description/proof of that for \(C^\infty\) manifold with boundary and \(C^\infty\) vectors field over manifold with boundary, integral curves that agree at parameters point agree over common parameters area
Topics
About: \(C^\infty\) manifold
The table of contents of this article
Starting Context
- The reader knows a definition of \(C^\infty\) manifold with boundary.
- The reader knows a definition of integral curve of \(C^\infty\) vectors field over \(C^\infty\) manifold with boundary.
- The reader knows a definition of map between arbitrary subsets of Euclidean \(C^\infty\) manifolds \(C^k\) at point, where \(k\) excludes \(0\) and includes \(\infty\).
- The reader admits the proposition that any \(C^1\) map from any open set on any Euclidean normed \(C^\infty\) manifold to any Euclidean normed \(C^\infty\) manifold satisfies the Lipschitz condition in any convex open set whose closure is bounded and contained in the original open set.
- The reader admits the proposition that the image of any continuous map from any compact topological space to the \(\mathbb{R}\) Euclidean topological space has the minimum and the maximum.
- The reader admits the local unique solution existence for a closed interval domain for a Euclidean-normed Euclidean vectors space ordinary differential equation with an initial condition with a clarification on the solution domain area.
- The reader admits the local uniqueness of solution for Euclidean-normed Euclidean vectors space ordinary differential equation with initial condition.
Target Context
- The reader will have a description and a proof of the proposition that for any \(C^\infty\) manifold with boundary and any \(C^\infty\) vectors field over the manifold with boundary, any integral curves that agree at any parameters point agree over the common parameters area.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\(M\): \(\in \{\text{ the } d \text{ -dimensional } C^\infty \text{ manifolds with boundary }\}\)
\((TM, M, \pi)\): \(= \text{ the tangent vectors bundle over } M\)
\(s\): \(\in \{\text{ the } C^\infty \text{ sections of } \pi\}\)
\(J_1\): \(\in \{\text{ the intervals of } \mathbb{R}\}\)
\(J_2\): \(\in \{\text{ the intervals of } \mathbb{R}\}\)
\(J\): \(= J_1 \cap J_2 \subseteq \mathbb{R}\)
\(\gamma_1\): \(: J_1 \to M\), \(\in \{\text{ the integral curves of } s\}\)
\(\gamma_2\): \(: J_2 \to M\), \(\in \{\text{ the integral curves of } s\}\)
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Statements:
\(\exists r_0 \in J (\gamma_1 (r_0) = \gamma_2 (r_0))\)
\(\implies\)
\(\gamma_1 \vert_J = \gamma_2 \vert_J\)
//
2: Proof
Whole Strategy: Step 0: suppose that \(J\) is not any 1 point interval hereafter; Step 1: when \(r_0\) is not in the interior of \(J\), take another \(r_0\) in the interior of \(J\); Step 2: suppose that \(J\) is not upper bounded; Step 3: suppose that \(\gamma_1 \vert_{[r_0, \infty)} \neq \gamma_2 \vert_{[r_0, \infty)}\), and find a contradiction; Step 4: suppose that \(J\) is upper bounded; Step 5: suppose that \(r' := Sup (\{r \in J \vert r_0 \le r \land \gamma_1 \vert_{[r_0, r]} = \gamma_2 \vert_{[r_0, r]}\})\) was not the upper boundary, and find a contradiction; Step 6: suppose that \(J\) is not lower bounded; Step 7: suppose that \(\gamma_1 \vert_{(- \infty, r_0]} \neq \gamma_2 \vert_{(- \infty, r_0]}\), and find a contradiction; Step 8: suppose that \(J\) is lower bounded; Step 9: suppose that \(r' := Inf (\{r \in J \vert r \le r_0 \land \gamma_1 \vert_{[r, r_0]} = \gamma_2 \vert_{[r, r_0]}\})\) was not the lower boundary, and find a contradiction; Step 10: conclude the proposition.
Step 0:
When \(J\) is any 1 point interval, the proposition obviously holds.
So, let us suppose that \(J\) is not any 1 point interval, hereafter.
Step 1:
When \(r_0\) is not in the interior of \(J\), let us take another \(r_0\) such that \(\gamma_1 (r_0) = \gamma_2 (r_0)\) in the interior of \(J\), as follows.
Let \(m_0 := \gamma_1 (r_0) = \gamma_2 (r_0)\).
\(r_0\) is the lower closed boundary of \(J\) or is the upper closed boundary of \(J\).
Let us suppose that \(r_0\) is the lower closed boundary of \(J\).
As \(J\) is not 1-point, there is a \(\delta'' \in \mathbb{R}\) such that \(0 \lt \delta''\) and \([r_0, r_0 + \delta'') \subseteq J \subseteq J_j\).
As \(\gamma_j\) is an integral curve of \(s\), over \([r_0, r_0 + \delta'')\), \(d \gamma_j / d r (r) = s (\gamma_j (r))\).
Let us take any chart around \(m_0\), \((U_{m_0} \subseteq M, \phi_{m_0})\), and the induced chart, \((\pi^{-1} (U_{m_0}) \subseteq TM, \widetilde{\phi_{m_0}})\): when \(m_0\) is any interior point of \(M\), let us take any interior chart around \(m_0\); when \(m_0\) is any boundary point of \(M\), we inevitably take any boundary chart around \(m_0\).
As \(\gamma_j\) is continuous, there is a \([r_0, r_0 + \delta') \subseteq [r_0, r_0 + \delta'')\) such that \(\gamma_1 ([r_0, r_0 + \delta')) \subseteq U_{m_0}\) and \(\gamma_2 ([r_0, r_0 + \delta')) \subseteq U_{m_0}\).
Over \([r_0, r_0 + \delta')\), \(\partial_1 (\phi_{m_0} \circ \gamma_j) (r) = \pi_1 \circ \widetilde{\phi_{m_0}} \circ s \circ {\phi_{m_0}}^{-1} (\phi_{m_0} \circ \gamma_j (r))\), where \(\pi_1: \widetilde{\phi_{m_0}} (\pi^{-1} (U_{m_0})) = \mathbb{R}^d \times \phi_{m_0} (U_{m_0}) \to \mathbb{R}^d\) is the projection.
Let \(f: \phi_{m_0} (U_{m_0}) \to \mathbb{R}^d, p \mapsto \pi_1 \circ \widetilde{\phi_{m_0}} \circ s \circ {\phi_{m_0}}^{-1} (p)\), which is \(C^\infty\), because \(s\) is \(C^\infty\).
