## proof

Here is a simple direct proof.

The others being obvious, only (iv) needs proof.

The proof will only be indicated briefly.

We can assume that p is as close to q as is necessary for the following proof to work.

The proof follows very closely the proof of (2), except for the appearance of the factor x2.

The proof proper [= The actual proof] will consist of establishing the following statements in sequence.

The standard proofs proceed via the Cauchy formula.

An ingenious alternative proof, shorter but still complicated, can be found in [MR].

Kim announces that (by a tedious proof) the upper bound can be reduced to 10.

A close inspection of the proof reveals that......

This finishes <completes> the proof.

The method of proof carries over to domains satisfying......

This sort of proof will recur frequently in what follows.

We end this section by stating without proof an analogue of......

It seems reasonable to expect that......, but we have no proof of this.

a laborious <tedious/complicated/routine/straightforward/ingenious> proof

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