theorem
  carr(H) c= (a + H) + (-a + H) & carr(H) c= (-a + H) + (a + H)
proof
A1: -a + a + H = 0_G + H by Def5
    .= carr(H) by Th37;
  a + -a + H = 0_G + H by Def5
    .= carr(H) by Th37;
  hence thesis by A1,Th116;
end;
