∀ n m, O < n ⇒ (mod m n) = O ⇒ n | m

Theorem mod_O_to_divides : forall (n:nat.nat), forall (m:nat.nat), (nat.lt nat.O n) -> (logic.eq (nat.nat) (div_mod.mod m n) nat.O) -> divides n m.

theorem mod_O_to_divides : \forall (n:nat). \forall (m:nat). ((lt) (O) n) -> ((eq) (nat) ((mod) m n) (O) ) -> (divides) n m.

theorem mod_O_to_divides : forall (n:nat.nat) , forall (m:nat.nat) , ((((nat.lt_) ) ((nat.O) )) (n)) -> ((((logic.eq_) (nat.nat)) ((((div_mod.mod) ) (m)) (n))) ((nat.O) )) -> (((primes.divides) ) (n)) (m).

mod_O_to_divides : LEMMA (FORALL(n:nat_sttfa_th.sttfa_nat):(FORALL(m:nat_sttfa_th.sttfa_nat):(nat_sttfa_th.lt(nat_sttfa_th.sttfa_O)(n) => (logic_sttfa_th.eq[nat_sttfa_th.sttfa_nat](div_mod_sttfa_th.mod(m)(n))(nat_sttfa_th.sttfa_O) => primes_sttfa.sttfa_divides(n)(m)))))

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