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_1:134
  x1 is non-empty & A is non-empty & [|x1,A|] = [|x2,B|]
  implies x1 = x2 & A = B
proof
  assume that
A1: x1 is non-empty and
A2: A is non-empty and
A3: [|x1,A|] = [|x2,B|];
  now
    let i be object;
    assume
A4: i in I;
    then
A5: x1.i is non empty by A1;
A6: A.i is non empty by A2,A4;
    [:x1.i,A.i:] = [|x1,A|].i by A4,PBOOLE:def 16
      .= [:x2.i,B.i:] by A3,A4,PBOOLE:def 16;
    hence x1.i = x2.i by A5,A6,ZFMISC_1:110;
  end;
  hence x1 = x2;
  now
    let i be object;
    assume
A7: i in I;
    then
A8: x1.i is non empty by A1;
A9: A.i is non empty by A2,A7;
    [:x1.i,A.i:] = [|x1,A|].i by A7,PBOOLE:def 16
      .= [:x2.i,B.i:] by A3,A7,PBOOLE:def 16;
    hence A.i = B.i by A8,A9,ZFMISC_1:110;
  end;
  hence thesis;
end;
