Let \(F\) be a conservative vector field on an open and connected domain and let \(f\) and \(g\) be functions such that \(∇f=F\) and \(∇g=F\).
Since \(f\) and \(g\) are both potential functions for \(\textbf{F}\), then:
$$ ∇(f-g)=∇f-∇g=\textbf{F}-\textbf{F}=0 $$
Let \(h=f-g\), then we have \(∇h=0\):
Assume \(h\) is a function of \(x\) and \(y\). Since \(∇h=0\), we have \(h_x=0\) and \(h_y=0\). The expression \(h_x=0\) implies that \(h\) is a constant function with respect to \(x\). Similarly, \(h\) is a constant function with respect to \(y\). Thus, \(h(x,y)=C\) for some constant C on the connected domain of \(\textbf{F}\).