Inductive divP : Term A n -> Term A n -> Prop :=
    divTerm_def :
      forall a b : Term A n,
      ~ zeroP (A:=A) A0 eqA (n:=n) a ->
      forall nZb : ~ zeroP (A:=A) A0 eqA (n:=n) b,
      eqTerm (A:=A) eqA (n:=n) a
        (multTerm (A:=A) multA (n:=n) (divTerm a nZb) b) -> 
      divP a b.Inductive divP : Term A n -> Term A n -> Prop :=
    divTerm_def :
      forall a b : Term A n,
      ~ zeroP (A:=A) A0 eqA (n:=n) a ->
      forall nZb : ~ zeroP (A:=A) A0 eqA (n:=n) b,
      eqTerm (A:=A) eqA (n:=n) a
        (multTerm (A:=A) multA (n:=n) (divTerm a nZb) b) -> 
      divP a b.

Theorem divP_plusTerm : forall a b c : Term A n, divP a c -> divP b c -> eqT a b -> ~ zeroP (A:=A) A0 eqA (n:=n) (plusTerm (A:=A) plusA (n:=n) a b) -> divP (plusTerm (A:=A) plusA (n:=n) a b) c.