Equivalence of Definitions of Locally Path-Connected Space/Definition 2 implies Definition 1

Theorem
Let $T = \struct{S, \tau}$ be a topological space. Let each point of $T$ have a neighborhood basis consisting of path-connected sets in $T$.

Then
 * each point of $T$ has a local basis consisting entirely of path-connected sets in $T$.

Proof
Let $T = \struct{S, \tau}$ be locally path-connected by Definition 2:
 * each point of $T$ has a neighborhood basis consisting of path-connected sets of $S$ in $T$.

Let $x \in S$ and $x \in U \in \tau$.

By definition of local basis, we have to show that there exists a path-connected open set $V\in\tau$ with $x\in V\subset U$.

Let $V = \map {\operatorname{PC}_x} T$ be the path component of $x$ in $U$.

By definition of topological subspace, it suffices to show that $V$ is open in $U$.

By Set is Open iff Neighborhood of all its Points, we may do this by showing that $V$ is a neighborhood in $U$ of all of its points.

Let $y \in V$.

By assumption, there exists a path-connected neighborhood $W$ of $y$ in $T$ with $W\subset U$.

By Neighborhood in Topological Subspace, $W$ is a neighborhood of $y$ in $U$.

By definition of path component:
 * $W \subset \map {\operatorname{PC}_y} U = \map {\operatorname{PC}_x} U = V$

By Neighborhood iff Contains Neighborhood, $V$ is a neighborhood of $y$ in $U$.

Because $y$ was arbitrary, $V$ is open in $U$.

Because $U$ was arbitrary, $x$ has a local basis of path-connected (open) sets.