reserve A,B,C for Ordinal,
        o for object,
        x,y,z,t,r,l for Surreal,
        X,Y for set;

theorem Th33:
  for X1,X2,Y be set holds
    (X1\/X2) ++ Y = (X1 ++ Y) \/ (X2++Y)
proof
  let X1,X2,Y be set;
  thus (X1\/X2) ++ Y c= (X1 ++ Y) \/ (X2++Y)
  proof
    let xy be object;
    assume xy in (X1\/X2)++Y;
    then consider x,y be Surreal such that
    A1:  x in X1\/X2 & y in Y & xy=x+y by Def8;
    x in X1 or x in X2 by A1,XBOOLE_0:def 3;
    then xy in (X1 ++ Y) or xy in (X2++Y) by A1,Def8;
    hence thesis by XBOOLE_0:def 3;
  end;
  let xy be object;
  assume xy in (X1 ++ Y) \/ (X2++Y);
  then per cases by XBOOLE_0:def 3;
  suppose xy in X1++Y;
    then consider x,y be Surreal such that
    A2:  x in X1 & y in Y & xy=x+y by Def8;
    x in X1\/X2 by A2,XBOOLE_0:def 3;
    hence thesis by A2,Def8;
  end;
  suppose xy in X2++Y;
    then consider x,y be Surreal such that
    A3:  x in X2 & y in Y & xy=x+y by Def8;
    x in X1\/X2 by A3,XBOOLE_0:def 3;
    hence thesis by A3,Def8;
  end;
end;
