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 $\mathbb{Z}^n$.

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.

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.)