∀ {R : Type u_3} {S : Type u_4} [inst : CommRing S] [inst_1 : PartialOrder S] [IsOrderedRing S] [NoZeroDivisors S]
  [inst_4 : Ring R] (abv : R → S) [IsAbsoluteValue abv] (a b : R), abv (a - b) = abv (b - a)