theorem ThB78:
  a,0_G are_conjugated or 0_G,a are_conjugated implies a = 0_G
proof
  assume
A1: a,0_G are_conjugated or 0_G,a are_conjugated;
  now
    per cases by A1;
    suppose
      a,0_G are_conjugated;
      then ex g st 0_G = a * g by Th74;
      hence thesis by ThB18;
    end;
    suppose
      0_G,a are_conjugated;
      hence thesis by ThB18;
    end;
  end;
  hence thesis;
end;
