  If we have an ultrafilter U over I, then we can construct a reduced product from C by making two elements of C equivalent if and only if the set of indices at which they are equal is a set in the ultrafilter U. This is indeed an equivalence relation on the domain of C, and the resulting reduced product is called an ultraproduct of the factors of C. If C is a power of A then this ultraproduct is called an ultrapower of A, and it is sometimes written U-prod A. A theorem called Łoś’s theorem describes what sentences are true in an ultraproduct. Its most useful consequence is the following: 