\algnewcommand\Output{\item[\algorithmicoutput]}
\newtheorem{proposition}{Proposition}
+\newtheorem{proof}{Proof}
\begin{document}
%
% use a multiple column layout for up to two different
% affiliations
-\author{\IEEEauthorblockN{Rapha\"el Couturier\IEEEauthorrefmark{1}, Lilia Ziane Khodja \IEEEauthorrefmark{2}, and Christophe Guyeux\IEEEauthorrefmark{1}}
+\author{\IEEEauthorblockN{Rapha\"el Couturier\IEEEauthorrefmark{1}, Lilia Ziane Khodja\IEEEauthorrefmark{2}, and Christophe Guyeux\IEEEauthorrefmark{1}}
\IEEEauthorblockA{\IEEEauthorrefmark{1} Femto-ST Institute, University of Franche Comte, France\\
Email: \{raphael.couturier,christophe.guyeux\}@univ-fcomte.fr}
\IEEEauthorblockA{\IEEEauthorrefmark{2} INRIA Bordeaux Sud-Ouest, France\\
\begin{equation}
||r_m|| \leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{m}{2}} ||r_0|| ,
\end{equation}
-where $\alpha = \lambda_min(M)^2$ and $\beta = \lambda_max(A^T A)$, which proves
+where $\alpha = \lambda_{min}(M)^2$ and $\beta = \lambda_{max}(A^T A)$, which proves
the convergence of GMRES($m$) for all $m$ under that assumption regarding $A$.
\end{proposition}
+We can now claim that,
+\begin{proposition}
+If $A$ is a positive real matrix, then the TSIRM algorithm is convergent.
+\end{proposition}
+\begin{proof}
+\end{proof}
%%%*********************************************************
%%%*********************************************************
% that's all folks
\end{document}
-
