The Lemma said: "If $f$ is a linear functional on $H$ and $f(x\_{0}) \neq 0$ for some $ x\_{0} \in H $, then every $x \in H$ has the form $$x = \beta x\_{0} + z, \beta \in \mathbb{C}, z \in Ker f.$$ "

And its proof as written in the book is: 

" **Proof:**

Take $\beta = \frac{f(x)}{f(x\_{0})}$ and $z = x - \beta x\_{0}.$ "

But I can not understand the importance of the $\beta$ given in the proof, Also it is not clear for me the total idea of the proof, could anyone explain this for me please?