ZFMISC_1:97
      .= [|Z,X|].i \/ [:Z.i,Y.i:] by A2,PBOOLE:def 16
      .= [|Z,X|].i \/ [|Z,Y|].i by A2,PBOOLE:def 16
      .= ([|Z,X|] (\/) [|Z,Y|]).i by A2,PBOOLE:def 4;
  end;
  hence thesis;
end;
