# Use of the adverb first in conjunction with then

Are the use and the positions of the adverbs first and then correct in the following two sentences?

We prove, first, two preliminary properties, and, then, the whole theorem.

We first show the formula, and then prove that its value is positive.

If the sentences are incorrect, or if they do not flow smoothly, how could they be fixed or improved?

Thanks.

Too many commas — old-fashioned. Try:

We prove first two preliminary properties and then the whole theorem.

First[,] we prove two preliminary properties and then the whole theorem.

You don't need comma before "and," as these are not independent clauses (they have the same subject, "we").

e.g.,

CRM Proceedings & Lecture Notes - Volumes 14-15 - Page 131 John P. Harnad, ‎Alex Kasman - 1998 -

... rigidity theorems, we solve for harmonic spinors which have the correct behavior at infinity and then integrate by parts. ... To prove Theorem 2, we prove first the existence of a harmonic spinor which is asymptotically parallel in the sense that ...