  We consider bisingular pseudodifferential operators which are
pseudodifferential operators of tensor product type. These operators are
defined on the product manifold $M_1 \times M_2$, for closed manifolds $M_1$
and $M_2$. We prove a topological index theorem of product type. In addition,
we show that the Fredholm index of elliptic bisingular operators equals the
topological index, whenever the operator takes the form of an external tensor
product of pseudodifferential operators, up to equivalence. To this end we
construct a suitable double deformation groupoid and a Poincar\'e duality type
homomorphism.
