E { ∫ ... ∫ U$_{x}$$_{1}$$_{,y}$$_{1}$U$_{x}$$_{2}$$_{,y}$$_{2}$ [ y$_{1}$ψ ′ v $_{1}$( y$_{1}$ ) + ψ$_{v}$$_{1}$ ( y$_{1}$ )][ y$_{2}$ψ ′ v $_{2}$( y$_{2}$ ) + ψ$_{v}$$_{2}$ ( y$_{2}$ )] dy$_{1}$dy$_{2}$ · ∂ $^{d$_{x}$}$ψ ( F$_{x}$ ( x$_{1}$ )) d ( F$_{x}$ ( x$_{1}$ )) ∂ $^{d$_{x}$}$ψ ( F$_{x}$ ( x$_{2}$ )) d ( F$_{x}$ ( x$_{2}$ )) } = E$_{1}$ − E$_{1}$$_{,}$$_{2}$ with E$_{1}$ = { ∫ ... ∫ F ( x$_{1}$, y$_{1}$ )[ y$_{1}$ψ ′ v $_{1}$( y$_{1}$ ) + ψ$_{v}$$_{1}$ ( y$_{1}$ )] [∫ y 1 [ y$_{2}$ψ ′ v $_{2}$( y$_{2}$ ) + ψ$_{v}$$_{2}$ ( y$_{2}$ )] dy$_{2}$ ] dy$_{1}$ · ∂ $^{d$_{x}$}$ψ ( F$_{x}$ ( x$_{1}$ )) d ( F$_{x}$ ( x$_{1}$ )) ∫ x 1 ∂ $^{d$_{x}$}$ψ ( F$_{x}$ ( x$_{2}$ )) d ( F$_{x}$ ( x$_{2}$ )) } ˜ E$_{i}$ = ∫ ... ∫ F ( x$_{i}$, y$_{i}$ )[ y$_{i}$ψ ′ $_{v}$( y$_{i}$ ) + ψ$_{v}$ ( y$_{i}$ )] ∂ $^{d$_{x}$}$ψ ( F$_{x}$ ( x$_{i}$ )) d ( F$_{x}$ ( x$_{i}$ )) dy$_{i}$. ∫ ... ∫ F ( x, y )[ y $^{2}$ψ ′ v $_{1}$( y ) ψ$_{v}$$_{2}$ ( y ) + yψ$_{v}$$_{1}$ ( y ) ψ$_{v}$$_{2}$ ( y )] dy · 1 2 ∂ $^{d$_{x}$}$ψ $^{2}$( F$_{x}$ ( x )) dF ( x ) . ∣ ∫ ... ∫ F ( x, y )[ y $^{2}$ψ ′ v $_{1}$( y ) ψ$_{v}$$_{2}$ ( y ) + yψ$_{v}$$_{1}$ ( y ) ψ$_{v}$$_{2}$ ( y )] dy · 1 2 ∂ $^{d$_{x}$}$ψ $^{2}$( F$_{x}$ ( x )) dF ( x ) ∣ ≤ ˜ B 2 [ ( E$_{|}$$_{x}$ ( y 2 ) , ψ 2 ) + ∣ ( E$_{|}$$_{x}$y, ψ 2 )∣ ∣ ] . 42 Note that ψ 2 ∈ D ( W ) . By Assumption 5 then this contribution to the covariance is bounded. By construction of the partition of unity ∣ Σ ψ ′ v $_{1}$( y ) ψ$_{v}$$_{2}$ ( y ) ∣ as well as Σ ψ$_{v}$$_{1}$ ( y ) ψ$_{v}$$_{2}$ ( y ) are uniformly bounded, say both by some ˜ B. We get For E$_{1}$ integrating we get (dropping the subsript 1 on variables) and E$_{1}$$_{,}$$_{2}$ = ˜ E$_{1}$ ˜ E$_{2}$ where for i = 1 , 2