In an article by Lubich, I came across a decomposition for points on the straight line between two points lying in an embedded submanifold $M$ of $R^{n}$.

To be precise, it is proposed that for $X$, $\tilde X \in M$ and $\tau$ small enough, any point $X + \tau(\tilde X -X)$ may be decomposed into $$ X + \tau(\tilde X -X) = Y(\tau) + Z(\tau)$$ with $Y \in M$ and $Z \bot T_XM$, the tangent space at $X$.

To me, this looks like an application of some implicit function theorem or something, but as I am no expert in manifold theory, I just cannot get my head around it. Is there anyone who can help?