Determinantal formula for reproducing integral kernel How do I prove the following?
$$\int\det\left(K(x_{i},x_{j})\right)_{1\leq i,j\leq n}dx_{1} \cdots dx_{N}=\underset{i=1}{\overset{n}{\prod}}\left(\int K(x_{i},x_{i})\;dx_{i}-(i-1)\right)$$
where
$$K(x,y)=\sum_{l=1}^n \psi_l(x)\overline{\psi_l}(y)$$
and
$$\{\psi_l(x)\}_{l=1}^n$$
is an ON-sequence in $L^2$.
One may note that $$\int K(x_i,x_j)K(x_j,x_i) \; d\mu(x_i)=K(x_j,x_j)$$
and also that $$\int K(x_a,x_b)K(x_b,x_c)d\mu(x_b)=K(x_a,x_c).$$

Iterated use of Lemma 5.27 (p. 103) in Orthogonal polynomials and random matrices: a Riemann-Hilbert approach by Percy Deift. (Google Books link.)
