Talk:Arcsine as Integral/Lemma 1

Here is the proof as I have it now.

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

Consider $\int_{0}^{x}\frac{1}{\sqrt{1-x^2} }\d{x}$

Let $x=\sin_A\left(\Theta\right)\iff x=\arcsin_A\left(\Theta\right)$

I would like to change this proof so it reads as follows:

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

But in establishing the above, only the analytic sine and Arc Sine, $\sin_A$, and $\arcsin_A$ should be used. This avoids circular reasoning in proving Derivative of Sine Function in the geometric case.

The problem is, that if I made this change, I am not sure that the reader would understand what I am saying and why. It may be the original version, even if longer and more cumbersome, may be better understood. I would welcome suggestions and help here.