In Connected Smooth Manifold Any Two Points can be Joined by Admissible Curve

Theorem
Let $M$ be a connected smooth manifold with or without a boundary.

Let $p, q \in M$ be points.

Let $\gamma : \closedint a b \to M$ be an admissible curve.

Then:


 * $\forall p, q \in M : \exists \gamma \subset M : \paren {\map \gamma a = p} \land \paren {\map \gamma b = q}$