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_1:118
  X c= Y implies [|X,Z|] c= [|Y,Z|] & [|Z,X|] c= [|Z,Y|]
proof
  assume
A1: X c= Y;
  thus [|X,Z|] c= [|Y,Z|]
  proof
    let i;
    assume
A2: i in I;
    then X.i c= Y.i by A1;
    then [:X.i,Z.i:] c= [:Y.i,Z.i:] by ZFMISC_1:95;
    then [|X,Z|].i c= [:Y.i,Z.i:] by A2,PBOOLE:def 16;
    hence thesis by A2,PBOOLE:def 16;
  end;
  let i;
  assume
A3: i in I;
  then X.i c= Y.i by A1;
  then [:Z.i,X.i:] c= [:Z.i,Y.i:] by ZFMISC_1:95;
  then [|Z,X|].i c= [:Z.i,Y.i:] by A3,PBOOLE:def 16;
  hence thesis by A3,PBOOLE:def 16;
end;
