I am reading the paper here and have a small doubt in Lemma 1. The proof (on page 1) begins with:
Lemma 1. Let $M$ and $N$ be cardinal numbers. Let $S$ be $N$-regular in $X$, a set of cardinality $M$, and let $T$ be $N^M$-regular in $Y$. Then if $S\otimes T$ is N-regular in $X\times Y$.
Proof: Let $P$ be a set of cardinality $N$. Then a partition of $X \times Y$ into $N$ parts can be represented by a function from $X \times Y$ into $P$. For each $y \in Y$, $f$ defines a function $f_y$ from $X$ into $P$ given by $f_y(x) = f(x,y)$. Since there are $N^M$ such functions the mapping $y \rightarrow f_y$ induces a partition of $Y$ into $N^M$ parts.
My doubt is that the proof should not the proof say "Since there are at least $N^M$ such functions the mapping $y \rightarrow f_y$ induces a partition of $Y$ into at least $N^M$ parts." since each of the $N^M$ functions from $X$ to $P$ may correspond to more then 1 $f_y$.
If my doubt is correct (and I expect that the proof will still go through) then should small omissions like this not be entirely unexpected in research papers. I am asking this because I have little experience in reading research papers.