When \(m_0\) is any interior point of \(M\), \(\phi_{m_0} (U_{m_0})\) is an open neighborhood of \(\phi_{m_0} (m_0)\) on \(\mathbb{R}^d\), and let us take any \(B_{\phi_{m_0} (m_0), K} \subseteq \mathbb{R}^d\) such that \(\overline{B_{\phi_{m_0} (m_0), K}} \subseteq \phi_{m_0} (U_{m_0})\) and \(g: B_{\phi_{m_0} (m_0), K} \to \mathbb{R}^d = f \vert_{B_{\phi_{m_0} (m_0), K}}\).
When \(m_0\) is any boundary point of \(M\), there is a \(C^\infty\) extension of \(f\), \(f': U'_{\phi_{m_0} (m_0)} \to \mathbb{R}^d\) where \(U'_{\phi_{m_0} (m_0)} \subseteq \mathbb{R}^d\) is an open neighborhood of \(\phi_{m_0} (m_0)\) on \(\mathbb{R}^d\), by the definition of map between arbitrary subsets of Euclidean \(C^\infty\) manifolds \(C^k\) at point, where \(k\) excludes \(0\) and includes \(\infty\), and let us take any \(B_{\phi_{m_0} (m_0), K} \subseteq \mathbb{R}^d\) such that \(\overline{B_{\phi_{m_0} (m_0), K}} \subseteq U'_{\phi_{m_0} (m_0)}\), and let us take \(g: B_{\phi_{m_0} (m_0), K} \to \mathbb{R}^d = f' \vert_{B_{\phi_{m_0} (m_0), K}}\).
As \(B_{\phi_{m_0} (m_0), K} \cap \phi_{m_0} (U_{m_0})\) is an open neighborhood of \(\phi_{m_0} (m_0)\) on \(\phi_{m_0} (U_{m_0})\) and \(\phi_{m_0} \circ \gamma_1 \vert_{[r_0, r_0 + \delta')}: [r_0, r_0 + \delta') \to \phi_{m_0} (U_{m_0})\) and \(\phi_{m_0} \circ \gamma_2 \vert_{[r_0, r_0 + \delta')}: [r_0, r_0 + \delta') \to \phi_{m_0} (U_{m_0})\) are continuous, there is a \([r_0, r_0 + \delta) \subseteq [r_0, r_0 + \delta')\) such that \(\phi_{m_0} \circ \gamma_1 ([r_0, r_0 + \delta)) \subseteq B_{\phi_{m_0} (m_0), K} \cap \phi_{m_0} (U_{m_0}) \subseteq B_{\phi_{m_0} (m_0), K}\) and \(\phi_{m_0} \circ \gamma_2 ([r_0, r_0 + \delta)) \subseteq B_{\phi_{m_0} (m_0), K} \cap \phi_{m_0} (U_{m_0}) \subseteq B_{\phi_{m_0} (m_0), K}\).
Anyway, over \([r_0, r_0 + \delta)\), \(\partial_1 (\phi_{m_0} \circ \gamma_j) (r) = g (\phi_{m_0} \circ \gamma_j (r))\) with the initial condition, \(\phi_{m_0} \circ \gamma_j (r_0) = \phi_{m_0} (m_0)\).
\(g\) is Lipschitz, by the proposition that any \(C^1\) map from any open set on any Euclidean normed \(C^\infty\) manifold to any Euclidean normed \(C^\infty\) manifold satisfies the Lipschitz condition in any convex open set whose closure is bounded and contained in the original open set.
\(g \le M'\) for an \(M' \in \mathbb{R}\) over \(B_{\phi_{m_0} (m_0), K}\), by the proposition that the image of any continuous map from any compact topological space to the \(\mathbb{R}\) Euclidean topological space has the minimum and the maximum: \(f \le M'\) or \(f' \le M'\) over \(\overline{B_{\phi_{m_0} (m_0), K}}\).
So, by the local unique solution existence for a closed interval domain for a Euclidean-normed Euclidean vectors space ordinary differential equation with an initial condition with a clarification on the solution domain area, there is the unique solution for the ordinary differential equation with initial condition over \([r_0 - \epsilon_1, r_0 + \epsilon_2]\), where \(0 \lt \epsilon_1, \epsilon_2\), \(\phi_{m_0} \circ \gamma: [r_0 - \epsilon_1, r_0 + \epsilon_2] \to B_{\phi_{m_0} (m_0), K}\).
Taking \(\epsilon_1 = 0\) and \(\epsilon_2 \lt \delta\) (we are free to choose any small enough \(\epsilon_1\) and \(\epsilon_2\)), \(\phi_{m_0} \circ \gamma\) is the unique solution, but as \(\phi_{m_0} \circ \gamma_1 \vert_{[r_0, r_0 + \epsilon_2]}\) and \(\phi_{m_0} \circ \gamma_2 \vert_{[r_0, r_0 + \epsilon_2]}\) are some solutions (as they are some solutions for \(f\), they are some solutions for \(g\)), \(\phi_{m_0} \circ \gamma = \phi_{m_0} \circ \gamma_1 \vert_{[r_0, r_0 + \epsilon_2]} = \phi_{m_0} \circ \gamma_2 \vert_{[r_0, r_0 + \epsilon_2]}\), which implies that \(\gamma = \gamma_1 \vert_{[r_0, r_0 + \epsilon_2]} = \gamma_2 \vert_{[r_0, r_0 + \epsilon_2]}\).
So, especially, \(\gamma_1 (r_0 + \epsilon_2 / 2) = \gamma_2 (r_0 + \epsilon_2 / 2)\).
So, let us take \(r_0 + \epsilon_2 / 2\) as new \(r_0\).
Let us suppose that \(r_0\) is the upper closed boundary of \(J\).
As \(J\) is not 1-point, there is a \(\delta'' \in \mathbb{R}\) such that \(0 \lt \delta''\) and \((r_0 - \delta'', r_0] \subseteq J \subseteq J_j\).
As \(\gamma_j\) is an integral curve of \(s\), over \((r_0 - \delta'', r_0]\), \(d \gamma_j / d r (r) = s (\gamma_j (r))\).
Let us take any chart around \(m_0\), \((U_{m_0} \subseteq M, \phi_{m_0})\), and the induced chart, \((\pi^{-1} (U_{m_0}) \subseteq TM, \widetilde{\phi_{m_0}})\): when \(m_0\) is any interior point of \(M\), let us take any interior chart around \(m_0\); when \(m_0\) is any boundary point of \(M\), we inevitably take any boundary chart around \(m_0\).
As \(\gamma_j\) is continuous, there is a \((r_0 - \delta', r_0] \subseteq (r_0 - \delta'', r_0]\) such that \(\gamma_1 ((r_0 - \delta', r_0]) \subseteq U_{m_0}\) and \(\gamma_2 ((r_0 - \delta', r_0]) \subseteq U_{m_0}\).
