A theorem about the dimension of affine subspaces

Question

The related theorem and its proof is:

My problem is that from $E\cap{F}\not=\emptyset $, how it is deduced that $E$ and $F$ are reduced to their associated vector subspaces? Also, what is the relationship between $<E,F>$, i.e, the smallest affine subspace that contains $E$ and $F$, and $E'+F'$?

Answer 1

The following Lemma provides the necessary tools, the rest is linear algebra.