  The gap probability generating function has as its coefficients the
probability of an interval containing exactly $k$ eigenvalues. For scaled
random matrices with orthogonal symmetry, and the interval at the hard or soft
spectrum edge, the gap probability generating functions have the special
property that they can be evaluated in terms of Painlev\'e transcendents. The
derivation of these results makes use of formulas for the same generating
function in certain scaled, superimposed ensembles expressed in terms of its
correlation functions. It is shown that by a judicious choice of the
superimposed ensembles, the scaled limit necessary to derive these formulas can
be rigorously justified by a straight forward analysis.
