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The r.h.s. of (12) provides 

<img src='content_image/315060.jpg'>

and writing this in more concise notation 

<img src='content_image/315059.jpg'>

Since f$_{x}$ is continuous, then ˜ ψ ( x ) = ψ ( F ( x )) f ( x ) is continuous on R $^{d$_{x}$}$. For an arbitrary ˜ ψ ∈ D$_{0}$ ( R d$_{x}$ ) consider 

Do the transformation, then 

<img src='content_image/315057.jpg'>

<img src='content_image/315058.jpg'>

Define a continuous function ψ ( F$_{x}$$_{1}$, ..., F$_{x}$$_{d}$$_{x}$ ) = ˜ ψ ( x 1 ,...,x$_{d}$$_{x}$ ) f$_{x}$ ( x 1 ,...,x$_{d}$$_{x}$ ) on (0 , 1) d$_{x}$ , then this equals (13) . 

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Suppose now that F$_{x}$ is absolutely continuous with continuous density function, f$_{y}$$_{|}$$_{x}$ ; then the support of the density function is an open set in R $^{d$_{x}$}$, S$_{y}$$_{|}$$_{x}$. The Lemma applies by considering ˜ ψ ( x ) = f$_{x}$ ( x ) ψ ( F ( x )) ∈ D$_{0}$ ( S$_{y}$$_{|}$$_{x}$ ) 

