how to prove that multiplication of points of an elliptic curve can be done with division polynomials

Question

I'm trying to solve an exercise in the book The Arithmetic of Elliptic Curves by Joseph H. Silverman on the page 106. The exercise asks to prove that \begin{equation} nP=\left( x-\frac{\psi_{n-1}\psi_{n+1}}{\psi_{n}^2},\frac{\psi_{n+2}\psi_{n-1}^2-\psi_{n-2}\psi_{n+1}^2}{4y \psi_n^3}\right) \end{equation} where P is a point of an elliptic curve E given by $y^2=x^3+ax+b$ and \begin{align*} \psi_0 & =0\\ \psi_1 & =1\\ \psi_2 & =2y\\ \psi_3 & =3x^4+6ax^2+12bx-a^2\\ \psi_4 & =4y(x^6+5ax^4+20bx^3-5a^2x^2-4abx-8b^2-a^3)\\ \psi_{2m+1} & =\psi_{m+2}\psi_{m}^3-\psi_{m-1}\psi_{m+1}^3 \ \ m\geqslant2\\ \psi_{2m} & =(2y)^{-1}(\psi_m)(\psi_{m+2}\psi_{m-1}^2-\psi_{m-2}\psi_{m+1}^2) \ m\geqslant3\\ \end{align*} Does anyone have an idea how to prove this? I tried to do some kind of recursive reasoning however I did not really succeed.