theorem Th122:
  H * a * b c= (H * a) * (H * b)
proof
  let x be object;
  1_G in H by Th46;
  then
A1: 1_G * b in H * b by Th104;
  assume x in H * a * b;
  then x in H * (a * b) by Th34;
  then consider g such that
A2: x = g * (a * b) and
A3: g in H by Th104;
A4: x = g * a * b by A2,GROUP_1:def 3
    .= g * a * (1_G * b) by GROUP_1:def 4;
  g * a in H * a by A3,Th104;
  hence thesis by A1,A4;
end;