Over \((r_0 - \delta', r_0]\), \(\partial_1 (\phi_{m_0} \circ \gamma_j) (r) = \pi_1 \circ \widetilde{\phi_{m_0}} \circ s \circ {\phi_{m_0}}^{-1} (\phi_{m_0} \circ \gamma_j (r))\).
Let \(f: \phi_{m_0} (U_{m_0}) \to \mathbb{R}^d, p \mapsto \pi_1 \circ \widetilde{\phi_{m_0}} \circ s \circ {\phi_{m_0}}^{-1} (p)\), which is \(C^\infty\), because \(s\) is \(C^\infty\).
When \(m_0\) is any interior point of \(M\), \(\phi_{m_0} (U_{m_0})\) is an open neighborhood of \(\phi_{m_0} (m_0)\) on \(\mathbb{R}^d\), and let us take any \(B_{\phi_{m_0} (m_0), K} \subseteq \mathbb{R}^d\) such that \(\overline{B_{\phi_{m_0} (m_0), K}} \subseteq \phi_{m_0} (U_{m_0})\) and \(g: B_{\phi_{m_0} (m_0), K} \to \mathbb{R}^d = f \vert_{B_{\phi_{m_0} (m_0), K}}\).
When \(m_0\) is any boundary point of \(M\), there is a \(C^\infty\) extension of \(f\), \(f': U'_{\phi_{m_0} (m_0)} \to \mathbb{R}^d\) where \(U'_{\phi_{m_0} (m_0)} \subseteq \mathbb{R}^d\) is an open neighborhood of \(\phi_{m_0} (m_0)\) on \(\mathbb{R}^d\), by the definition of map between arbitrary subsets of Euclidean \(C^\infty\) manifolds \(C^k\) at point, where \(k\) excludes \(0\) and includes \(\infty\), and let us take any \(B_{\phi_{m_0} (m_0), K} \subseteq \mathbb{R}^d\) such that \(\overline{B_{\phi_{m_0} (m_0), K}} \subseteq U'_{\phi_{m_0} (m_0)}\), and let us take \(g: B_{\phi_{m_0} (m_0), K} \to \mathbb{R}^d = f' \vert_{B_{\phi_{m_0} (m_0), K}}\).
As \(B_{\phi_{m_0} (m_0), K} \cap \phi_{m_0} (U_{m_0})\) is an open neighborhood of \(\phi_{m_0} (m_0)\) on \(\phi_{m_0} (U_{m_0})\) and \(\phi_{m_0} \circ \gamma_1 \vert_{(r_0 - \delta', r_0]}: (r_0 - \delta', r_0] \to \phi_{m_0} (U_{m_0})\) and \(\phi_{m_0} \circ \gamma_2 \vert_{(r_0 - \delta', r_0]}: (r_0 - \delta', r_0] \to \phi_{m_0} (U_{m_0})\) are continuous, there is a \((r_0 - \delta, r_0] \subseteq (r_0 - \delta', r_0]\) such that \(\phi_{m_0} \circ \gamma_1 ((r_0 - \delta, r_0]) \subseteq B_{\phi_{m_0} (m_0), K} \cap \phi_{m_0} (U_{m_0}) \subseteq B_{\phi_{m_0} (m_0), K}\) and \(\phi_{m_0} \circ \gamma_2 ((r_0 - \delta, r_0]) \subseteq B_{\phi_{m_0} (m_0), K} \cap \phi_{m_0} (U_{m_0}) \subseteq B_{\phi_{m_0} (m_0), K}\).
Anyway, over \((r_0 - \delta, r_0]\), \(\partial_1 (\phi_{m_0} \circ \gamma_j) (r) = g (\phi_{m_0} \circ \gamma_j (r))\) with the initial condition, \(\phi_{m_0} \circ \gamma_j (r_0) = \phi_{m_0} (m_0)\).
\(g\) is Lipschitz, by the proposition that any \(C^1\) map from any open set on any Euclidean normed \(C^\infty\) manifold to any Euclidean normed \(C^\infty\) manifold satisfies the Lipschitz condition in any convex open set whose closure is bounded and contained in the original open set.
\(g \le M'\) for an \(M' \in \mathbb{R}\) over \(B_{\phi_{m_0} (m_0), K}\), by the proposition that the image of any continuous map from any compact topological space to the \(\mathbb{R}\) Euclidean topological space has the minimum and the maximum: \(f \le M'\) or \(f' \le M'\) over \(\overline{B_{\phi_{m_0} (m_0), K}}\).
So, by the local unique solution existence for a closed interval domain for a Euclidean-normed Euclidean vectors space ordinary differential equation with an initial condition with a clarification on the solution domain area, there is the unique solution for the ordinary differential equation with initial condition over \([r_0 - \epsilon_1, r_0 + \epsilon_2]\), where \(0 \lt \epsilon_1, \epsilon_2\), \(\phi_{m_0} \circ \gamma: [r_0 - \epsilon_1, r_0 + \epsilon_2] \to B_{\phi_{m_0} (m_0), K}\).
Taking \(\epsilon_1 \lt \delta\) and \(\epsilon_2 = 0\) (we are free to choose any small enough \(\epsilon_1\) and \(\epsilon_2\)), \(\phi_{m_0} \circ \gamma\) is the unique solution, but as \(\phi_{m_0} \circ \gamma_1 \vert_{[r_0 - \epsilon_1, r_0]}\) and \(\phi_{m_0} \circ \gamma_2 \vert_{[r_0 - \epsilon_1, r_0]}\) are some solutions (as they are some solutions for \(f\), they are some solutions for \(g\)), \(\phi_{m_0} \circ \gamma = \phi_{m_0} \circ \gamma_1 \vert_{[r_0 - \epsilon_1, r_0]} = \phi_{m_0} \circ \gamma_2 \vert_{[r_0 - \epsilon_1, r_0]}\), which implies that \(\gamma = \gamma_1 \vert_{[r_0 - \epsilon_1, r_0]} = \gamma_2 \vert_{[r_0 - \epsilon_1, r_0]}\).
So, especially, \(\gamma_1 (r_0 - \epsilon_1 / 2) = \gamma_2 (r_0 - \epsilon_1 / 2)\).
So, let us take \(r_0 - \epsilon_1 / 2\) as new \(r_0\).
So, hereafter, we suppose that \(r_0\) is in the interior of \(J\).
Step 2:
\(J\) may be not upper bounded or be upper bounded.
Let us suppose that \(J\) is not upper bounded.
Step 3:
Let us suppose that \(\gamma_1 \vert_{[r_0, \infty)} \neq \gamma_2 \vert_{[r_0, \infty)}\).
Let us define \(r' := Sup (\{r \in J \vert r_0 \le r \land \gamma_1 \vert_{[r_0, r]} = \gamma_2 \vert_{[r_0, r]}\})\), where the supremum was taken in \(\mathbb{R}\), which would exist as \(r' \in \mathbb{R}\), by the supposition.
