The theorem to be proved is the following. [= which will be proved]
The uniqueness of $f$ is easily proved, since ......
If $n=1$, there is nothing to prove.
However, this cannot be proved of the cardinal function $d(X)$.
Much to our surprise, even for $k=1$ the problem proved quite difficult.
This result will prove extremely useful in Section 2.
The conjecture will be disproved by exhibiting ......