Affine space/Affine subspace/Characterization/Fact

Let ${}E$ be an affine space over the ${}K$-vector space ${}V$. For a subset ${}F\subseteq E$, the following conditions are equivalent.

1. ${}F$ is an affine subspace of ${}E$.
2. For ${}P_{1},\ldots ,P_{n}\in F$ and numbers ${}a_{1},\ldots ,a_{n}$ satisfying ${}\sum _{i=1}^{n}a_{i}=1$, we have ${}\sum _{i=1}^{n}a_{i}P_{i}\in F$.
3. For two points ${}P,Q\in F$ and numbers ${}r,s\in K$ with ${}r+s=1$, we have ${}rP+sQ\in F$.

Proof, Write another proof