simple induction a.
replace (p * 0) with 0.
intros.
cut (p = 0 \/ b = 0).
intro.
elim H0; auto with arith.
cut (p * b = 0).
elim (mult_eq_zero p b); auto with arith.
auto with arith.
apply mult_n_O.
intros n H_rec.
simple induction b.
replace (p * 0) with 0.
intro.
apply (mult_eq_zero p (S n) H).
apply mult_n_O.
intros m H.
replace (p * S n) with (p * n + p).
replace (p * S m) with (p * m + p).
2: apply mult_n_Sm.
2: apply mult_n_Sm.
clear a; clear b.
intro.
elim (H_rec m); auto with arith.
apply (fun a b : nat => plus_reg_l a b p).
elim (plus_comm (p * n) p).
elim (plus_comm (p * m) p); trivial with arith.
Qed.
Hint Resolve mult_reg_l.

Lemma mult_reg_l_bis : forall a b p : nat, 0 < p -> p * a = p * b -> a = b.
intros a b p H H1.