On $n$ dimensional manifolds and lattices EDIT (By Tara B):  I'd still be interested in a reference to a proof of this, as I had to prove it myself for my own paper.

I'm looking for the proof of Theorem 4 that appears in this paper:

An Infinite Hierarchy of Intersections of Context-Free Languages by Liu and Weiner.


  Theorem 4: An $n$-dimensional affine manifold is not expressible as a finite union of affine manifolds each of which is of dimension $n-1$ or less.



Does anyone knows a reference to the proof?
If the manifold is finite and we define a natural order on the elements, is there any similar statement in terms of lattices?




Some background to understand the theorem:

Definition: Let $\mathbb{Q}$ be the set of rational numbers. A subset $M\subseteq \mathbb{Q}^n$ is an affine manifold if $(\lambda x+(1-\lambda)y)\in M$ when $x\in M$, $y\in M$, and $\lambda\in\mathbb{Q}$.

Definition: An affine manifold $M'$ is said to be parallel to an affine manifold $M$ if $M'=M+a$ for some $a\in \mathbb{Q}^n$.

Theorem: Each non-empty affine manifold $M\subseteq \mathbb{Q}^n$ is parallel to a unique subspace $K$. This $K$ is given by $K=\{x-y:x,y\in M\}$

Definition: The dimension of a non-empty affine manifold is the dimension of the subspace parallel to it.
Here is a measure-free proof which works for affine manifolds over an arbitrary infinite field $\mathbb F$ (the result is false for finite fields).

By induction on $n\ge0$, we will show that an affine manifold $A\subseteq\mathbb F^m$ of dimension $n$ is not a finite union of affine manifolds of dimension less than $n$.

The statement is clear for $n=0$: a point is not a (finite) union of empty sets.

Assume the statement holds for $n$, we will show it for $n+1$. Let $A=\bigcup_{i&lt;k}A_i$, where $\dim(A)=n+1$ and $\dim(A_i)\le n$. Consider an arbitrary affine submanifold $B\subset A$ of dimension $n$. Since $B=\bigcup_i(B\cap A_i)$, the induction hypothesis implies that $\dim(B\cap A_i)=n$ for some $i&lt;k$, i.e., $B=A_i$. Since there are only $k$ sets $A_i$, and $B$ was arbitrary, it follows that $A$ has only finitely many submanifolds of dimension $n$. However, this is a contradiction: if we fix any such submanifold $B_0$ and a vector $v$ parallel to $A$ but not to $B_0$, there are infinitely many affine submanifolds of $A$ of the form $B_0+av$, where $a\in\mathbb F$.