Equivalence of Definitions of Incomplete Elliptic Integral of the First Kind

Proof
Let $\map K {k, \phi}$ be the incomplete elliptic integral of the first kind by definition $1$.

Let $v := \sin \phi$.

Then we have:

So as $0 \le \phi \le \dfrac \pi 2$, we have that $0 \le v \le 1$.

Hence:

Thus $\map K {k, \phi}$ is the incomplete elliptic integral of the first kind by definition $2$.