[see also: show, establish, demonstrate, evidence, substantiate, validate]

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 ......

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