Jordan Curve and Jordan Arc form Two Jordan Curves/Corollary

Corollary to Jordan Curve and Jordan Arc form Two Jordan Curves
Let $\left[{a \,.\,.\, b}\right]$ denote the closed real interval between $a \in \R, b \in \R: a \le b$.

Let $\gamma: \left[{a \,.\,.\, b}\right] \to \R^2$ be a Jordan curve.

Let the interior of $\gamma$ be denoted $\operatorname{Int} \left({\gamma}\right)$.

Let the image of $\gamma$ be denoted $\operatorname{Im} \left({\gamma}\right)$.

Let $\sigma: \left[{c \,.\,.\, d}\right] \to \R^2$ be a Jordan arc such that:
 * $\sigma \left({c}\right) \ne \sigma \left({d}\right)$
 * $\sigma \left({c}\right), \sigma \left({d}\right) \in \operatorname{Im} \left({\gamma}\right)$

and:
 * $\forall t \in \left({c \,.\,.\, d}\right): \sigma \left({t}\right) \in \operatorname{Int} \left({\gamma}\right)$

Let $t_1 = \gamma^{-1} \left({ \sigma \left({c}\right) }\right)$.

Let $t_2 = \gamma^{-1} \left({ \sigma \left({d}\right) }\right)$.

Let $t_1 < t_2$.

Define:
 * $-\sigma: \left[{c \,.\,.\, d}\right] \to \operatorname{Im} \left({\sigma}\right)$ by $-\sigma \left({t}\right) = \sigma \left({c + d - t}\right)$

Let $*$ denote concatenation of paths.

Let $\gamma \restriction_{\left[{a \,.\,.\, t_1}\right] }$ denote the restriction of $\gamma$ to $\left[{a \,.\,.\, t_1}\right]$.

Define:
 * $\gamma_1 = \gamma {\restriction_{\left[{a \,.\,.\, t_2}\right] } } * \left({-\sigma }\right) * \gamma{\restriction_{ \left[{t_1 \,.\,.\, b}\right] } }$

Define:
 * $\gamma_2 = \gamma {\restriction_{\left[{t_2 \,.\,.\, t_1}\right] } } * \sigma$

Then $\gamma_1$ and $\gamma_2$ are Jordan curves such that:
 * $\operatorname{Int} \left({\gamma_1}\right) \subseteq \operatorname{Int} \left({\gamma}\right)$

and:
 * $\operatorname{Int} \left({\gamma_2}\right) \subseteq \operatorname{Int} \left({\gamma}\right)$

Proof
This is proved in the same way as Jordan Curve and Jordan Arc form Two Jordan Curves.