Affine space/Affine basis/Hyperplane realization/Fact/Proof

Due to fact, there exists a uniquely determined affine-linear mapping

${\displaystyle E\longrightarrow K^{n+1}}$

sending ${\displaystyle {}P_{i}}$ to the ${\displaystyle {}i}$-th standard vector ${\displaystyle {}e_{i}}$. Because of fact, the point

${\displaystyle {}P=\sum _{i=1}^{n+1}a_{i}P_{i}\,}$

is sent to

${\displaystyle \sum _{1=1}^{n+1}a_{i}e_{i}.}$

Because of

${\displaystyle {}\sum _{i=1}^{n+1}a_{i}=1\,,}$

this image point belongs to ${\displaystyle {}F}$. Bijectivity is clear.