# Prove $S_{m}^{m}(\Delta)$={$s:s\in C^{m}[a,b]$ and $s$ is a polynomial of order $m$ in each $[x_{i},x_{i+1}]$}=$P_{m}$

Basically, I don't understand clearly. The point is to prove that these two spaces are equal or that the polynomial $$s \in S_{m}^{m}(\Delta)$$ is unique/the same in each $$[x_{i},x_{i+1}]$$? where $$(\Delta)$$ is a segmentation of [a,b]

Let $$s_{i-1}$$ be the polynomial in the $$[x_{i-1},x_{i}]$$ region of $$[a,b]$$ and $$s_{i}$$ the polynomial in the $$[x_{i},x_{i+1}]$$. Then by the continuity of all the $$m$$ derivatives of $$s$$ in the internal nodes of $$[a,b]$$ we have that:
$$s_{i-1}^{(m)}(x_{i}) = c_{i-1}\quad$$ and$$\quad$$ $$s_{i} ^{(m)}(x_{i}) = c_{i}$$ $$\quad$$ $$\rightarrow$$ $$\quadc_{i-1}= c_{i}$$
$$s_{i-1}^{(m-1)}(x_{i}) =a_{i-1}x_{i}+c_{i}\quad$$ and$$\quad$$ $$s_{i} ^{(m-1)}(x_{i}) = a_{i}x_{i}+c_{i}\quad\rightarrow\quad$$ $$a_{i-1}=a_{i}$$
So we have that $$s_{i-1}=s_{i}$$ i.e. we have that $$s$$ is the same polynomial in each $$[x_{i},x_{i+1}]$$.