Recursion Property of Elementary Symmetric Function/Proof 2

Proof
Recall the definition of elementary symmetric function:

Consider the summands of $\map {e_m} {\set {z_1, z_2, \ldots, z_n, z_{n + 1} } }$:
 * $z_{j_1} z_{j_2} \cdots z_{j_m}$

where $1 \le j_1 < j_2 < \cdots j_m \le n + 1$.

They consist of $2$ types:
 * Type $(1)$: such that $j_m < n + 1$
 * Type $(2)$: such that $j_m = n + 1$.

We have that:
 * the summands of Type $(1)$ are exactly the summands of $\map {e_m} {\set {z_1, z_2, \ldots, z_n} }$
 * the summands of Type $(2)$ consist of the summands of $\map {e_{m - 1} } {\set {z_1, z_2, \ldots, z_n} }$ multiplied by $z_{n + 1}$.

Hence the result.