Recall the standard stratification of $X$. In particular, if we consider the subset $\mathcal{S}^{n-3}\subset X$ we have that $\dim\mathcal{S}^{n-3}\leq n-3$, and that for every point $x\not\in\mathcal{S}^{n-3}$ there exists some tangent cone at $x$ which is isometric to $\mathds{R}^{n-2}\times C(S^{1}_{\beta})$. That is, there exists $r_{a}\to 0$ such that

$\displaystyle(X,r_{a}^{-1}d,x)\to\mathds{R}^{n-2}\times C(S^{1}_{\beta})\,.$

(5.8)

However by Theorem 5.1 we then have $\beta=2\pi$, which is to say that

$\displaystyle(X,r_{a}^{-1}d,x)\to\mathds{R}^{n}\,.$

(5.9)

Thus, for $a\in\mathds{N}$ sufficiently large, we can apply the standard $\epsilon$-regularity theorem, Theorem 2.3, to see that a neighborhood of $x$ is a $C^{1,\alpha}$ Riemannian manifold, which proves the corollary. ∎