Theorem 11 (*Conway & Sloane, Sphere Packings, Lattices and Groups, 3rd Edition, pp 383, Ch 15*). If a system of putative $p$-adic symbols for each $p$ satisfies the determinant, oddity, and $p$-adic existence conditions, then there exists an integral quadratic form with these $p$-adic symbols.

The book does not provide any references or pointers on where to find a proof. We checked the obvious reference i.e., B. W. Jones, Cassels, O'Meara, and could not find the relevant theorem. Unfortunately, we find it far from obvious and hence are unable to prove it ourselves.

We have two quick questions.

1. Can someone please recommend a reference where a proof appears ?
2. Is the Theorem if and only if ?

Thank you for your time.