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_1:126
  [|x1,x2|] (\) [|A,B|] = [|x1(\)A,x2|] (\/) [|x1,x2(\)B|]
proof
  now
    let i be object;
    assume
A1: i in I;
    hence ([|x1,x2|] (\) [|A,B|]).i = [|x1,x2|].i \ [|A,B|].i by PBOOLE:def 6
      .= [:x1.i,x2.i:] \ [|A,B|].i by A1,PBOOLE:def 16
      .= [:x1.i,x2.i:] \ [:A.i,B.i:] by A1,PBOOLE:def 16
      .= [:x1.i\A.i,x2.i:] \/ [:x1.i,x2.i\B.i:] by ZFMISC_1:103
      .= [:(x1(\)A).i,x2.i:] \/ [:x1.i,x2.i\B.i:] by A1,PBOOLE:def 6
      .= [:(x1(\)A).i,x2.i:] \/ [:x1.i,(x2(\)B).i:] by A1,PBOOLE:def 6
      .= [|x1(\)A,x2|].i \/ [:x1.i,(x2(\)B).i:] by A1,PBOOLE:def 16
      .= [|x1(\)A,x2|].i \/ [|x1,x2(\)B|].i by A1,PBOOLE:def 16
      .= ([|x1(\)A,x2|] (\/) [|x1,x2(\)B|]).i by A1,PBOOLE:def 4;
  end;
  hence thesis;
end;
