  We give a different and possibly more accessible proof of a general
Borsuk--Ulam theorem for a product of spheres, originally due to Ramos. That
is, we show the non-existence of certain $(\mathbb{Z}/2)^k$-equivariant maps
from a product of $k$ spheres to the unit sphere in a real
$(\mathbb{Z}/2)^k$-representation of the same dimension. Our proof method
allows us to derive Borsuk--Ulam theorems for certain equivariant maps from
Stiefel manifolds, from the corresponding results about products of spheres,
leading to alternative proofs and extensions of some results of Fadell and
Husseini.
