Definition reducep (L : list (poly A0 eqA ltM)) (p q : poly A0 eqA ltM) :
  Prop :=
  reduce A A0 A1 eqA invA minusA multA divA eqA_dec n ltM ltM_dec L
    (s2p A A0 eqA n ltM p) (s2p A A0 eqA n ltM q).

Theorem divp_reduce1 : forall (p : poly A0 eqA ltM) (L1 L2 : list (poly A0 eqA ltM)), (forall r1 : poly A0 eqA ltM, In r1 L1 -> ~ zerop A A0 eqA n ltM r1 -> exists r2 : poly A0 eqA ltM, In r2 L2 /\ divp A A0 eqA multA divA n ltM r1 r2) -> forall q : poly A0 eqA ltM, reducep L1 p q -> exists r : poly A0 eqA ltM, reducep L2 p r.