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_1:114
  X is non-empty & Y is non-empty & [|X,Y|] = [|Y,X|] implies X = Y
proof
  assume that
A1: X is non-empty and
A2: Y is non-empty and
A3: [|X,Y|] = [|Y,X|];
  now
    let i be object;
    assume
A4: i in I;
    then
A5: X.i is non empty by A1;
A6: Y.i is non empty by A2,A4;
    [:X.i,Y.i:] = [|X,Y|].i by A4,PBOOLE:def 16
      .= [:Y.i,X.i:] by A3,A4,PBOOLE:def 16;
    hence X.i = Y.i by A5,A6,ZFMISC_1:91;
  end;
  hence thesis;
end;
