Two parallel lines on $\mathbb R^2$ have the same slope. I'm trying to generalize this result to higher dimension. Could you have a check on my attempt?
Let $X$ be a vector space and $A$ an affine subspace of $X$. Let $f,g: A \to \mathbb R$ be affine maps. There is $a \in X$ and a vector subspace $V$ of $X$ such that $a+V = A$. We define $f', g':V \to \mathbb R$ by $f' (x) := f(x+a)-f(a)$ and $g'(x) := g(x+a)-g(a)$. Then $f',g'$ are linear.
Theorem: If $f \le g$, then $f' = g'$.
Proof: Assume the contrary that $f'(b) - g'(b) = \alpha \neq 0$ for some $b \in V$. Then $$ f'(kb) - g'(kb) = f(kb+a) -f(a) -g(kb+a) + g(a) = k \alpha, \quad k \in \mathbb N. $$
Then $f(kb+a)-g(kb+a) = k \alpha +[f(a)-g(a)]$ can be arbitrarily large or arbitrarily small. This contradicts the fact that $f \le g$. This completes the proof.
