theorem
  for V being Abelian add-associative right_zeroed right_complementable
  non empty addLoopStr, v being Element of V holds
  Sum<* v,- v *> = 0.V & Sum<*- v,v *> = 0.V
proof
  let V be Abelian add-associative right_zeroed right_complementable non
  empty addLoopStr, v be Element of V;
  thus Sum<* v,- v *> = v + (- v) by Th45
    .= 0.V by Th5;
  hence thesis by Th54;
end;
