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_1:53, 63
  x in X & y in X iff {x,y} (/\) X = {x,y}
proof
  thus x in X & y in X implies {x,y} (/\) X = {x,y}
  proof
    assume that
A1: x in X and
A2: y in X;
    now
      let i be object;
      assume
A3:   i in I;
      then
A4:   x.i in X.i by A1;
A5:   y.i in X.i by A2,A3;
      thus ({x,y} (/\) X).i = {x,y}.i /\ X.i by A3,PBOOLE:def 5
        .= {x.i,y.i} /\ X.i by A3,Def2
        .= {x.i,y.i} by A4,A5,ZFMISC_1:47
        .= {x,y}.i by A3,Def2;
    end;
    hence thesis;
  end;
  assume
A6: {x,y} (/\) X = {x,y};
  thus x in X
  proof
    let i;
    assume
A7: i in I;
    then {x.i,y.i} /\ X.i = {x,y}.i /\ X.i by Def2
      .= ({x,y} (/\) X).i by A7,PBOOLE:def 5
      .= {x.i,y.i} by A6,A7,Def2;
    hence thesis by ZFMISC_1:55;
  end;
  let i;
  assume
A8: i in I;
  then {x.i,y.i} /\ X.i = {x,y}.i /\ X.i by Def2
    .= ({x,y} (/\) X).i by A8,PBOOLE:def 5
    .= {x.i,y.i} by A6,A8,Def2;
  hence thesis by ZFMISC_1:55;
end;
