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_1:47, 48
  {x,y} (\/) A = A iff x in A & y in A
proof
  thus {x,y} (\/) A = A implies x in A & y in A
  proof
    assume
A1: {x,y} (\/) A = A;
    thus x in A
    proof
      let i;
      assume
A2:   i in I;
      then {x.i,y.i} \/ A.i = {x,y}.i \/ A.i by Def2
        .= A.i by A1,A2,PBOOLE:def 4;
      hence thesis by ZFMISC_1:41;
    end;
    let i;
    assume
A3: i in I;
    then {x.i,y.i} \/ A.i = {x,y}.i \/ A.i by Def2
      .= A.i by A1,A3,PBOOLE:def 4;
    hence thesis by ZFMISC_1:41;
  end;
  assume that
A4: x in A and
A5: y in A;
  now
    let i be object;
    assume
A6: i in I;
    then
A7: x.i in A.i by A4;
A8: y.i in A.i by A5,A6;
    thus ({x,y} (\/) A).i = {x,y}.i \/ A.i by A6,PBOOLE:def 4
      .= {x.i,y.i} \/ A.i by A6,Def2
      .= A.i by A7,A8,ZFMISC_1:42;
  end;
  hence thesis;
end;
