and T 2 jj = For every vector h and s = 1 , 2 It follows by substituting into T 2 jj and expanding ψ$_{s}$ that T 2 jj = − Eψ$_{1}$ ( x ) Eψ$_{2}$ ( x )+ ¯ hR$_{2}$. Similarly, where after the change of variable ψ$_{s}$ ( x$_{j}$ − hw ) is expanded around the point x$_{j}$. Next we establish that | R$_{1}$ | < ∞ , | R$_{2}$ | < ∞ . 34 T 1 jj = ∫ (∫ 1 Π h$_{i}$ K ( x$_{j}$ − x h ) ψ$_{1}$ ( x ) dx ) (∫ 1 Π h$_{i}$ K ( x$_{j}$ − x h ) ψ$_{2}$ ( x ) dx ) dF ( x$_{j}$ ) = K ( w ) ψ$_{1}$ ( x$_{j}$ − hw ) dw ∫ K ( w ) ψ$_{2}$ ( x$_{j}$ − hw ) dw dF ( x$_{j}$ ) = = ∫ (∫ K ( w ) dwψ$_{1}$ ( x$_{j}$ ) − ¯ h ∫ K ( w ) [ Σ k i =1 ∂ψ$_{1}$ ∂x$_{i}$ ( x$_{j}$ − h ˜ w ) w$_{i}$ h$_{i}$ ¯ $_{h}$] dw ) × K ( w ) dwψ$_{2}$ ( x$_{j}$ ) − ¯ h ∫ K ( w ) [ Σ k i =1 ∂ψ$_{2}$ ∂x$_{i}$ ( x$_{j}$ − h ˜ w ) w$_{i}$ h$_{i}$ ¯ dw ) dF ( x$_{j}$ ) = Eψ$_{1}$ ( x ) ψ$_{2}$ ( x ) + ¯ hR$_{1}$ ; ∫ 1 Π h$_{i}$ K ( x$_{j}$ − x h ) ψ$_{s}$ ( x ) dx = ∫ K ( w ) ψ$_{s}$ ( x$_{j}$ − hw ) dw. − ∫ [ ∫ 1 Π h$_{i}$ K ( x$_{j}$ − x h ) ψ$_{1}$ ( x ) dx ] dF ( x$_{j}$ ) × ( f, ψ$_{1}$ ) − [ 1 Π h$_{i}$ K ( x$_{j}$ − x h ) ψ$_{2}$ ( x ) dx ] dF ( x$_{j}$ ) × ( f, ψ$_{2}$ ) +( f, ψ$_{1}$ ) × ( f, ψ$_{2}$ ) .