theorem
  carr(H) c= (H * a) * (H * a") & carr(H) c= (H * a") * (H * a)
proof
A1: H * a"* a = H * (a"* a) by Th34
    .= H * 1_G by GROUP_1:def 5
    .= carr(H) by Th37;
  H * a * a" = H * (a * a") by Th34
    .= H * 1_G by GROUP_1:def 5
    .= carr(H) by Th37;
  hence thesis by A1,Th122;
end;
