Syzygies and tensor product of modules

We give an application of the New Intersection Theorem and prove the following: let $R$ be a local complete intersection ring of codimension $c$ and let $M$ and $N$ be nonzero finitely generated $R$ -modules. Assume $n$ is a nonnegative integer and that the tensor product $M\otimes _{R}N$ is an $(n+c)$ th syzygy of some finitely generated $R$ -module. If ${{\mathrm{Tor}}}^{R}_{>0}(M,N)=0$ , then both $M$ and $N$ are $n$ th syzygies of some finitely generated $R$ -modules.