Nonempty topologically convex implies equiconnected

This article gives the statement and possibly, proof, of an implication relation between two topological space properties. That is, it states that every topological space satisfying the first topological space property (i.e., nonempty topologically convex space) must also satisfy the second topological space property (i.e., equiconnected space)
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Statement

Topological version

Any nonempty topologically convex space is an equiconnected space.

Realized version

Any nonempty convex subset of Euclidean space is (topologically) an equiconnected space.

Definitions used

Nonempty topologically convex space

Further information: nonempty topologically convex space

A topological space is called a nonempty topologically convex space if it is nonempty and homeomorphic to a convex subset of Euclidean space.

Convex subset of Euclidean space

Further information: convex subset of Euclidean space

A convex subset of Euclidean space is a subset in $\mathbb {R} ^{n}$ for some $n$, with the property that given any two points in the subset, the line segment joining those two points also lies completely within the subset.

The definition may also apply to infinite-dimensional Euclidean spaces, and the proof would work even in that case.