theorem Th116:
  a * b * H c= (a * H) * (b * H)
proof
  let x be object;
  assume x in a * b * H;
  then consider g such that
A1: x = a * b * g and
A2: g in H by Th103;
A3: x = a * 1_G * b * g by A1,GROUP_1:def 4
    .= (a * 1_G) * (b * g) by GROUP_1:def 3;
  1_G in H by Th46;
  then
A4: a * 1_G in a * H by Th103;
  b * g in b * H by A2,Th103;
  hence thesis by A3,A4;
end;
