Differentiating more, it's easily seen that
Where on the left side we have differentiations and on the right side are polynomials of degree .
As such, form a basis to the vector space of polynomials of degree at most . Therefore,
is a linear combination of
up to times differentiation.
But since the zero of at is of order , we have that all the functions in (1) evaluated at vanish, and thus their linear combination evaluated at that point,