On page 294, Andrews, Askey and Roy - Special functions. For sequences of (independent) functions $\lbrace \phi(x) \rbrace_{n=0}^{\infty}$ and $\lbrace \psi(x) \rbrace_{n=0}^{\infty}$, a sequence $\lbrace p_n(x) \rbrace_{n=0}^{\infty}$ of functions is introduced, where
\begin{equation} p_n(x)=\begin{vmatrix} \mu_{0,0} & \mu_{0,1} & ... & \mu_{0,n} \\ \mu_{0,0} & \mu_{1,1} & ... & \mu_{1,n} \\ ... & ... & & ... \\ \mu_{n-1,0} & \mu_{n-1,1} & ... & \mu_{n-1,n} \\ \phi_0 & \phi_1 & ... & \phi_n \end{vmatrix}, \quad \mu_{i,j}=\int_{a}^{b} \psi_i \phi_j d\alpha(x), \end{equation}
with respect to some measure $\alpha$ such that the integrals exist. A corollary of theirs is that
$$ \int_{a}^{b} \psi_m(x) p_n(x) d\alpha(x)=0, \quad m \leq n-1. $$
They go on to show that, for the choice of $\alpha'(x)=(1-x)^{\alpha}(1+x)^{\beta}$ over $(-1,1)$, with $\phi_k(x)=(1-x)^k$ and $\psi_k(x)=(1+x)^k$, the polynomials $p_n(x)$ are constant multiples of the Jacobi polynomials $$ P_n^{(\alpha,\beta)}(x)=\frac{(\alpha+1)_n}{n!}F(-n,n+\beta+1+\alpha;\alpha+1;\frac{1-x}{2}). $$
My question: Does the orthogonality of the Jacobi polynomials follow from the above? I cannot see how. May add that this is not explicitly stated by the authors, but little is, and I would have thought this would be one of the reasons for giving the result. Can also add that the corollary follows from the corresponding result for $\psi=\phi$.