  Assuming projective determinacy, we extend Spector's strong version of the
Spector-Gandy Theorem to all odd levels of the projective hierarchy:
  Theorem. For every space $X$ which is a finite product of the natural numbers
$N$ and Baire space $N^N$ and for every n, if $P$ is a $\Pi^1_{2n+1}$ subset of
$X$, then there is a $\Pi^1_{2n}$ set $Q$ such that
  $P(x) \Longleftrightarrow (\exists!\alpha)Q(x,\alpha) \Longleftrightarrow
(\exists\alpha\in\Delta^1_{2n+1}(x))Q(x,\alpha)$.