\(r' \in Int (J)\), obviously.
\(\gamma_1 (r') = \gamma_2 (r')\), because if \(r' = r_0\), \(\gamma_1 (r') = \gamma_1 (r_0) = \gamma_2 (r_0) = \gamma_2 (r')\), and otherwise, if \(\gamma_1 (r') \neq \gamma_2 (r')\), there would be an open neighborhood of \(\gamma_1 (r')\), \(U_{\gamma_1 (r')} \subseteq M\), and an open neighborhood of \(\gamma_2 (r')\), \(U_{\gamma_2 (r')} \subseteq M\), such that \(U_{\gamma_1 (r')} \cap U_{\gamma_2 (r')} = \emptyset\), because \(M\) was Hausdorff, and there would be a \(B_{r', \delta} \subseteq (r_0, \infty)\) such that \(\gamma_1 (B_{r', \delta}) \subseteq U_{\gamma_1 (r')}\) and \(\gamma_2 (B_{r', \delta}) \subseteq U_{\gamma_2 (r')}\), because \(\gamma_j\) was continuous, which would mean that \(\gamma_1 (B_{r', \delta}) \cap \gamma_2 (B_{r', \delta}) = \emptyset\), which would mean that \(\gamma_1 (r' - \delta / 2) \neq \gamma_2 (r' - \delta / 2)\), a contradiction against that \(r'\) was the supremum.
So, let \(m' := \gamma_1 (r') = \gamma_2 (r')\).
There would be a chart around \(m'\), \((U_{m'} \subseteq M, \phi_{m'})\), and the induced chart, \((\pi^{-1} (U_{m'}) \subseteq TM, \widetilde{\phi_{m'}})\): when \(m'\) was an interior point, let \((U_{m'} \subseteq M, \phi_{m'})\) be an interior chart; when \(m'\) was a boundary point, \((U_{m'} \subseteq M, \phi_{m'})\) would be inevitably a boundary chart.
As \(\gamma_j \vert_J\) was continuous and \(r' \in Int (J)\), there would be a positive \(\delta'\) such that \(B_{r', \delta'} \subseteq J\) and \(\gamma_j (B_{r', \delta'}) \subseteq U_{m'}\). Then, \(\phi_{m'} \circ \gamma_j (B_{r', \delta'}) \subseteq \phi_{m'} (U_{m'})\).
As \(\gamma_j\) was an integral curve of \(s\), over \(B_{r', \delta'}\), \(d \gamma_j / d r (r) = s (\gamma_j (r))\), which was components-wise, \(\partial_1 (\phi_{m'} \circ \gamma_j) (r) = \pi_1 \circ \widetilde{\phi_{m'}} \circ s \circ {\phi_{m'}}^{- 1} \circ \phi_{m'} \circ \gamma_j (r)\), where \(\pi_1: \widetilde{\phi_{m'}} (\pi^{-1} (U_{m'})) = \mathbb{R}^d \times \phi_{m'} (U_{m'}) \to \mathbb{R}^d\) was the projection, with the initial condition, \(\phi_{m'} \circ \gamma_j (r') = \phi_{m'} (m')\), which was the ordinary differential equation for \(\phi_{m'} \circ \gamma_j\) with initial condition with \(f: \phi_{m'} (U_{m'}) \to \mathbb{R}^d, p \mapsto \pi_1 \circ \widetilde{\phi_{m'}} \circ s \circ {\phi_{m'}}^{- 1} (p)\), \(C^\infty\), because \(s\) was \(C^\infty\).
When \(m'\) was an interior point of \(M\), \(\phi_{m'} (U_{m'})\) would be an open neighborhood of \(\phi_{m'} (m')\) on \(\mathbb{R}^d\), and let us take any \(B_{\phi_{m'} (m'), K} \subseteq \mathbb{R}^d\) such that \(\overline{B_{\phi_{m'} (m'), K}} \subseteq \phi_{m'} (U_{m'})\) and \(g: B_{\phi_{m'} (m'), K} \to \mathbb{R}^d = f \vert_{B_{\phi_{m'} (m'), K}}\).
When \(m'\) was a boundary point of \(M\), there would be a \(C^\infty\) extension of \(f\), \(f': U'_{\phi_{m'} (m')} \to \mathbb{R}^d\) where \(U'_{\phi_{m'} (m')} \subseteq \mathbb{R}^d\) was an open neighborhood of \(\phi_{m'} (m')\) on \(\mathbb{R}^d\), by the definition of map between arbitrary subsets of Euclidean \(C^\infty\) manifolds \(C^k\) at point, where \(k\) excludes \(0\) and includes \(\infty\), and let us take any \(B_{\phi_{m'} (m'), K} \subseteq \mathbb{R}^d\) such that \(\overline{B_{\phi_{m'} (m'), K}} \subseteq U'_{\phi_{m'} (m')}\), and let us take \(g: B_{\phi_{m'} (m'), K} \to \mathbb{R}^d = f' \vert_{B_{\phi_{m'} (m'), K}}\).
As \(B_{\phi_{m'} (m'), K} \cap \phi_{m'} (U_{m'})\) was an open neighborhood of \(\phi_{m'} (m')\) on \(\phi_{m'} (U_{m'})\) and \(\phi_{m'} \circ \gamma_j \vert_{B_{r', \delta'}}: B_{r', \delta'} \to \phi_{m'} (U_{m'})\) was continuous, there would be a \(B_{r', \delta} \subseteq B_{r', \delta'}\) such that \(\phi_{m'} \circ \gamma_1 (B_{r', \delta}) \subseteq B_{\phi_{m'} (m'), K} \cap \phi_{m'} (U_{m'}) \subseteq B_{\phi_{m'} (m'), K}\) and \(\phi_{m'} \circ \gamma_2 (B_{r', \delta}) \subseteq B_{\phi_{m'} (m'), K} \cap \phi_{m'} (U_{m'}) \subseteq B_{\phi_{m'} (m'), K}\).
Then, over \(B_{r', \delta}\), \(\partial_1 (\phi_{m'} \circ \gamma_j) (r) = g (\phi_{m'} \circ \gamma_j (r))\) with the initial condition, \(\phi_{m'} \circ \gamma_j (r') = \phi_{m'} (m')\), which would be the ordinary differential equation for \(\phi_{m'} \circ \gamma_j\) with initial condition with \(g\).
\(g\) would be Lipschitz, by the proposition that any \(C^1\) map from any open set on any Euclidean normed \(C^\infty\) manifold to any Euclidean normed \(C^\infty\) manifold satisfies the Lipschitz condition in any convex open set whose closure is bounded and contained in the original open set.
