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

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

Then
 * each point of $T$ has a neighborhood basis consisting of connected sets of $T$.

Proof
From Local Basis for Open Sets Implies Neighborhood Basis of Open Sets, it follows directly that:
 * each point of $T$ has a neighborhood basis consisting entirely of connected sets in $T$.