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_1:132
  [|{x,y},X|] = [|{x},X|] (\/) [|{y},X|] &
  [|X,{x,y}|] = [|X,{x}|] (\/) [|X,{y}|]
proof
  now
    let i be object;
    assume
A1: i in I;
    hence [|{x,y},X|].i = [:{x,y}.i,X.i:] by PBOOLE:def 16
      .= [:{x.i,y.i},X.i:] by A1,Def2
      .= [:{x.i},X.i:] \/ [:{y.i},X.i:] by ZFMISC_1:109
      .= [:{x}.i,X.i:] \/ [:{y.i},X.i:] by A1,Def1
      .= [:{x}.i,X.i:] \/ [:{y}.i,X.i:] by A1,Def1
      .= [|{x},X|].i \/ [:{y}.i,X.i:] by A1,PBOOLE:def 16
      .= [|{x},X|].i \/ [|{y},X|].i by A1,PBOOLE:def 16
      .= ([|{x},X|] (\/) [|{y},X|]).i by A1,PBOOLE:def 4;
  end;
  hence [|{x,y},X|] = [|{x},X|] (\/) [|{y},X|];
  now
    let i be object;
    assume
A2: i in I;
    hence [|X,{x,y}|].i = [:X.i,{x,y}.i:] by PBOOLE:def 16
      .= [:X.i,{x.i,y.i}:] by A2,Def2
      .= [:X.i,{x.i}:] \/ [:X.i,{y.i}:] by ZFMISC_1:109
      .= [:X.i,{x}.i:] \/ [:X.i,{y.i}:] by A2,Def1
      .= [:X.i,{x}.i:] \/ [:X.i,{y}.i:] by A2,Def1
      .= [|X,{x}|].i \/ [:X.i,{y}.i:] by A2,PBOOLE:def 16
      .= [|X,{x}|].i \/ [|X,{y}|].i by A2,PBOOLE:def 16
      .= ([|X,{x}|] (\/) [|X,{y}|]).i by A2,PBOOLE:def 4;
  end;
  hence thesis;
end;
