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

theorem
  for X be surreal-membered set holds X ++ {0_No} = X
proof
  let X be surreal-membered set;
  thus X ++ {0_No} c= X
  proof
    let xy be object such that A1:xy in X ++ {0_No};
    consider x,y such that
    A2: x in X & y in {0_No} & xy= x+y by A1,Def8;
    y=0_No by A2,TARSKI:def 1;
    hence thesis by A2;
  end;
  let xy be object such that A3:xy in X;
  reconsider x=xy as Surreal by A3,SURREAL0:def 16;
  x+0_No = x &0_No in {0_No} by TARSKI:def 1;
  hence thesis by A3,Def8;
end;
