If all order of normal derivatives on some nbhd of point $x_0$ is same then all derivatives on the boundary at $x_0$ is also same.

Question

I was reading https://doi.org/10.1002/cpa.3160370302. In that author wanted to prove $$D^k\gamma_1(x_0)=D^k\gamma(x_0)$$ for all $k=(k_1,\cdots,k_n)$. But he says that it is enough to show for each $k$ $$\left(\frac{\partial}{\partial \nu}\right)^k\gamma_1(x)=\left(\frac{\partial}{\partial \nu}\right)^k\gamma_2(x)$$ for all $x$ on boundary in some nbhd of $x_0$.

I could not able to argue above fact.

Any help or hint will be greatly appreciated.