So, by the local uniqueness of solution for Euclidean-normed Euclidean vectors space ordinary differential equation with initial condition, \(\phi_{m'} \circ \gamma_1 \vert_{B_{r', \delta}} = \phi_{m'} \circ \gamma_2 \vert_{B_{r', \delta}}\), which would mean that \(\gamma_1 \vert_{[r_0, r' + \delta / 2]} = \gamma_2 \vert_{[r_0, r' + \delta / 2]}\), a contradiction against that \(r'\) was the supremum.
So, \(\gamma_1 \vert_{[r_0, \infty)} = \gamma_2 \vert_{[r_0, \infty)}\).
Step 4:
Let us suppose that \(J\) is upper bounded with the upper boundary, \(r_2\).
Step 5:
Note that \(J\) may be upper open or upper closed.
What we will do is parallel to the case that \(J\) is not upper bounded.
Let us define \(r' := Sup (\{r \in J \vert r_0 \le r \land \gamma_1 \vert_{[r_0, r]} = \gamma_2 \vert_{[r_0, r]}\})\), where the supremum is taken in \(\mathbb{R}\), which exists as \(r' \in \mathbb{R}\), because \(J\) is upper bounded.
Let us suppose that \(r'\) was not the upper boundary, which would mean \(r' \lt r_2\).
\(r' \in Int (J)\) whether \(J\) is upper open or upper closed, obviously.
\(\gamma_1 (r') = \gamma_2 (r')\), because if \(r' = r_0\), \(\gamma_1 (r') = \gamma_1 (r_0) = \gamma_2 (r_0) = \gamma_2 (r')\), and otherwise, if \(\gamma_1 (r') \neq \gamma_2 (r')\), there would be an open neighborhood of \(\gamma_1 (r')\), \(U_{\gamma_1 (r')} \subseteq M\), and an open neighborhood of \(\gamma_2 (r')\), \(U_{\gamma_2 (r')} \subseteq M\), such that \(U_{\gamma_1 (r')} \cap U_{\gamma_2 (r')} = \emptyset\), because \(M\) was Hausdorff, and there would be a \(B_{r', \delta} \subseteq (r_0, r_2)\) such that \(\gamma_1 (B_{r', \delta}) \subseteq U_{\gamma_1 (r')}\) and \(\gamma_2 (B_{r', \delta}) \subseteq U_{\gamma_2 (r')}\), because \(\gamma_j\) was continuous, which would mean that \(\gamma_1 (B_{r', \delta}) \cap \gamma_2 (B_{r', \delta}) = \emptyset\), which would mean that \(\gamma_1 (r' - \delta / 2) \neq \gamma_2 (r' - \delta / 2)\), a contradiction against that \(r'\) was the supremum.
So, let \(m' := \gamma_1 (r') = \gamma_2 (r')\).
There would be a chart around \(m'\), \((U_{m'} \subseteq M, \phi_{m'})\), and the induced chart, \((\pi^{-1} (U_{m'}) \subseteq TM, \widetilde{\phi_{m'}})\): when \(m'\) was an interior point, let \((U_{m'} \subseteq M, \phi_{m'})\) be an interior chart; when \(m'\) was a boundary point, \((U_{m'} \subseteq M, \phi_{m'})\) would be inevitably a boundary chart.
As \(\gamma_j \vert_J\) was continuous and \(r' \in Int (J)\), there would be a positive \(\delta'\) such that \(B_{r', \delta'} \subseteq J\) and \(\gamma_j (B_{r', \delta'}) \subseteq U_{m'}\). Then, \(\phi_{m'} \circ \gamma_j (B_{r', \delta'}) \subseteq \phi_{m'} (U_{m'})\).
As \(\gamma_j\) was an integral curve of \(s\), over \(B_{r', \delta'}\), \(d \gamma_j / d r (r) = s (\gamma_j (r))\), which was components-wise, \(\partial_1 (\phi_{m'} \circ \gamma_j) (r) = \pi_1 \circ \widetilde{\phi_{m'}} \circ s \circ {\phi_{m'}}^{- 1} \circ \phi_{m'} \circ \gamma_j (r)\), where \(\pi_1: \widetilde{\phi_{m'}} (\pi^{-1} (U_{m'})) = \mathbb{R}^d \times \phi_{m'} (U_{m'}) \to \mathbb{R}^d\) was the projection, with the initial condition, \(\phi_{m'} \circ \gamma_j (r') = \phi_{m'} (m')\), which was the ordinary differential equation for \(\phi_{m'} \circ \gamma_j\) with initial condition with \(f: \phi_{m'} (U_{m'}) \to \mathbb{R}^d, p \mapsto \pi_1 \circ \widetilde{\phi_{m'}} \circ s \circ {\phi_{m'}}^{- 1} (p)\), \(C^\infty\), because \(s\) was \(C^\infty\).
When \(m'\) was an interior point of \(M\), \(\phi_{m'} (U_{m'})\) would be an open neighborhood of \(\phi_{m'} (m')\) on \(\mathbb{R}^d\), and let us take any \(B_{\phi_{m'} (m'), K} \subseteq \mathbb{R}^d\) such that \(\overline{B_{\phi_{m'} (m'), K}} \subseteq \phi_{m'} (U_{m'})\) and \(g: B_{\phi_{m'} (m'), K} \to \mathbb{R}^d = f \vert_{B_{\phi_{m'} (m'), K}}\).
When \(m'\) was a boundary point of \(M\), there would be a \(C^\infty\) extension of \(f\), \(f': U'_{\phi_{m'} (m')} \to \mathbb{R}^d\) where \(U'_{\phi_{m'} (m')} \subseteq \mathbb{R}^d\) was an open neighborhood of \(\phi_{m'} (m')\) on \(\mathbb{R}^d\), by the definition of map between arbitrary subsets of Euclidean \(C^\infty\) manifolds \(C^k\) at point, where \(k\) excludes \(0\) and includes \(\infty\), and let us take any \(B_{\phi_{m'} (m'), K} \subseteq \mathbb{R}^d\) such that \(\overline{B_{\phi_{m'} (m'), K}} \subseteq U'_{\phi_{m'} (m')}\), and let us take \(g: B_{\phi_{m'} (m'), K} \to \mathbb{R}^d = f' \vert_{B_{\phi_{m'} (m'), K}}\).
