Let $A$ be an affine subspace of $V$, say $A = x + U$ for some vector $x$ and some subspace $U$, and suppose $B$ is an affine hyperplane of $V$, say $B = y + H$ for some vector $y$ and some hyperplane $H$. Suppose $A \cap B = \emptyset$. Does this mean that $A = z + H'$ for some vector $z$ (where $z - y \notin H$) and some subspace $H'$ of $H$?
What about the converse, is it also true? That is, if I have an affine subspace $z + H'$ for some vector $z$ with $z - y \notin H$ and some subspace $H'$ of $H$, does this mean that $z+H'$ is disjoint from $B$?
It seems to be true at least when I try to picture it in $\mathbb{R}^3$.