I am trying to figure out something concerning the index of lattices.
The question came about after reading the paper of W.Fulton and B.Sturmfels, (*"Intersection theorey on toric varieties"*). To paraphrase one of their constructions: in their paper they define the intersection between rational subspaces $V,W \subset \mathbb{R}^n$ and say that the multiplicity between these
should be $$ [\mathbb{Z}^n:M+ M']$$
where $M=V\cap\mathbb{Z}^n$ and $M'=W\cap\mathbb{Z}^n$. Here, the notation means the index of the sublattice $M+M'$ in $\mathbb{Z}^n \cap affine(M+M')$, where $affine(M)$ is the affine hull of the lattice $M$.

Now, let $L=V^{\perp}\cap\mathbb{Z}^n$ and $L'=W^{\perp}\cap\mathbb{Z}^n$ where $V^{\perp}$ denotes the orthogonal complement of $V$ in $\mathbb{R}^n$.

**Question:**
Is it true that $$ [\mathbb{Z}^n:M+ M']=[\mathbb{Z}^n:L+ L']?$$

This is true if $n=2$, and all the examples I have considered in $n=3$ satisfy this as well. I have very little familliarity with lattice theory, so I would really appreciate any advice, or where to look.

**Example:** Here is a simple example for $n=3$ which hopefully clarifies the question. Let $V$ be the line determined by the vector $(0,0,1)$ (i.e. $V=\mathbb{R} \cdot (0,0,1)$) and let $W$ be the line determined by $(k,k,1)$, for $k \in \mathbb{N}$.
Then, with the notation above, $M + M'=\mathbb{Z}< (0,0,1)> + \mathbb{Z} <(k,k,1)> $ and $$ [\mathbb{Z}^3:M+M']=[\mathbb{Z}^3 \cap affine(M+M'):M+M' ])=k,$$
since a basis for $\mathbb{Z}^3 \cap affine(M+M')$ is given by $(0,0,1)$ and $(1,1,0)$.

On the other hand $L+L'=\mathbb{Z} <(1,0,0),(0,1,0)> + \mathbb{Z} <(0,1,-k),(1,-1,0) > =$ $ =\mathbb{Z}<(1,0,0),(0,1,0),(0,0,-k)>$ and thus $$ [\mathbb{Z}^n:L+L']=k,$$ which tells us that the answer to the question is yes in this case.

Thank you very much for your consideration!

(ps. I posted this question for $n=3$ at math.stackexchange.com a couple of days ago, but there were no takers, so I hope it is ok to post it here as well.)