theorem Th110:
  (0).G + a = {a} & a + (0).G = {a}
proof
A1: the carrier of (0).G = {0_G} by Def7;
  hence (0).G + a = {0_G + a} by Th18
    .= {a} by Def4;
  thus a + (0).G = {a + 0_G} by A1,Th18
    .= {a} by Def4;
end;
