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

Proof

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

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

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

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

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

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

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

${}Q=P+u\,$

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

To fact