Now, if we assume that is a linear combination of for all then noting that
Allows us to conclude that is a linear combination of the and since each of these are a linear combination of the the induction follows.
From the fact that every element of a generating set of is a linear combination of allows us to conclude that is a generating set for . What we now claim though is that is a linearly indepdent set. Indeed, the basic idea is that if we could write
then this tells us that but this is ridiculous since .
From all of this we may conclude that is a basis for and so we may conclude that .
Thus, if and only if which by the above is true if and only if .
Remark: Why you should have thought this to be true was to look at and look at polynomials with and then use the CRT to conclude that .