Affine space/Finite point family/Affinely independent/Characterization/Fact

Let $E$ be an affine space over a $K$-vector space $V$, and let

$P_1,\dots ,P_n$

denote a finite family of points in $E$. Then the following statements are equivalent.

1. The points $P_1,\dots ,P_n$ are affinely independent.
2. For every $i\in \{1,\dots ,n\}$, the family of vectors

   $\overrightarrow{P_i{P}_1},\dots ,\overrightarrow{P_i{P}_{i-1}},\overrightarrow{P_i{P}_{i+1}},\dots ,\overrightarrow{P_i{P}_n}$

   is linearly independent.

3. There exists some $i\in \{1,\dots ,n\}$ such that the family of vectors

   $\overrightarrow{P_i{P}_1},\dots ,\overrightarrow{P_i{P}_{i-1}},\overrightarrow{P_i{P}_{i+1}},\dots ,\overrightarrow{P_i{P}_n}$

   is linearly independent.

4. The points $P_1,\dots ,P_n$ form an affine basis in the affine subspace generated by them.

Write a proof