For simplicity sake, I will work with only two vector spaces. . And the more general case will follow easily and similarly.
I did not confirm this because of laziness, but I believe that if is a basis for , and is a basis for . Then, is not necessarily a basis for but it is a spanning set of vectors. And hence, since the size of the basis is less than the size of the spanning set we have,
Because is the size of spanning set and is the size (cardinality) of the dimension (basis) of those spanning vectors.