theorem Th44:
  1_H = 1_G
proof
  set h = the Element of H;
  reconsider g = h, g9 = 1_H as Element of G by Th42;
  h * 1_H = h by GROUP_1:def 4;
  then g * g9 = g by Th43;
  hence thesis by GROUP_1:7;
end;
