riemannian geometry - Injectivity radius and the cut locus

Consider a connected, complete and compact Riemannian manifold $M$. Is it correct that the following equality holds: $text{inj}(x)=text{dist}left(x,text{CuL}(x)right)$? Or in words that the injectivity radius of a point is the distance from the point to its cut locus.

Here is my explanation: As the manifold is compact and complete, then the cut locus $text{CuL}(x)$ is compact as well[1]. Thus, there exists a point $yin text{CuL}(x)$ such that $text{dist}left(x,text{CuL}(x)right)=text{dist}(x,y)$. Since $y$ is a cut point of $x$, there exists a tangent vector $xi_0in T_x M$ such that $y=exp_xleft(c(xi_0)xi_0right)$[2], where $c(xi_0)$ is the distance from $x$ to its cut point in the $xi_0$ direction. This in turn means that $text{dist}(x,y) = c(xi_0)$.

Recall that $text{inj}(x)=inf_{xiin T_x M}(c(xi))$. This means that $text{inj}(x) leq c(xi_0) = text{dist}(x,y)=text{dist}left(x,text{CuL}(x)right)$. If $text{inj}(x)< c(xi_0)$, then since $M$ is compact, it means that there exists some other tangent vector $xiin T_x M$ with $c(xi) < c(xi_0)$. But this means that $exp_x(c(xi)xi)$ is a cut point of $x$ closer to it then $y$, and this is a contradiction.

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[1] See Contributions to Riemannian Geometry in the Large by W. Klingenberg

[2] Here I'm using the notation of I. Chavel in his book Riemannian Geometry - Modern Introduction.

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Update(@dror)
Today I finally found a copy of the book *Riemannian Geometry" by Takashi Sakai, and there the above is stated as proposition 4.13 in chapter 3.
Thanks anyway.