(* Proof by case analysis: 81 cases in 6 line :-) *)

Lemma Qquadratic_sign_neg_1 :
forall (a b c d e f g h : Z) (p1 p2 : Qpositive)
(H_Qquadratic_sg_denom_nonzero : Qquadratic_sg_denom_nonzero e f g h p1 p2)
(na nb nc nd ne nf ng nh : Z) (np1 np2 : Qpositive),
Qquadratic_sign a b c d e f g h p1 p2 H_Qquadratic_sg_denom_nonzero =
((-1)%Z, (na, (nb, (nc, nd)), (ne, (nf, (ng, nh))), (np1, np2))) ->
{(0 < na + nb + nc + nd)%Z /\ (ne + nf + ng + nh < 0)%Z} +
{(na + nb + nc + nd < 0 < ne + nf + ng + nh)%Z}.