What do mathematicians mean by "push through a proof"?

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I sometimes hear mathematicians say they want to "push through a proof". What does this mean?

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It isn't an idiom, it applies the usual figurative meaning of "push through" (force passage when it is hard) to a proof. As in Hawkins, The Mathematics of Frobenius in Context, p. 265:

"Frobenius was able to prove the answer was affirmative... but it turned out far more difficult for him to prove than the above proposition. To push through a proof he was forced to develop a “mod-$k$” analogue of his reduction theorem and its corollary on transformation to normal form for any positive integer $k$..."