As \(B_{\phi_{m'} (m'), K} \cap \phi_{m'} (U_{m'})\) was an open neighborhood of \(\phi_{m'} (m')\) on \(\phi_{m'} (U_{m'})\) and \(\phi_{m'} \circ \gamma_j \vert_{B_{r', \delta'}}: B_{r', \delta'} \to \phi_{m'} (U_{m'})\) was continuous, there would be a \(B_{r', \delta} \subseteq B_{r', \delta'}\) such that \(\phi_{m'} \circ \gamma_1 (B_{r', \delta}) \subseteq B_{\phi_{m'} (m'), K} \cap \phi_{m'} (U_{m'}) \subseteq B_{\phi_{m'} (m'), K}\) and \(\phi_{m'} \circ \gamma_2 (B_{r', \delta}) \subseteq B_{\phi_{m'} (m'), K} \cap \phi_{m'} (U_{m'}) \subseteq B_{\phi_{m'} (m'), K}\).
Then, over \(B_{r', \delta}\), \(\partial_1 (\phi_{m'} \circ \gamma_j) (r) = g (\phi_{m'} \circ \gamma_j (r))\) with the initial condition, \(\phi_{m'} \circ \gamma_j (r') = \phi_{m'} (m')\), which would be the ordinary differential equation for \(\phi_{m'} \circ \gamma_j\) with initial condition with \(g\).
\(g\) would be Lipschitz, by the proposition that any \(C^1\) map from any open set on any Euclidean normed \(C^\infty\) manifold to any Euclidean normed \(C^\infty\) manifold satisfies the Lipschitz condition in any convex open set whose closure is bounded and contained in the original open set.
So, by the local uniqueness of solution for Euclidean-normed Euclidean vectors space ordinary differential equation with initial condition, \(\phi_{m'} \circ \gamma_1 \vert_{B_{r', \delta}} = \phi_{m'} \circ \gamma_2 \vert_{B_{r', \delta}}\), which would mean that \(\gamma_1 \vert_{[r_0, r' + \delta / 2]} = \gamma_2 \vert_{[r_0, r' + \delta / 2]}\), a contradiction against that \(r'\) was the supremum.
So, \(r'\) is the upper boundary, which means \(r' = r_2\).
That means that \(\gamma_1 \vert_{[r_0, r_2)} = \gamma_2 \vert_{[r_0, r_2)}\) or \(\gamma_1 \vert_{[r_0, r_2]} = \gamma_2 \vert_{[r_0, r_2]}\) according to whether \(J\) is upper open or is upper closed: \(\gamma_1 (r_2) = \gamma_2 (r_2)\), because otherwise, there would be an open neighborhood of \(\gamma_1 (r_2)\), \(U_{\gamma_1 (r_2)} \subseteq M\), and an open neighborhood of \(\gamma_2 (r_2)\), \(U_{\gamma_2 (r_2)} \subseteq M\), such that \(U_{\gamma_1 (r_2)} \cap U_{\gamma_2 (r_2)} = \emptyset\), and there would be a \((r_2 - \delta, r_2] \subseteq J\) such that \(\gamma_1 ((r_2 - \delta, r_2]) \subseteq U_{\gamma_1 (r_2)}\) and \(\gamma_2 ((r_2 - \delta, r_2]) \subseteq U_{\gamma_2 (r_2)}\), which would mean that \(\gamma_1 (r_2 - \delta / 2) \neq \gamma_2 (r_2 - \delta / 2)\), a contradiction against \(\gamma_1 \vert_{[r_0, r_2)} = \gamma_2 \vert_{[r_0, r_2)}\).
Step 6:
For the area smaller than \(r_0\), the logic is parallel.
\(J\) may be not lower bounded or be lower bounded.
Let us suppose that \(J\) is not lower bounded.
Step 7:
Let us suppose that \(\gamma_1 \vert_{(- \infty, r_0]} \neq \gamma_2 \vert_{(- \infty, r_0]}\).
Let us define \(r' := Inf (\{r \in J \vert r \le r_0 \land \gamma_1 \vert_{[r, r_0]} = \gamma_2 \vert_{[r, r_0]}\})\), where the infimum was taken in \(\mathbb{R}\), which would exist as \(r' \in \mathbb{R}\), by the supposition.
\(r' \in Int (J)\), obviously.
\(\gamma_1 (r') = \gamma_2 (r')\), because if \(r' = r_0\), \(\gamma_1 (r') = \gamma_1 (r_0) = \gamma_2 (r_0) = \gamma_2 (r')\), and otherwise, if \(\gamma_1 (r') \neq \gamma_2 (r')\), there would be an open neighborhood of \(\gamma_1 (r')\), \(U_{\gamma_1 (r')} \subseteq M\), and an open neighborhood of \(\gamma_2 (r')\), \(U_{\gamma_2 (r')} \subseteq M\), such that \(U_{\gamma_1 (r')} \cap U_{\gamma_2 (r')} = \emptyset\), because \(M\) was Hausdorff, and there would be a \(B_{r', \delta} \subseteq (- \infty, r_0)\) such that \(\gamma_1 (B_{r', \delta}) \subseteq U_{\gamma_1 (r')}\) and \(\gamma_2 (B_{r', \delta}) \subseteq U_{\gamma_2 (r')}\), because \(\gamma_j\) was continuous, which would mean that \(\gamma_1 (B_{r', \delta}) \cap \gamma_2 (B_{r', \delta}) = \emptyset\), which would mean that \(\gamma_1 (r' + \delta / 2) \neq \gamma_2 (r' + \delta / 2)\), a contradiction against that \(r'\) was the infimum.
So, let \(m' := \gamma_1 (r') = \gamma_2 (r')\).
There would be a chart around \(m'\), \((U_{m'} \subseteq M, \phi_{m'})\), and the induced chart, \((\pi^{-1} (U_{m'}) \subseteq TM, \widetilde{\phi_{m'}})\): when \(m'\) was an interior point, let \((U_{m'} \subseteq M, \phi_{m'})\) be an interior chart; when \(m'\) was a boundary point, \((U_{m'} \subseteq M, \phi_{m'})\) would be inevitably a boundary chart.
As \(\gamma_j \vert_J\) was continuous and \(r' \in Int (J)\), there would be a positive \(\delta'\) such that \(B_{r', \delta'} \subseteq J\) and \(\gamma_j (B_{r', \delta'}) \subseteq U_{m'}\). Then, \(\phi_{m'} \circ \gamma_j (B_{r', \delta'}) \subseteq \phi_{m'} (U_{m'})\).
As \(\gamma_j\) was an integral curve of \(s\), over \(B_{r', \delta'}\), \(d \gamma_j / d r (r) = s (\gamma_j (r))\), which was components-wise, \(\partial_1 (\phi_{m'} \circ \gamma_j) (r) = \pi_1 \circ \widetilde{\phi_{m'}} \circ s \circ {\phi_{m'}}^{- 1} \circ \phi_{m'} \circ \gamma_j (r)\) with the initial condition, \(\phi_{m'} \circ \gamma_j (r') = \phi_{m'} (m')\), which was the ordinary differential equation for \(\phi_{m'} \circ \gamma_j\) with initial condition with \(f: \phi_{m'} (U_{m'}) \to \mathbb{R}^d, p \mapsto \pi_1 \circ \widetilde{\phi_{m'}} \circ s \circ {\phi_{m'}}^{- 1} (p)\), \(C^\infty\), because \(s\) was \(C^\infty\).
