 $$\theta_0^{(i,\ell_{ k_{0,{\mathfrak{m}}}+j+0})}  \geq  \theta_0^{(i,\ell_{ k_{0,{\mathfrak{m}}}+j})}  + \bar{r}_{\Theta_i,0} \left({\boldsymbol{\theta}}^{(i,\ell_{ k_{0,{\mathfrak{m}}}+j})}, {\boldsymbol{\theta}}'^{(i,\ell_{ k_{0,{\mathfrak{m}}}+j})}\right)