Let $\phi:[a,b] \to \mathbb{H}$ be a path in $\mathbb{H}$ such that $t \to x(t) + y(t)i$
We define the length of $\phi$ by $lon(\phi) := \int_a^b \frac{1}{y} \sqrt{(x')^2 + (y')^2}dt$.
I want to prove that this longth does not depent of injective parametrizations.
MY IDEA: Let $\epsilon: [a,b] \to \mathbb{H} $ such that $t \to x_{\epsilon}(t) + y_{\epsilon}(t)i$ and
$\rho: [c,d] \to \mathbb{H} $ such that $t \to x_{\rho}(t) + y_{\rho}(t)i$
be parametrizations of $\mathcal{C}$, then I have:
$1)$ $\epsilon([a,b]) = \rho([c,d]) = \mathcal{C}$
$2)$ $x_{\epsilon}(a) = x_{\rho}(c)$, $y_{\epsilon}(a) = y_{\rho}(c)$, $x_{\epsilon}(b) = x_{\rho}(d)$ and $y_{\epsilon}(b) = y_{\rho}(d)$
I tried to calculate $\int_a^b \frac{1}{y_{\epsilon}} \sqrt{(x_{\epsilon}')^2 + (y_{\epsilon}')^2}dt$ and to finish replace by $2)$, however I not obtain a good result
Some Hint Pliss