Sentence examples for the following proof from inspiring English sources

The idea of the following proof is followed by the second author [15].

For completeness, we give the following proof.

The following proof is similar to (i).

The following proof is divided into two steps.

The following proof is divided into two parts.

The following proof will be divided into two steps.

The following proof will be decomposed into two cases.

The following proof is based on the approach of [14].

For the sake of completeness, we give the following proof.

The following proof for the assertion is similar and omitted.

We assume that (nge5) in the following proof.