  Assume <aleph_0,aleph_1>-><lambda,lambda^+>. Assume M is a model of a first
order theory T of cardinality at most lambda^+ in a vocabulary L(T) of
cardinality <= lambda . Let N be a model with the same vocabulary. Let Delta be
a set of first order formulas in L(T) and let D be a regular filter on lambda.
Then M is Delta-embeddable into the reduced power N^lambda/D, provided that
every Delta-existential formula true in M is true also in N. We obtain the
following corollary: for M as above and D a regular ultrafilter over lambda,
M^lambda/D is lambda^{++}-universal. Our second result is as follows: For i<mu
let M_i and N_i be elementarily equivalent models of a vocabulary which has has
cardinality <=lambda. Suppose D is a regular filter on mu and
<aleph_0,aleph_1>-><lambda,lambda^+> holds. We show that then the second player
has a winning strategy in the Ehrenfeucht-Fraisse game of length lambda^+ on
prod_i M_i/D and prod_i N_i/D. This yields the following corollary: Assume GCH
and lambda regular). For L, M_i and N_i as above, if D is a regular filter on
lambda, then prod_i M_i/D cong prod_i N_i/D .
