Affine space/Affine subspace/Intersection property/Fact/Proof

If the intersection is empty, then the statement holds by definition. So let ${\displaystyle {}P\in \bigcap _{i\in I}F_{i}}$. We may write the affine subspaces as

${\displaystyle {}F_{i}=P+U_{i}\,}$

with linear subspaces ${\displaystyle {}U_{i}\subseteq V}$. Let

${\displaystyle {}U=\bigcap _{i\in I}U_{i}\,,}$

which is a linear subspace, due to fact  (1). We claim that

${\displaystyle {}\bigcap _{i\in I}F_{i}=P+U\,.}$

From ${\displaystyle {}Q\in \bigcap _{i\in I}F_{i}}$, we can deduce

${\displaystyle {}Q=P+u\,}$

with ${\displaystyle {}u\in \bigcap _{i\in I}U_{i}}$, so that ${\displaystyle {}Q\in P+U}$ holds. If ${\displaystyle {}Q\in P+U}$ holds, then directly ${\displaystyle {}Q\in P+U_{i}=F_{i}}$ follows.