In the paper "A reflection phenomenon in descriptive set theory" by J.P. Burgess, it is stated:

Moschovakis [11] has proved the following important Uniform Boundedness Theorem: There is a continuous function \Gamma: Y \rightarrow Z such that for all y \in Y, if U_y \subseteq W, then \Gamma(y) \in W and z \prec \Gamma(y) for all z \in U_y,

where Y=2^(\omega \times \omega), Z is the set of codes for linear orders on \omega and W=WO.

But the reference [11] hadn't yet appeared and I'm able to find only the Boundedness Theorem in [11], but not the uniform version. How can we show this theorem? Have you ever seen this uniform version somewhere?