When \(m'\) was an interior point of \(M\), \(\phi_{m'} (U_{m'})\) would be an open neighborhood of \(\phi_{m'} (m')\) on \(\mathbb{R}^d\), and let us take any \(B_{\phi_{m'} (m'), K} \subseteq \mathbb{R}^d\) such that \(\overline{B_{\phi_{m'} (m'), K}} \subseteq \phi_{m'} (U_{m'})\) and \(g: B_{\phi_{m'} (m'), K} \to \mathbb{R}^d = f \vert_{B_{\phi_{m'} (m'), K}}\).
When \(m'\) was a boundary point of \(M\), there would be a \(C^\infty\) extension of \(f\), \(f': U'_{\phi_{m'} (m')} \to \mathbb{R}^d\) where \(U'_{\phi_{m'} (m')} \subseteq \mathbb{R}^d\) was an open neighborhood of \(\phi_{m'} (m')\) on \(\mathbb{R}^d\), by the definition of map between arbitrary subsets of Euclidean \(C^\infty\) manifolds \(C^k\) at point, where \(k\) excludes \(0\) and includes \(\infty\), and let us take any \(B_{\phi_{m'} (m'), K} \subseteq \mathbb{R}^d\) such that \(\overline{B_{\phi_{m'} (m'), K}} \subseteq U'_{\phi_{m'} (m')}\), and let us take \(g: B_{\phi_{m'} (m'), K} \to \mathbb{R}^d = f' \vert_{B_{\phi_{m'} (m'), K}}\).
As \(B_{\phi_{m'} (m'), K} \cap \phi_{m'} (U_{m'})\) was an open neighborhood of \(\phi_{m'} (m')\) on \(\phi_{m'} (U_{m'})\) and \(\phi_{m'} \circ \gamma_j \vert_{B_{r', \delta'}}: B_{r', \delta'} \to \phi_{m'} (U_{m'})\) was continuous, there would be a \(B_{r', \delta} \subseteq B_{r', \delta'}\) such that \(\phi_{m'} \circ \gamma_1 (B_{r', \delta}) \subseteq B_{\phi_{m'} (m'), K} \cap \phi_{m'} (U_{m'}) \subseteq B_{\phi_{m'} (m'), K}\) and \(\phi_{m'} \circ \gamma_2 (B_{r', \delta}) \subseteq B_{\phi_{m'} (m'), K} \cap \phi_{m'} (U_{m'}) \subseteq B_{\phi_{m'} (m'), K}\).
Then, over \(B_{r', \delta}\), \(\partial_1 (\phi_{m'} \circ \gamma_j) (r) = g (\phi_{m'} \circ \gamma_j (r))\) with the initial condition, \(\phi_{m'} \circ \gamma_j (r') = \phi_{m'} (m')\), which would be the ordinary differential equation for \(\phi_{m'} \circ \gamma_j\) with initial condition with \(g\).
\(g\) would be Lipschitz, by the proposition that any \(C^1\) map from any open set on any Euclidean normed \(C^\infty\) manifold to any Euclidean normed \(C^\infty\) manifold satisfies the Lipschitz condition in any convex open set whose closure is bounded and contained in the original open set.
So, by the local uniqueness of solution for Euclidean-normed Euclidean vectors space ordinary differential equation with initial condition, \(\phi_{m'} \circ \gamma_1 \vert_{B_{r', \delta}} = \phi_{m'} \circ \gamma_2 \vert_{B_{r', \delta}}\), which would mean that \(\gamma_1 \vert_{[r' - \delta / 2, r_0]} = \gamma_2 \vert_{[r' - \delta / 2, r_0]}\), a contradiction against that \(r'\) was the infimum.
So, \(\gamma_1 \vert_{(- \infty, r_0]} = \gamma_2 \vert_{(- \infty, r_0]}\).
Step 8:
Let us suppose that \(J\) is lower bounded with the lower boundary, \(r_1\).
Step 9:
Note that \(J\) may be lower open or lower closed.
What we will do is parallel to the case that \(J\) is not lower bounded.
Let us define \(r' := Inf (\{r \in J \vert r \le r_0 \land \gamma_1 \vert_{[r, r_0]} = \gamma_2 \vert_{[r, r_0]}\})\), where the infimum is taken in \(\mathbb{R}\), which exists as \(r' \in \mathbb{R}\), because \(J\) is lower bounded.
Let us suppose that \(r'\) was not the lower boundary, which would mean \(r_1 \lt r'\).
\(r' \in Int (J)\) whether \(J\) is lower open or lower closed, obviously.
\(\gamma_1 (r') = \gamma_2 (r')\), because if \(r' = r_0\), \(\gamma_1 (r') = \gamma_1 (r_0) = \gamma_2 (r_0) = \gamma_2 (r')\), and otherwise, if \(\gamma_1 (r') \neq \gamma_2 (r')\), there would be an open neighborhood of \(\gamma_1 (r')\), \(U_{\gamma_1 (r')} \subseteq M\), and an open neighborhood of \(\gamma_2 (r')\), \(U_{\gamma_2 (r')} \subseteq M\), such that \(U_{\gamma_1 (r')} \cap U_{\gamma_2 (r')} = \emptyset\), because \(M\) was Hausdorff, and there would be a \(B_{r', \delta} \subseteq (r_1, r_0)\) such that \(\gamma_1 (B_{r', \delta}) \subseteq U_{\gamma_1 (r')}\) and \(\gamma_2 (B_{r', \delta}) \subseteq U_{\gamma_2 (r')}\), because \(\gamma_j\) was continuous, which would mean that \(\gamma_1 (B_{r', \delta}) \cap \gamma_2 (B_{r', \delta}) = \emptyset\), which would mean that \(\gamma_1 (r' + \delta / 2) \neq \gamma_2 (r' + \delta / 2)\), a contradiction against that \(r'\) was the infimum.
So, let \(m' := \gamma_1 (r') = \gamma_2 (r')\).
There would be a chart around \(m'\), \((U_{m'} \subseteq M, \phi_{m'})\), and the induced chart, \((\pi^{-1} (U_{m'}) \subseteq TM, \widetilde{\phi_{m'}})\): when \(m'\) was an interior point, let \((U_{m'} \subseteq M, \phi_{m'})\) be an interior chart; when \(m'\) was a boundary point, \((U_{m'} \subseteq M, \phi_{m'})\) would be inevitably a boundary chart.
