Show that if $|f(x)| \leq \phi(x) + \psi (x)$, exist $g,h$ such that $f(x) = g(x)+h(x),\, |g(x)| \leq \phi (x), \, \, \, |h(x)| \leq \psi(x)$

Given $ \phi$ and $ \psi$ two seminorms in a vector space X, and a functional $ f:X \rightarrow \mathbb{K}$ , where $ \mathbb{K} = \mathbb{R}$ or $ \mathbb{C}$ , such that $ |f(x)| \leq \phi(x) + \psi (x) \, \, \forall x \in X$ , show that there exist two linear functionals $ g,h: X \rightarrow \mathbb{K}$ such that: $ f(x) = g(x) + h(x), \, \, \, |g(x)| \leq \phi (x), \, \, \, |h(x)| \leq \psi(x) \, \, \, \forall x \in X$ .

It is obvious that if $ g,h$ exist, with the indicated bound, then $ |f(x)| \leq \phi(x) + \psi (x)$ , but how can I show that these functionals exist?