theorem Th120:
  H * a = H * b iff b * a" in H
proof
  thus H * a = H * b implies b * a" in H
  proof
    assume
A1: H * a = H * b;
    carr(H) = H * 1_G by Th37
      .= H * (a * a") by GROUP_1:def 5
      .= H * b * a" by A1,Th34
      .= H * (b * a") by Th34;
    hence thesis by Th119;
  end;
  assume b * a" in H;
  hence H * a = H * (b * a") * a by Th119
    .= H * (b * a" * a) by Th34
    .= H * (b * (a" * a)) by GROUP_1:def 3
    .= H * (b * (1_G)) by GROUP_1:def 5
    .= H * b by GROUP_1:def 4;
end;
