that if $\min(\mathtt{a}^{\Omega_C}, \mathtt{a}^{\Omega_I}, \alpha)\geq \alpha_0$ then, $$\label{elip}
\text{Re}\left[ (1 - \imath) A_h(({\boldsymbol{v}}, \varphi,m), (\overline {\boldsymbol{v}}, \overline \varphi,\overline m))\right] \geq \dfrac{1}{2} {\lVert({\boldsymbol{v}}, \varphi,m)\rVert}^2 \qquad \forall ({\boldsymbol{v}}, \varphi,m)\in \mathbf{X}_h\times V_h\times {\mathbb C}.$$

By definition of $A_h(\cdot, \cdot)$, $$\label{elip0}
\begin{array}{l}
\text{Re}\left[(1 - \imath) A_h(({\boldsymbol{v}}, \varphi,m), (\overline {\boldsymbol{v}}, \overline \varphi,\overline m))\right] = \omega {\lVert \mu^{1/2} {\boldsymbol{v}}\rVert}_{0,\Omega_C}^2 + 
 {\lVert\sigma^{-1/2} {\mathbf{curl}}_h {\boldsymbol{v}}\rVert}_{0,\Omega_C}^2\\[.1cm] 
\qquad+ 2  \text{Re} \left( {\{\sigma^{-1} {\mathbf{curl}}_h {\boldsymbol{v}}\}}_{{\mathcal{F}}}, {\llbracket (\overline {\boldsymbol{v}}, \overline \varphi,\overline m) \rrbracket}_{{\mathcal{F}}} \right)_{{\mathcal{F}}_h^{\Omega_C}} 
+  \mathtt{a}^{\Omega_C} {\lVerth_{{\mathcal{F}}}^{-1/2}{\llbracket ({\boldsymbol{v}}, \varphi,m) \rrbracket}_{{\mathcal{F}}}\rVert}_{0,{\mathcal{F}}_h^{\Omega_C}}^2\\[.1cm] 
 \qquad +\omega \mu_0 {\lVert\nabla_h \varphi+m \boldsymbol \rho\rVert}_{0,\Omega_C}^2 - 2 \omega \mu_0 \text{Re} \left( 
 {\{\nabla_h \varphi+ m \boldsymbol \rho\}}_{{\mathcal{F}}}, {\llbracket \overline \varphi {\boldsymbol{n}}\rrbracket}_{{\mathcal{F}}}\right)_{{\mathcal{F}}_h^{\Omega_I}} \\[.1cm] 
 \qquad+  \mathtt{a}^{\Omega_I}{\lVerth_{{\mathcal{F}}}^{-1/2}{\llbracket \varphi{\boldsymbol{n}}\rrbracket}_{{\mathcal{F}}}\rVert}^2_{0,{\mathcal{F}}_h^{\Omega_I}} 
- 2 \text{Re}  \left( {\{\sigma^{-1} {\mathbf{curl}}_h {\boldsymbol{v}}\}}_{{\mathcal{E}}}, {\llbracket \overline \varphi {\boldsymbol{t}}\rrbracket}_{{\mathcal{E}}} \right)_{{\mathcal{E}}_h} 
+  \alpha {\lVerth_{{\mathcal{E}}}^{-1}{\llbracket \varphi{\boldsymbol{t}}\rrbracket}_{{\mathcal{E}}}\rVert}^2_{0, {\mathcal{E}}_h}.
\end{array}$$

It follows from the Cauchy-Schwarz inequality and that, $$\label{elip1}
\begin{array}{l}
2 |  \text{Re} \left( {\{\sigma^{-1} {\mathbf{curl}}_h {\boldsymbol{v}}\}}_{{\mathcal{F}}}, {\llbracket (\overline {\boldsymbol{v}}, \overline \varphi,\overline m) \rrbracket}_{{\mathcal{F}}} \right)_{{\mathcal{F}}_h^{\Omega_C}}|
\\[.1cm]
\qquad \leq 2 {\lVert\mathtt{s}_{{\mathcal{F}}}^{1/2} h_{{\mathcal{F}}}^{1/2} {\{\sigma^{-1} {\mathbf{curl}}_h {\boldsymbol{v}}\}}_{{\mathcal{F}}}\rVert}_{0,{\mathcal{F}}_h^{\Omega_C}} 
{\lVert\mathtt{s}_{{\mathcal{F}}}^{-1/2} h_{{\mathcal{F}}}^{-1/2} {\llbracket ( {\boldsymbol{v}},  \varphi,m)  \rrbracket}_{{\mathcal{F}}}\rVert}_{0,{\mathcal{F}}_h^{\Omega_C}} \\[.1cm]
\qquad\leq 2C_{\Omega_C} {\lVert\sigma^{-1/2}{\mathbf{curl}}_h {\boldsymbol{v}}\rVert}_{0,\Omega_C} {\lVert\mathtt{s}_{{\mathcal{F}}}^{-1/2} h_{{\mathcal{F}}}^{-1/2} {\llbracket ( {\boldsymbol{v}},  \varphi, m)  \rrbracket}_{{\mathcal{F}}}\rVert}_{0,{\mathcal{F}}_h^{\Omega_C}} \\[.1cm]
\qquad\leq \frac{1}{4} {\lVert\sigma^{-1/2}{\mathbf{curl}}_h {\boldsymbol{v}}\rVert}_{0,\Omega_C}^2 + 4 C_{\Omega_C}^2   
{\lVert\mathtt{s}_{{\mathcal{F}}}^{-1/2} h_{{\mathcal{F}}}^{-1/2}  {\llbracket ( {\boldsymbol{v}},  \varphi, m)  \rrbracket}_{{\mathcal{F}}}\rVert}_{0,{\mathcal{F}}_h^{\Omega_C}}^2.
\end{array}$$ Similarly, by virtue of , $$\label{elip2}
\begin{array}{l}
