The "elementary symmetric polynomials" $\sigma_{n;k}$ are: $$\sigma_{n;k}(x_1,\ldots,x_n)=\sum_{1\le i_1\lt\cdots\lt i_k\le n}x_{i_1}\cdots x_{i_k}$$ where $k=1,\ldots,n$. (...) Every polynomial $p$ in $n$ variables can be written as a finite sum of monomials $c\cdot x_1^{a_1}\cdot\ldots\cdot x_n^{a_n}$ with $c\ne 0$. We sort these polynomials lexicographic by exponent, comparing at first only the exponents of $x_1$, then those of $x_2$ etc. With the "leading monomial" $\operatorname{LM}(p)$ we denote the lexicographically largest monomial of $p$.
(This is taken from https://arxiv.org/abs/2003.14035v2).
On page 4, the following is used without proof: $$\operatorname{LM}\left(\prod_{1\le k\le n}\sigma_{n;k}\right)=\prod_{1\le k\le n}\operatorname{LM}(\sigma_{n;k})$$ and $$\operatorname{LM}(\sigma_{n;k}^a)=\operatorname{LM}(\sigma_{n;k})^a$$ where $a\in\mathbb{N}$ and $\sigma_{n;k}=\sigma_{n;k} (x_1,\ldots,x_n)$.
To prove it, I tried to use induction. The base case is trivial and for the induction step I got $$\operatorname{LM}\left(\prod_{1\le k\le n+1}\sigma_{n+1;k}\right)=\operatorname{LM}\left(\prod_{1\le k\le n+1}\left(\sigma_{n;k}+\prod_{1\le r\le n+1}x_r\right)\right)$$ but I'm stuck there.