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_1:123
  [|x1 (/\) x2, A (/\) B|] = [|x1,A|] (/\) [|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 16
      .= [:x1.i /\ x2.i,(A (/\) B).i:] by A1,PBOOLE:def 5
      .= [:x1.i /\ x2.i,A.i /\ B.i:] by A1,PBOOLE:def 5
      .= [:x1.i,A.i:] /\ [:x2.i,B.i:] by ZFMISC_1:100
      .= [|x1,A|].i /\ [:x2.i,B.i:] by A1,PBOOLE:def 16
      .= [|x1,A|].i /\ [|x2,B|].i by A1,PBOOLE:def 16
      .= ([|x1,A|] (/\) [|x2, B|]).i by A1,PBOOLE:def 5;
  end;
  hence thesis;
end;
