  The main result of this paper, Theorem 1.5, establishes a conjecture of Lyons
and Peres: for a determinantal point process governed by a reproducing kernel,
the system of kernels sampled at the particles of a random configuration is
complete in the range of the kernel. A key step in the proof, Lemma 1.11,
states that conditioning on the configuration in a subset preserves the
determinantal property, and the main Lemma 1.12 is a new local property for
kernels of conditional point processes. In Theorem 1.7 we prove the triviality
of the tail sigma-algebra for determinantal point processes governed by
self-adjoint kernels.
