Arcsine as Integral/Lemma 1

Lemma
Let $sin_A$ be the analytic sine function for real numbers, the one defined by Definition:Sine/Real Numbers. $\arcsin_A$ is the inverse of this function.


 * $\displaystyle \map {\arcsin_A} x = \int_0^x \frac {\d x} {\sqrt {1 - x^2} }$

Proof
For this proof only, let $\sin_A$ be the analytic sine function from Definition:Sine/Real Numbers.

Consider:
 * $\displaystyle \int_0^x \frac {\d x} {\sqrt {1 - x^2} }$

Let:
 * $x = \sin_A \theta \iff x = \map {\arcsin_A} \theta$

Then: