Affin-lineare mapping/Values on affine basis/Fact/Proof

Let ${\displaystyle {}i_{0}\in I}$. Due to fact, there exists a uniquely determined linear mapping

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

such that

${\displaystyle {}\varphi ({\overrightarrow {P_{i_{0}}P_{i}}})={\overrightarrow {Q_{i_{0}}Q_{i}}}\,}$

for all ${\displaystyle {}i\in I\setminus \{i_{0}\}}$. Therefore,

${\displaystyle {}\psi (R)=Q_{i_{0}}+\varphi ({\overrightarrow {P_{i_{0}}R}})\,}$

is an affine-linear mapping with the properties looked for. Note that

${\displaystyle {}\psi (P_{i_{0}})=Q_{i_{0}}+\varphi ({\overrightarrow {P_{i_{0}}P_{i_{0}}}})=Q_{i_{0}}\,}$

and

${\displaystyle {}\psi (P_{i})=Q_{i_{0}}+\varphi ({\overrightarrow {P_{i_{0}}P_{i}}})=Q_{i_{0}}+{\overrightarrow {Q_{i_{0}}Q_{i}}}=Q_{i}\,}$

holds. Such an affine mapping ${\displaystyle {}\psi }$ is uniquely determined by its linear part and the image of just one point, so that

${\displaystyle {}\psi _{0}=\varphi \,}$

must hold.