= Solution
For the canonical <Poisson bracket>, direct substitution of $F=q_i$ and $F=p_i$ gives
$$
\{q_i,G\}=\frac{\partial G}{\partial p_i},\qquad
\{p_i,G\}=-\frac{\partial G}{\partial q_i}.
$$
Therefore
$$
\sum_i\frac{\partial F}{\partial q_i}\{q_i,G\}
+\sum_i\frac{\partial F}{\partial p_i}\{p_i,G\}
=\sum_i\left(F_{q_i}G_{p_i}-F_{p_i}G_{q_i}\right)
=\boxed{\{F,G\}.}
$$
This expresses the bracket as a first-order derivative acting on $F$, with coefficients determined by $G$; it is the phase-space chain-rule form of the definition.
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