End DifferentDom.
Hypothesis aR_anti : antisymmetric aR.
Hypothesis rR_anti : antisymmetric rR.

(* Specifications: *)
Lemma mono_inj_in : {in D &, {mono f : x y / aR x y >-> rR x y}} ->
{in D &, injective f}.
Lemma total_homo_mono_in : total aR ->
{in D &, {homo f : x y / aR' x y >-> rR' x y}} ->
{in D &, {mono f : x y / aR x y >-> rR x y}}.