Definition same_ratio (a b c d e f g h : Z) :=
  (a * f)%Z = (b * e)%Z /\
  (b * g)%Z = (c * f)%Z /\
  (c * h)%Z = (d * g)%Z /\
  (a * g)%Z = (c * e)%Z /\ (a * h)%Z = (d * e)%Z /\ (b * h)%Z = (d * f)%Z.

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}.