∀ n m, O < n ⇒ n | m ⇒ div_mod_spec m n (div m n) O

Theorem divides_to_div_mod_spec : forall (n:nat.nat), forall (m:nat.nat), (nat.lt nat.O n) -> (divides n m) -> div_mod.div_mod_spec m n (div_mod.div m n) nat.O.

theorem divides_to_div_mod_spec : \forall (n:nat). \forall (m:nat). ((lt) (O) n) -> ((divides) n m) -> (div_mod_spec) m n ((div) m n) (O) .

theorem divides_to_div_mod_spec : forall (n:nat.nat) , forall (m:nat.nat) , ((((nat.lt_) ) ((nat.O) )) (n)) -> ((((primes.divides) ) (n)) (m)) -> (((((div_mod.div_mod_spec) ) (m)) (n)) ((((div_mod.div) ) (m)) (n))) ((nat.O) ).

divides_to_div_mod_spec : 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) => (primes_sttfa.sttfa_divides(n)(m) => div_mod_sttfa_th.div_mod_spec(m)(n)(div_mod_sttfa_th.div(m)(n))(nat_sttfa_th.sttfa_O)))))

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