In Borevich & Shafarevich's Number Theory, Theorem 6.2.3 claims that
Let $p \neq 2$ and $0<r<n$. The form $$F = F_0+pF_1$$ represents zero in the field $\mathbb{Q}_p$ iff at least one of the forms $F_0$ or $F_1$ reprensents zero, where $$F_0 = \epsilon_1 x_1^{2} + \dots + \epsilon_r x_r^{2}, F_1 = \epsilon_{r+1} x_{r+1}^{2} + \dots + \epsilon_n x_n^{2}, $$ $\epsilon_i$'s are units in $\mathbb{Z}_p$.
The authors omit the proof of sufficiency, claiming it 'obvious'. But somehow I cannot see how to show $F$ represents zero in the case only one of $F_i$'s represents zero. Can anyone help?