theorem Th78:
  a,1_G are_conjugated or 1_G,a are_conjugated implies a = 1_G
proof
  assume
A1: a,1_G are_conjugated or 1_G,a are_conjugated;
  now
    per cases by A1;
    suppose
      a,1_G are_conjugated;
      then ex g st 1_G = a |^ g by Th74;
      hence thesis by Th18;
    end;
    suppose
      1_G,a are_conjugated;
      then ex g st 1_G = a |^ g;
      hence thesis by Th18;
    end;
  end;
  hence thesis;
end;
