Affine mapping/One point suffices/Exercise

Let ${\displaystyle {}K}$ be a field, and let ${\displaystyle {}E}$ and ${\displaystyle {}F}$ denote affine spaces over the vector spaces ${\displaystyle {}V}$ and ${\displaystyle {}W}$, respectively. Let a mapping

${\displaystyle \psi \colon E\longrightarrow F,}$

a linear mapping

${\displaystyle \varphi \colon V\longrightarrow W,}$

and a point ${\displaystyle {}P\in E}$ be given such that

${\displaystyle {}\psi (P+v)=\psi (P)+\varphi (v)\,}$

holds for all ${\displaystyle {}v\in V}$. Show that ${\displaystyle {}\psi }$ is affine-linear.