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_1:115
  [|X,X|] = [|Y,Y|] implies X = Y
proof
  assume
A1: [|X,X|] = [|Y,Y|];
  now
    let i be object;
    assume
A2: i in I;
    then [:X.i,X.i:] = [|X,X|].i by PBOOLE:def 16
      .= [:Y.i,Y.i:] by A1,A2,PBOOLE:def 16;
    hence X.i = Y.i by ZFMISC_1:92;
  end;
  hence thesis;
end;
