Net flow out of $x$ equals net flow into $y$

Question

Let $D=(V, A)$ be a digraph and $f$ a real-valued function on $A$.

I am trying to show:

1. $\sum\left\{f^{+}(v): v \in V\right\}=\sum\left\{f^{-}(v): v \in V\right\}$,
2. if $f$ is an $(x, y)$-flow, then the net flow $f^{+}(x)-f^{-}(x)$ out of $x$ is equal flow $f^{-}(y)-f^{+}(y)$ into $y$.

My attempt:

I don't know how to do the first part, but for the second part I have the following

\begin{aligned} 0=& f(V, V)-f(V, V) \\ =& {[f(x, V)+f(y, V)]-[f(V, x)+f(V, y)]} \\ &+\sum_{v \neq x, y}(f(v, V)-f(V, v)) \\ =& {[f(x, V)+f(y, V)]-[f(V, x)+f(V, y)] } \end{aligned}

Note: $f(x, V)$ is the same as $f^{+}(x)$ (flow from $x$ to another vertex) and $f(V, x)$ is the same as $f^{-}(x)$ (flow from another vertex to $x$), and $f(x, V)$-$f(V, x)$ is the net flow out of $x$