\in
        (\widetilde{T^{'}} - \delta, \widetilde{T^{'}} + \delta) \cap [t_{l}, T]$ we have

        $$\begin{array}{l}
        \sup_{t \in [t_{l}, T^{'}]} E(Iw_{l}(t))  \leq \left( C_{E} N^{2(1-s)} A^{p+1} - l (1- \epsilon_{0}) (N^{s-1} \eta_{1} A^{-1})^{\alpha} \right)
        \left( 1+ \beta \right)
        \end{array}
        \label{Eqn:InducNrjThm}$$

        and

        $$\begin{array}{ll}
        \| I w_{l} \|_{L_{t}^{2(p-1)-} L_{x}^{2(p-1)+} ([t_{l},T^{'}])} & \leq 2 X_{C,Q}
        \end{array}
        \label{Eqn:BoundIwlXCQ}$$

        We divide $[t_{l},T^{'}]$ into subintervals $(J_{j})_{1 \leq j \leq m}$ such that $ \| I w_{l} \|_{L_{t}^{2(p-1)-} L_{x}^{2(p-1)+} (J_{j})} =
        \eta_{0} $ and $\| I w_{l} \|_{L_{t}^{2(p-1)-} L_{x}^{2(p-1)+} (J_{m}) } \leq \eta_{0}$. By Proposition \[prop:LocalBd\] and (\[Eqn:InducNrjThm\]) we have

        $$\begin{array}{ll}
        Z(w_{l},J_{j}) & \lesssim \left[ \left( C_{E} N^{2(1-s)} A^{p+1} - l(1- \epsilon_{0}) (N^{s-1} \eta_{1} A^{-1})^{\alpha} \right) (1+ \beta)
        \right]^{\frac{1}{2}}
        \end{array}$$

        Now, by applying Proposition \[Prop:Acl\], (\[Eqn:NControlVar\]) and (\[Eqn:BoundIwlXCQ\]) we see that in fact

        $$\begin{array}{ll}
        \sup_{t \in [0, T^{'}]} E(Iw_{l}(t))  \leq \left( C_{E} N^{2(1-s)} A^{p+1} - l (1- \epsilon_{0}) (N^{s-1} \eta_{1} A^{-1})^{\alpha} \right)
        \left( 1+ \frac{\beta}{2} \right)
        \end{array}$$

        Let $M$ be such that $ \frac{(N^{1-s} A)^{\alpha}}{M^{\bar{\alpha}}}:= \frac{1}{C_{2}^{C_{2} L(l+1) + 1}}$ ( with $\alpha$, $\bar{\alpha}$ and $C_{2}$ defined in Proposition \[prop:PerturbArg\]). If $ \| I w_{l} \|_{L_{t}^{2(p-1)-} L_{x}^{2(p-1)+} ([t_{l},T^{'}]) } \geq 2B $ then this implies that there exists $T_{0} \in [t_{l},T^{'}]$ such that $ \| I w_{l} \|_{L_{t}^{2(p-1)-} L_{x}^{2(p-1)+} ([t_{l},T_{0}])} \geq 