ds \\& +
\sum_{j=1}^{\infty} \int_{0}^{t} \int_{\mathcal{O}}
\varphi_n^{\prime} (u_s(x)-S'_s(x)) \,  \bar{h}_{j}( s,x, u_s-S'_s,\nabla (u_s-S'_s))\, dx dB_{s}^{j}\\
& + \frac{1}{2}\, \sum_{j=1}^{\infty} \int_{0}^{t}\int_{\mathcal{O}}
\varphi_n^{\prime \prime } (u_s(x)-S'_s(x)) \bar{h}_{j}^2(s,x,
u_s-S'_s,\nabla (u_s-S'_s)) \, dx\, ds\\&
+\int_0^t\int_{\ensuremath{\mathcal O}}\varphi_n^{\prime}(u_s(x)-S'_s(x))\nu(dxds) \, .\\
\end{split}$$ Since the support of $\nu$ is $\{u=S\}$, the last term is equal to $$\int_0^t\int_{\ensuremath{\mathcal O}}\varphi_n^{\prime}(S_s(x)-S'_s(x))\nu(dxds)$$and it is negative, because $$\int_0^t\int_{\ensuremath{\mathcal O}}\varphi_n^{\prime}(S_s(x)-S'_s(x)){{1} \hspace{-0.25 em}{\rm I}}_{\{\left|S-S'\right|\leq n\}}\nu(dxds)=l\int_0^t\int_{\ensuremath{\mathcal O}}sgn(S-S')\left|S_s(x)-S'_s(x)\right|^{l-1}\nu(dxds)\leq 0$$ and$$\begin{aligned}
\int_0^t\int_{\ensuremath{\mathcal O}}\varphi_n^{\prime}(S_s(x)&-&S'_s(x))I_{\{\left|S-S'\right|>n\}}\nu(dxds)\\&=&\int_0^t\int_{\ensuremath{\mathcal O}}n^{l-2}[l(l-1)(\left|S-S'\right|-n)sgn(S-S')+sgn(S-S')ln]\nu(dxds)\leq0\end{aligned}$$ By the uniform ellipticity of the operator $A$ we get $$\mathcal{E} \big(\varphi_n^{\prime} (u_s-S'_s), \, u_s-S'_s \big) \,
\geq \, \lambda  \, \int_{{\ensuremath{\mathcal O}}} \varphi_n^{\prime \prime} (u_s-S'_s)
|\nabla (u_s-S'_s)|^2 \, dx.$$ Let $\epsilon >0$ be fixed. Using the Lipschitz condition on $\bar{f}$ and the properties of the functions $(\varphi_n)_n$ we get $$\begin{split}   & |\varphi_n^{\prime} (u_s-S'_s)| \, |\bar{f}(s,x,u_s-S'_s, \nabla u_s-S'_s)| \\&
 \leq \,   |\varphi_n^{\prime} (u_s-S'_s)| \, \big( |\bar{f}^0 (s,x)| + C \, (|u_s-S'_s| +
 |\nabla (u_s-S'_s)|)\, \big)\\
 & \leq  \, |\varphi_n^{\prime} (u_s-S'_s)||\bar{f}^0 (s,x)|+ |u_s-S'_s||\varphi_n^{\prime\prime}
 (u_s-S'_s)| \, (C |u_s-S'_s| + C
 |\nabla (u_s-S'_s)|)\, )\\
 &\leq l (\varphi_n (u_s-S'_s) + 1) \, |\bar{f}^0(s,x) | + C|u_s-S'_s|^2|\varphi_n^{\prime\prime}
 (u_s-S'_s)| +C|u_s-S'_s| |\nabla (u_s-S'_s)||\varphi_n^{\prime\prime}
 (u_s-S'_s)|\\
 & \leq l (\varphi_n (u_s-S'_s) + 1) \, |\bar{f}^0(s,x) | + (C +
 c_{\epsilon})\, |u_s-S'_s|^2
 \varphi_n^{\prime\prime}(u_s-S'_s)  + \, \epsilon \varphi_n^{\prime\prime}(u_s-S'_s)|\nabla u_s-S'_s|^2 .\\
 \end{split}$$ Now using Cauchy-Schwarz inequality 