theorem
  carr(H) c= (H + a) + (H + -a) & carr(H) c= (H + -a) + (H + a)
proof
A1: H + -a+ a = H + (-a+ a) by ThB34
    .= H + 0_G by Def5
    .= carr(H) by Th37;
  H + a + -a = H + (a + -a) by ThB34
    .= H + 0_G by Def5
    .= carr(H) by Th37;
  hence thesis by A1,Th122;
end;
