Discrete algebraic Riccati equation: If Q and R are positive definite is P positive definite?

Question

Suppose I am trying to solve the discrete algebraic Riccati equation:

$P=Q_1+A^TPA-(B^TPA)^T(Q_2+B^TPB)^{-1}(B^TPA)$

with $Q_1=Q_1^T>0$ and $Q_2=Q_2^T>0$. How can I formally prove that $P>0$?

Thank you very much