### Abstract

We formalise the proof of an important theorem in additive
combinatorics due to Khovanskii, attesting that the cardinality of the
set of all sums of $n$ many elements of $A$, where $A$ is a finite
subset of an abelian group, is a polynomial in $n$ for all
sufficiently large $n$. We follow a proof due to Nathanson and Ruzsa
as presented in the notes âIntroduction to Additive Combinatoricsâ by
Timothy Gowers for the University of Cambridge.