= Solution
<Gaudin's lemma> is the <projection kernel determinant integration> identity. Suppose an <integral kernel> $K$ satisfies
$$
\int K(x,y)K(y,z)\,d\mu(y)=K(x,z),
\qquad \int K(x,x)\,d\mu(x)=N,
$$
with the relevant integrals absolutely convergent. Define $D_k(x_1,\ldots,x_k)=\det[K(x_i,x_j)]_{i,j=1}^k$ and $D_0=1$. Then
$$
\boxed{\int D_k(x_1,\ldots,x_k)\,d\mu(x_k)
=(N-k+1)D_{k-1}(x_1,\ldots,x_{k-1}).}
$$
In particular, a <finite-rank projection kernel> of rank $N$ has $\int D_N\,d\mu^N=N!$.
One can see the factor directly from the <permutation> expansion of the <determinant>. Terms whose <permutation> fixes $k$ integrate to $ND_{k-1}$. Every <permutation> on the other $k-1$ indices admits $k-1$ insertions of $k$ into a cycle. Each insertion reverses its sign, and the reproducing identity contracts its two adjacent kernel factors into one. These terms integrate to $-(k-1)D_{k-1}$, proving the formula.
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