Recursion Property of Elementary Symmetric Function

Theorem
Let $\set {z_1, z_2, \ldots, z_{n + 1} }$ be a set of $n + 1$ numbers, duplicate values permitted.

Then for $1 \le m \le n$:


 * $\map {e_m} {\set {z_1, \ldots, z_n, z_{n + 1} } } = z_{n + 1} \map {e_{m - 1} } {\set {z_1, \ldots, z_n} } + \map {e_m} {\set {z_1, \ldots, z_n} }$

where $\map {e_m} {\set {z_1, \ldots, z_n} }$ denotes the elementary symmetric function of degree $m$ on $\set {z_1, \ldots, z_n}$.