Thus, the value of sup ( ϕ,ψ ) ∈ Φ$_{c}$ J ( ϕ, ψ ) is unchanged even if one restricts the supremum to pairs of the form which implies that for each y$_{1}$ ∈$_{R}$ , Define $^{{}$A + k,D ( ϕ, s ) } ∞ k =0 , $^{{}$A − k,D ( ϕ, s ) } ∞ k =0 as follows: Also, according to the definitions above and Figure A.1, if y$_{1}$ ∈ A + ρ,D ( ϕ, s ) for some ρ ≥ 0 , then A + k,D ( ϕ, s ) = $^{{}$y$_{1}$ ∈$_{R}$|∃ y$_{0}$ ∈ A + k ( ϕ, s ) s.t. y$_{1}$ − y$_{0}$ ≥ δ and ( y$_{0}$, y$_{1}$ ) ∈ C } ∪ $^{{}$y$_{1}$ ∈$_{R}$|∃ y$_{0}$ ∈ A + k $_{+1}$( ϕ, s ) s.t. y$_{1}$ − y$_{0}$ < δ and ( y$_{0}$, y$_{1}$ ) ∈ C } for any integer k ≥ 0 , (A.6) A − 0 ,D ( ϕ, s ) = $^{{}$y$_{1}$ ∈$_{R}$|∀ y$_{0}$ ≤ y$_{1}$ − δ s.t. ( y$_{0}$, y$_{1}$ ) ∈ C, y$_{0}$ ∈ A − 0 ( ϕ, s ) } ∩ $^{{}$y$_{1}$ ∈$_{R}$|∀ y$_{0}$ > y$_{1}$ − δ s.t. ( y$_{0}$, y$_{1}$ ) ∈ C, y$_{0}$ ∈ $^{(}$A + 0 ( ϕ, s ) c $^{}}$, A − k,D ( ϕ, s ) = $^{{}$y$_{1}$ ∈$_{R}$|∀ y$_{0}$ ≤ y$_{1}$ − δ s.t. ( y$_{0}$, y$_{1}$ ) ∈ C, y$_{0}$ ∈ A − k ( ϕ, s ) } ∩ $^{{}$y$_{1}$ ∈$_{R}$|∀ y$_{0}$ > y$_{1}$ − δ s.t. ( y$_{0}$, y$_{1}$ ) ∈ C, y$_{0}$ ∈ A − k − $_{1}$( ϕ, s ) } for any integer k > 0 . sup y$_{0}$ ∈$_{R}$ [ ∞ ∑ k =0 $^{(}$1$_{A}$ + $_{k}$( ϕ,s $_{)}$( y$_{0}$ ) − 1$_{A}$ − k ( ϕ,s $_{)}$( y$_{0}$ ) $^{)}$− 1 { y$_{1}$ − y$_{0}$ < δ } − λ (1 − 1$_{C}$ ( y$_{0}$, y$_{1}$ ))$^{]}$ ≥ ρ + 1 , −∞ < sup y$_{0}$ ∈$_{R}$ [ ∞ ∑ k =0 $^{(}$1$_{A}$ + $_{k}$( ϕ,s $_{)}$( y$_{0}$ ) − 1$_{A}$ − k ( ϕ,s $_{)}$( y$_{0}$ ) $^{)}$− 1 ( y$_{1}$ − y$_{0}$ < δ ) − λ (1 − 1$_{C}$ ( y$_{0}$, y$_{1}$ ))$^{]}$ ≤ − ∞ ∑ k =0 $^{(}$1$_{A}$ + $_{k}$( ψ,s $_{)}$( y$_{1}$ ) − 1$_{A}$ − k ( ψ,s $_{)}$( y$_{1}$ ) $^{)}$. ( ∞ ∑ k =0 $^{(}$1$_{A}$ + $_{k}$( ϕ,s ) − 1$_{A}$ − k ( ϕ,s ) $^{)}$, ∞ ∑ k =0 $^{(}$1$_{A}$ + $_{k}$( ψ,s ) − 1$_{A}$ − k ( ψ,s ) ) ) . Hence for all ( y$_{0}$, y$_{1}$ ) ∈$_{R}$ $^{2}$, ∞ ∑ k =0 $^{(}$1$_{A}$ + $_{k}$( ϕ,s $_{)}$( y$_{0}$ ) − 1$_{A}$ − k ( ϕ,s $_{)}$( y$_{0}$ ) $^{)}$+ ∞ ∑ k =0 $^{(}$1$_{A}$ + $_{k}$( ψ,s $_{)}$( y$_{1}$ ) − 1$_{A}$ − k ( ψ,s $_{)}$( y$_{1}$ ) ≤ 1 { y$_{1}$ − y$_{0}$ < δ } + λ (1 − 1$_{C}$ ( y$_{0}$, y$_{1}$ )) , 59