Affine space/Point set/Generated affine subspace/Fact

Let ${}E$ be an affine space over the ${}K$-vector space ${}V$, and let ${}T\subseteq E$ denote a subset.

Then, the smallest

affine subspace ${}F\subseteq E$ of ${}E$, which contains ${}T$, consists of all barycentric combinations

$\sum _{i=1}^{n}a_{i}P_{i}{\text{ with }}P_{i}\in T{\text{ and }}\sum _{i=1}^{n}a_{i}=1.$

Proof, Write another proof