As \(\gamma_j \vert_J\) was continuous and \(r' \in Int (J)\), there would be a positive \(\delta'\) such that \(B_{r', \delta'} \subseteq J\) and \(\gamma_j (B_{r', \delta'}) \subseteq U_{m'}\). Then, \(\phi_{m'} \circ \gamma_j (B_{r', \delta'}) \subseteq \phi_{m'} (U_{m'})\).
As \(\gamma_j\) was an integral curve of \(s\), over \(B_{r', \delta'}\), \(d \gamma_j / d r (r) = s (\gamma_j (r))\), which was components-wise, \(\partial_1 (\phi_{m'} \circ \gamma_j) (r) = \pi_1 \circ \widetilde{\phi_{m'}} \circ s \circ {\phi_{m'}}^{- 1} \circ \phi_{m'} \circ \gamma_j (r)\), where \(\pi_1: \widetilde{\phi_{m'}} (\pi^{-1} (U_{m'})) = \mathbb{R}^d \times \phi_{m'} (U_{m'}) \to \mathbb{R}^d\) was the projection, with the initial condition, \(\phi_{m'} \circ \gamma_j (r') = \phi_{m'} (m')\), which was the ordinary differential equation for \(\phi_{m'} \circ \gamma_j\) with initial condition with \(f: \phi_{m'} (U_{m'}) \to \mathbb{R}^d, p \mapsto \pi_1 \circ \widetilde{\phi_{m'}} \circ s \circ {\phi_{m'}}^{- 1} (p)\), \(C^\infty\), because \(s\) was \(C^\infty\).
When \(m'\) was an interior point of \(M\), \(\phi_{m'} (U_{m'})\) would be an open neighborhood of \(\phi_{m'} (m')\) on \(\mathbb{R}^d\), and let us take any \(B_{\phi_{m'} (m'), K} \subseteq \mathbb{R}^d\) such that \(\overline{B_{\phi_{m'} (m'), K}} \subseteq \phi_{m'} (U_{m'})\) and \(g: B_{\phi_{m'} (m'), K} \to \mathbb{R}^d = f \vert_{B_{\phi_{m'} (m'), K}}\).
When \(m'\) was a boundary point of \(M\), there would be a \(C^\infty\) extension of \(f\), \(f': U'_{\phi_{m'} (m')} \to \mathbb{R}^d\) where \(U'_{\phi_{m'} (m')} \subseteq \mathbb{R}^d\) was an open neighborhood of \(\phi_{m'} (m')\) on \(\mathbb{R}^d\), by the definition of map between arbitrary subsets of Euclidean \(C^\infty\) manifolds \(C^k\) at point, where \(k\) excludes \(0\) and includes \(\infty\), and let us take any \(B_{\phi_{m'} (m'), K} \subseteq \mathbb{R}^d\) such that \(\overline{B_{\phi_{m'} (m'), K}} \subseteq U'_{\phi_{m'} (m')}\), and let us take \(g: B_{\phi_{m'} (m'), K} \to \mathbb{R}^d = f' \vert_{B_{\phi_{m'} (m'), K}}\).
As \(B_{\phi_{m'} (m'), K} \cap \phi_{m'} (U_{m'})\) was an open neighborhood of \(\phi_{m'} (m')\) on \(\phi_{m'} (U_{m'})\) and \(\phi_{m'} \circ \gamma_j \vert_{B_{r', \delta'}}: B_{r', \delta'} \to \phi_{m'} (U_{m'})\) was continuous, there would be a \(B_{r', \delta} \subseteq B_{r', \delta'}\) such that \(\phi_{m'} \circ \gamma_1 (B_{r', \delta}) \subseteq B_{\phi_{m'} (m'), K} \cap \phi_{m'} (U_{m'}) \subseteq B_{\phi_{m'} (m'), K}\) and \(\phi_{m'} \circ \gamma_2 (B_{r', \delta}) \subseteq B_{\phi_{m'} (m'), K} \cap \phi_{m'} (U_{m'}) \subseteq B_{\phi_{m'} (m'), K}\).
Then, over \(B_{r', \delta}\), \(\partial_1 (\phi_{m'} \circ \gamma_j) (r) = g (\phi_{m'} \circ \gamma_j (r))\) with the initial condition, \(\phi_{m'} \circ \gamma_j (r') = \phi_{m'} (m')\), which would be the ordinary differential equation for \(\phi_{m'} \circ \gamma_j\) with initial condition with \(g\).
\(g\) would be Lipschitz, by the proposition that any \(C^1\) map from any open set on any Euclidean normed \(C^\infty\) manifold to any Euclidean normed \(C^\infty\) manifold satisfies the Lipschitz condition in any convex open set whose closure is bounded and contained in the original open set.
So, by the local uniqueness of solution for Euclidean-normed Euclidean vectors space ordinary differential equation with initial condition, \(\phi_{m'} \circ \gamma_1 \vert_{B_{r', \delta}} = \phi_{m'} \circ \gamma_2 \vert_{B_{r', \delta}}\), which would mean that \(\gamma_1 \vert_{[r' - \delta / 2, r_0]} = \gamma_2 \vert_{[r' - \delta / 2, r_0]}\), a contradiction against that \(r'\) was the infimum.
So, \(r'\) is the lower boundary, which means \(r' = r_1\).
That means that \(\gamma_1 \vert_{(r_1, r_0]} = \gamma_2 \vert_{(r_1, r_0]}\) or \(\gamma_1 \vert_{[r_1, r_0]} = \gamma_2 \vert_{[r_1, r_0]}\) according to whether \(J\) is lower open or is lower closed: \(\gamma_1 (r_1) = \gamma_2 (r_1)\), because otherwise, there would be an open neighborhood of \(\gamma_1 (r_1)\), \(U_{\gamma_1 (r_1)} \subseteq M\), and an open neighborhood of \(\gamma_2 (r_1)\), \(U_{\gamma_2 (r_1)} \subseteq M\), such that \(U_{\gamma_1 (r_1)} \cap U_{\gamma_2 (r_1)} = \emptyset\), and there would be a \([r_1, r_1 + \delta) \subseteq J\) such that \(\gamma_1 ([r_1, r_1 + \delta)) \subseteq U_{\gamma_1 (r_1)}\) and \(\gamma_2 ([r_1, r_1 + \delta)) \subseteq U_{\gamma_2 (r_1)}\), which would mean that \(\gamma_1 (r_1 + \delta / 2) \neq \gamma_2 (r_1 + \delta / 2)\), a contradiction against \(\gamma_1 \vert_{(r_1, r_0]} = \gamma_2 \vert_{(r_1, r_0]}\).
Step 10:
So, \(\gamma_1 \vert_J = \gamma_2 \vert_J\).
