Without loss of generality $p=0$. Let $w_{0}$ be any first order deformation. By the maximality of $K$, we can choose real numbers $a_{i}$, such that the first order deformation $w_{0}-\sum_{1}^{K}a_{i}v_{i}$ vanishes at zero, so $w_{0}-\sum_{1}^{K}a_{i1}v_{i}=z^{k_{1}}w_{1}$ for some first order deformation $w_{1}$ which is nonzero at the origin. Finite dimensionality means this process can be repeated for only a finite number of times:

$$\begin{cases}w_{0}=\sum a_{i1}v_{i}+z^{k_{1}}w_{1},\\ w_{1}=\sum a_{i2}v_{i}+z^{k_{2}}w_{2},\\ \ldots\\ w_{N-1}=\sum a_{iN}v_{i}+z^{k_{N}}w_{N}.\end{cases}$$

We choose the smallest $N$ such that $v_{1},\ldots v_{K},w_{0},\ldots w_{N}$ are $\mathbb{R}$-linearly dependent as vector fields; notice $v_{1},\ldots v_{K}$ are linearly independent, so $N\geq 0$. We then get a linear relation

$$f_{0}(z)w_{N}=\sum_{1}^{K}f_{i}(z)v_{i},$$

where $f_{0},\ldots f_{K}$ are polynomials, and $f_{0}(0)\neq 0$. This implies the claim. ∎