theorem
  carr(H) c= (a * H) * (a" * H) & carr(H) c= (a" * H) * (a * H)
proof
A1: a" * a * H = 1_G * H by GROUP_1:def 5
    .= carr(H) by Th37;
  a * a" * H = 1_G * H by GROUP_1:def 5
    .= carr(H) by Th37;
  hence thesis by A1,Th116;
end;
