Therefore, by means of (6.7), we have for t ∈ [ θ, θ + τ ] . One can find numbers s$_{1}$, s$_{2}$, s$_{3}$ ∈ [ θ, θ + τ ] such that Inequality (6.8) yields Let us define the function f : R 3 → R 3 as f ( v ) = ( − σv$_{1}$ + σv$_{2}$, − v$_{1}$v$_{3}$ + rv$_{1}$ − v$_{2}$, v$_{1}$v$_{2}$ − bv$_{3}$ ) , where v = ( v$_{1}$, v$_{2}$, v$_{3}$ ) . One can confirm the presence of a positive number L$_{f}$ such that ‖ f ( v ) − f ( v ) ‖ ≤ L$_{f}$ ‖ v − v ‖ for all v, v ∈ U . The relation implies that According to the last inequality we have that Therefore, max t ∈ [ θ,θ + τ $_{]}$‖ y ( t ) − y ( t ) ‖ > τL$_{g}$ϵ$_{0}$ 6(2 + τL$_{f}$ ) . 11 | g$_{j}$$_{0}$ ( x ( t )) − g$_{j}$$_{0}$ ( x ( t )) | > | g$_{j}$$_{0}$ ( x ( s$_{0}$ )) − g$_{j}$$_{0}$ ( x ( s$_{0}$ )) | − L$_{g}$ϵ$_{0}$ 6 > L$_{g}$ϵ$_{0}$ 6 (6.8) < L$_{g}$ϵ$_{0}$ 6 . ∫ θ + τ θ [ g ( x ( s )) − g ( x ( s ))] ds = ( τ [ g$_{1}$ ( x ( s$_{1}$ )) − g$_{1}$ ( x ( s$_{1}$ ))] , τ [ g$_{2}$ ( x ( s$_{2}$ )) − g$_{2}$ ( x ( s$_{2}$ ))] , τ [ g$_{3}$ ( x ( s$_{3}$ )) − g$_{3}$ ( x ( s$_{3}$ ))] ) . ∥ ∫ θ + τ θ [ g ( x ( s )) − g ( x ( s ))] ds ∥ ≥ τ | g$_{j}$$_{0}$ ( x ( s$_{j}$$_{0}$ )) − g$_{j}$$_{0}$ ( x ( s$_{j}$$_{0}$ )) | > τL$_{g}$ϵ$_{0}$ 6 . y ( t ) − y ( t ) = ( y ( θ ) − y ( θ )) + ∫ t θ [ f ( y ( s )) − f ( y ( s ))] ds + ∫ t θ [ g ( x ( s )) − g ( x ( s ))] ds, t ∈ [ θ, θ + τ ] ‖ y ( θ + τ ) − y ( θ + τ ) ‖ ≥ ∥ ∫ θ + τ θ [ g ( x ( s )) − g ( x ( s ))] ds ∥ − ‖ y ( θ ) − y ( θ ) ‖ − ∫ θ + τ θ L$_{f}$ ‖ y ( s ) − y ( s ) ‖ ds > τL$_{g}$ϵ$_{0}$ 6 − ‖ y ( θ ) − y ( θ ) ‖ − ∫ θ + τ θ L$_{f}$ ‖ y ( s ) − y ( s ) ‖ ds. max t ∈ [ θ,θ + τ $_{]}$‖ y ( t ) − y ( t ) ‖ ≥ ‖ y ( θ + τ ) − y ( θ + τ ) ‖ > τL$_{g}$ϵ$_{0}$ 6 − (1 + τL$_{f}$ ) max t ∈ [ θ,θ + τ $_{]}$‖ y ( t ) − y ( t ) ‖ .