∀ n m q, O < q ⇒ (mod n q) = (mod m q) ⇒ q | (n - m)

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

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

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

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

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