∀ n m p, O < p ⇒ (mod (n × m) p) = (mod ((mod n p) × (mod m p)) p)

Theorem mod_times : forall (n:nat.nat), forall (m:nat.nat), forall (p:nat.nat), (nat.lt nat.O p) -> logic.eq (nat.nat) (div_mod.mod (nat.times n m) p) (div_mod.mod (nat.times (div_mod.mod n p) (div_mod.mod m p)) p).

theorem mod_times : \forall (n:nat). \forall (m:nat). \forall (p:nat). ((lt) (O) p) -> (eq) (nat) ((mod) ((times) n m) p) ((mod) ((times) ((mod) n p) ((mod) m p)) p).

theorem mod_times : forall (n:nat.nat) , forall (m:nat.nat) , forall (p:nat.nat) , ((((nat.lt_) ) ((nat.O) )) (p)) -> (((logic.eq_) (nat.nat)) ((((div_mod.mod) ) ((((nat.times) ) (n)) (m))) (p))) ((((div_mod.mod) ) ((((nat.times) ) ((((div_mod.mod) ) (n)) (p))) ((((div_mod.mod) ) (m)) (p)))) (p)).

mod_times : LEMMA (FORALL(n:nat_sttfa_th.sttfa_nat):(FORALL(m:nat_sttfa_th.sttfa_nat):(FORALL(p:nat_sttfa_th.sttfa_nat):(nat_sttfa_th.lt(nat_sttfa_th.sttfa_O)(p) => logic_sttfa_th.eq[nat_sttfa_th.sttfa_nat](div_mod_sttfa_th.mod(nat_sttfa_th.times(n)(m))(p))(div_mod_sttfa_th.mod(nat_sttfa_th.times(div_mod_sttfa_th.mod(n)(p))(div_mod_sttfa_th.mod(m)(p)))(p))))))

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