theorem Th67:
  (0).G * a = (0).G
proof
A1: the carrier of (0).G = {0_G} by Def7;
  the carrier of (0).G * a = (carr (0).G) * a by Def6A
    .= {(0_G) * a} by A1,ThB37
    .= {0_G} by Th17;
  hence thesis by Def7;
end;
