Lemma 19. If (A4), (A5) and (A6) hold, then there exists a function θ ( ξ ) , that to each ξ in R qp × vec ( S $_{q}$) assigns a pair ( B , Σ ) in R q × p × S $_{q}$, and a bounded subset C of R qp × vec ( S $_{q}$) , such that ( B$_{0}$ , σ 2 $_{0}$Σ$_{0}$ ) ∈ θ ( C ) $^{◦}$, for which each of the classes of functions where φ$_{kj}$ ( z ; θ ) = W ( d ( B , Σ )) ( y$_{k}$ − b ′ $_{k}$x ) x$_{j}$ and b$_{k}$ is the k th column vector of the matrix B , is Euclidean for certain envelope F$_{kj}$ with E$_{H}$$_{0}$F 2 kj < ∞ . Proof: For each ξ in R qp × vec( S $_{q}$) there exists a unique pair ( B , Σ ) in R q × p × S q such that ξ = ( vec( B ) $^{′}$, vec( Σ − 1 / $^{2}$) ′ ) , then define the function θ ( · ) as follows: θ (( vec( B ) $^{′}$, vec( Σ − 1 / $^{2}$) ′ )) = ( B , Σ ) . − 1 − 1 / 2 Let ε > 0 and δ = 2 ‖ σ 0 Σ 0 ‖ , considering the norm defined in (A.5), we denote by B ε to the ball of radius ε and center ( B$_{0}$ , σ 2 $_{0}$Σ$_{0}$ ), then define Let ξ and ξ ∗ be any two elements of C such that θ ( ξ ) = ( B , Σ ) and θ ( ξ $^{∗}$) = ( B $^{∗}$, Σ $^{∗}$), by the Mean Value Theorem there is a value c between d ( B , Σ ) and d ( B $^{∗}$, Σ $^{∗}$) such that Since W and its derivative are continuous and with compact support there exists a constant M such that | W ( u ) | ≤ M and | W $^{′}$( u ) | ≤ M for all u , using this and 52 | W ( d ( B $^{∗}$, Σ $^{∗}$)) − W ( d ( B , Σ )) | = | W $^{′}$( c ) || d ( B , Σ ) − d ( B $^{∗}$, Σ $^{∗}$) | . (A.51) C = θ − 1 ({ ( B , Σ ) ∈ R q × p × S q : ( B , Σ ) ∈ B ε and ‖ Σ − 1 / $^{2}$‖ ≤ δ }) . F$_{kj}$ = { φ$_{kj}$ ( z ; θ ( ξ )) : ξ ∈ C } , (A.50)