intros x y z mdiv_xy; inversion mdiv_xy.
intros mdiv_yz; inversion mdiv_yz; auto.
rewrite <- (inj_pair2 nat (fun x : nat => mon x) 0 n_0 z); auto.
rewrite <- (inj_pair2 nat (fun x : nat => mon x) 0 n_0 x); auto.
intros n Rec x y z mdiv_xy; inversion mdiv_xy.
rewrite <- (inj_pair2 nat (fun x : nat => mon x) (S n) (c_n n n' v') y); auto.
rewrite <- (inj_pair2 nat (fun x : nat => mon x) (S n) (c_n n n0 v) x); auto.
intros mdiv_yz; inversion mdiv_yz; auto.
rewrite <- (inj_pair2 nat (fun x : nat => mon x) (S n) (c_n n n'0 v'0) z);
auto.
apply mdiv_cons; auto.
apply le_trans with (m := n'); auto.
apply Rec with (y := v0); auto.
rewrite (inj_pair2 nat (fun x : nat => mon x) n v0 v'); auto.
Qed.
(*Division  of monomials (total function!). *)
Definition div_mon : forall d : nat, mon d -> mon d -> mon d.
intros d; elim d.
intros H' H'0; exact n_0.
intros n Rec S1 S2.
exact
(c_n n (pmon1 (S n) S1 - pmon1 (S n) S2)
(Rec (pmon2 (S n) S1) (pmon2 (S n) S2))).
Defined.

Theorem mdiv_div :
forall (d : nat) (a b : mon d),
mdiv d b a -> mult_mon d (div_mon d a b) b = a.