I am asking for a rigorous proof of the following:
Theorem: Let $R$ be a region and $C$ be a simple closed curve so that $C=\partial R$. If $\gamma:[a,b]\to \mathbb{R}^2$ is parametrization of $C$ then $\gamma$ is either positive or negative orientiated with respect to $R$.
Definition of positive orientation: Define $n(t)=(-y^{\prime}(t),x^{\prime}(t))$ where $\gamma(t)=(x(t),y(t))$. Then $\gamma$ is positive orientiated with respect to $R$ if $\forall t\in [a,b]\ \exists \epsilon>0$ so that $\gamma(t)+\epsilon n(t)\in R$. Negative orientation is defined similarly but with $n(t)=(y^{\prime}(t),-x^{\prime}(t))$.
Note that I request a proof that doesn't utilise the Jordan Curve Theorem and this is the reason the definitions and the theorem are stated this way.