Affine-linear mapping/Barycentric combination/Fact

Let $E$ and $F$ denote affine spaces over a field $K$, and let

$\psi :E⟶F$

denote a mapping.

Then, $\psi$ is

affine-linear if and only if for every barycentric combination $\sum _{i\in I}{a}_i{P}_i$ with $P_i\in E$, the equality

$\psi \left(\sum _{i\in I}{a}_i{P}_i\right)=\sum _{i\in I}{a}_i\psi (P_i)$

holds.

Proof, Write another proof