intuition.
destruct (leRat_difference_exists H).
destruct (leRat_difference_exists H0).
eapply eqRat_trans.
apply ratSubtract_eqRat_compat.
eapply eqRat_trans.
apply ratMult_eqRat_compat.
apply H1.
apply H2.
eapply eqRat_trans.
apply ratMult_distrib_r.
eapply eqRat_trans.
eapply ratAdd_eqRat_compat.
apply ratMult_comm.
apply ratMult_comm.
apply ratAdd_eqRat_compat.
apply ratMult_distrib_r.
apply ratMult_distrib_r.
apply eqRat_refl.
eapply eqRat_trans.
eapply ratSubtract_eqRat_compat.
apply ratAdd_assoc.
apply ratMult_comm.
eapply eqRat_trans.
apply ratSubtract_ratAdd_inverse.
rewrite <- ratAdd_assoc.
apply ratAdd_eqRat_compat.
rewrite ratAdd_comm.
apply ratAdd_eqRat_compat.
rewrite ratMult_comm.
apply ratMult_eqRat_compat.
eapply eqRat_symm.
eapply eqRat_trans.
eapply ratSubtract_eqRat_compat.
apply H1.
apply eqRat_refl.
apply ratSubtract_ratAdd_inverse.
intuition.
apply ratMult_eqRat_compat.
apply eqRat_symm.
eapply eqRat_trans.
apply ratSubtract_eqRat_compat.
apply H2.
apply eqRat_refl.
apply ratSubtract_ratAdd_inverse.
intuition.
rewrite ratMult_comm.
apply ratMult_eqRat_compat.
apply eqRat_symm.
eapply eqRat_trans.
apply ratSubtract_eqRat_compat.
apply H1.
apply eqRat_refl.
apply ratSubtract_ratAdd_inverse.
apply eqRat_symm.
eapply eqRat_trans.
apply ratSubtract_eqRat_compat.
apply H2.
apply eqRat_refl.
apply ratSubtract_ratAdd_inverse.
Qed.

Lemma ratSubtract_eq_r : forall r1 r2 r3,
r2 <= r1 ->
r3 <= r1 ->
(ratSubtract r1 r2) == (ratSubtract r1 r3) ->
r2 == r3.