Proof

Obviously, this relationship holds at for arbitrary . So for , define a function in as

This function and its first derivatives must then be continuous over . Now at , we obtain that

which can also be written as or for all .

Also, or or .

This means that at the points . Then the Generalised Rolle's Theorem (see Calculus/Analysis) says that there exists a in ( ) for which .

Now,

Remember that is a polynomial of degree and that the last term is a polynomial in of degree , so that

Therefore,

which can be rewritten as

This is the result we were looking for