Faà di Bruno's Formula/Example/0/Proof

Theorem
Consider Faà di Bruno's Formula:

When $n = 0$ we have:

Proof
In the summation:
 * $\displaystyle \sum_{j \mathop = 0}^0 D_u^j w \sum_{\substack {\sum_{p \mathop \ge 0} k_p \mathop = j \\ \sum_{p \mathop \ge 0} p k_p \mathop = 0 \\ \forall p \ge 0: k_p \mathop \ge 0} } 0! \prod_{m \mathop = 0}^1 \dfrac {\paren {D_x^m u}^{k_m} } {k_m! \paren {m!}^{k_m} }$

the only element appearing is for $j = 0$, and the product is vacuous.

Thus:
 * $\displaystyle \prod_{m \mathop = 0}^1 \dfrac {\paren {D_x^m u}^{k_m} } {k_m! \paren {m!}^{k_m} } = 1$

and we are left with: