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N$_{1}$ ≡ N (Λ , ζ$_{1}$, ζ$_{2}$ ) such that for n > N$_{1}$ 

Consider Λ that satisfies inf Λ | ε$_{1}$ | > 0 . Set ζ$_{1}$ ≤ inf Λ | ε$_{1}$ | , then 

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Under 6(c’) for a singular γ$_{g}$ there exists a deterministic sequence of regular functions, ( γ$_{g}$ ) n that converges to γ$_{g}$ in S ∗ and such that support of ( γ$_{g}$ ) n is in a compact set Λ$_{gn}$ that is a proper subset of support of γ ; it can be selected such that for some sequence ζ$_{gn}$ → 0 the Lebesgue measure of Λ$_{gn}$ , λ (Λ$_{gn}$) < ζ$_{gn}$ ; Λ$_{gn}$$_{1}$ ⊂ Λ$_{gn}$$_{2}$ for n$_{1}$ > n$_{2}$ and on any Λ$_{gn}$ for some fixed ζ$_{0}$ we get inf Λ$_{gn}$ ∣ ( γ$_{g}$ ) n ∣ > ζ$_{0}$. For the corresponding sequence ( ε$_{g}$ )$_{n}$ = ( γ$_{g}$ ) n φ on Λ$_{gn}$ positive lower bounds on modulus exist and are no less than ζ$_{0}$ inf Λ$_{gn}$ | φ | . 

Under 6(c’) consider the sequence of deterministic functions ( ε$_{g}$ )$_{n}$ = ( γ$_{g}$ ) n φ and the difference ( ε$_{L}$$_{1}$ )$_{n}$ = ε$_{1}$ − ( ε$_{g}$ )$_{n}$ . This is a sequence of piece-wise deterministic continuous functions that converges in S ∗ to the continuous function ε$_{1}$$_{L}$$_{1}$ = γ$_{L}$$_{1}$φ. Then on a compact set Λ for ζ$_{1}$ ≤ min ( inf Λ | ε$_{1}$$_{L}$$_{1}$ | , ζ$_{0}$ inf Λ$_{gn}$ | φ | ) we can find the corresponding N so that for ˆ ε$_{1}$$_{n}$ = ˆ ε$_{1}$$_{n}$ − ( ε$_{g}$ )$_{n}$ + ( ε$_{g}$ )$_{n}$ 

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