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%___________________ PARAGRAPHES ___________________

\let\sx\section
\let\ssx\subsection
\let\sssx\subsubsection

%___________________ STYLES __________________________

\let\ts=\textstyle
\let\ds=\displaystyle
\let\ssc=\scriptstyle
\let\sss=\scriptscriptstyle

\let\dl=\displaylines

\let\sou=\underline
\let\sur=\overline

% \font\large=cmr12
\font\grand=cmr12
\font\giga=cmr17

%______ EXPOSANTS, INDICES EN MODE NON MATH _________

\def\up#1{\raisebox{1ex}{{\scriptsize #1}}}
\def\down#1{\raisebox{-1ex}{{\scriptsize #1}}}
\def\no{n\up{$\circ$}}

%__________ ENCADREMENTS , TRAITS DIVERS _____________________

\def\entoure#1#2{\setbox1=\hbox{\kern#1{#2}\kern#1}%
\dimen1=\ht1 \advance\dimen1 by #1 \dimen2=\dp1
\advance\dimen2 by #1
\setbox1=\hbox{\vrule height\dimen1
depth\dimen2\box1\vrule}%
\setbox1=\vbox{\hrule\box1\hrule}%
\advance\dimen1 by .4pt \ht1=\dimen1
\advance\dimen2 by .4pt \dp1=\dimen2 \box1\relax}

\def\x#1{\entoure{4pt}{#1}}

\def\cvirp{\raise 2pt\hbox{,}}

\def\cqfd{\unskip\kern 6pt\penalty 500\raise
-2pt\hbox{\vrule\vbox to 10pt{\hrule width
4pt\vfill\hrule}\vrule}\par}

\def\clv{\hbox{\vrule\vbox to 6 pt{\hrule width
4pt\vfill\hrule}\vrule}}

\def\trait {\hrule height 1pt depth 0pt}

\def\traith {\hrule height 1pt depth 0pt}
\def\traitb {\hrule height 0pt depth 1pt}

\def\tvi#1#2{\vrule height #1 pt depth #2 pt width 0pt}
\def\tvii#1{\vrule height #1 pt depth 5pt width 0pt}
\def\tv{\tvii{12}\vrule}
\def\ttv{\left\Vert\tvi{12}\right.}

%____________ INF OU EGAL __________________________

\let\infeg\leqslant
\let\supeg\geqslant

%____________ SUPERPOSITION DE SYMBOLES _____________

\def\superpose#1_#2^#3{\mathrel{\mathop{\kern
0pt#1}\limits_{#2}^{#3}}}

\def\fr #1#2{\ds {\raise -2pt\hbox{$#1$}\over 
                            \raise 2pt\hbox{$#2$}}}

%______ Lignes de titre _________________

\def\ligne#1#2{#1 \hfill #2}
\def\titre#1{\centerline{\entoure{5pt}{\bf #1}}}

%__________ N, Z, Q, R, C __________

\def\nmat{\hbox{\it I\hskip -2pt N}}
\def\zmat{\hbox{\it Z\hskip -4pt Z}}
\def\qmat{\hbox{\it l\hskip -5.5pt Q}}
\def\rmat{\hbox{\it I\hskip -2pt R}}
\def\cmat{\hbox{\it l\hskip -5.5pt C\/}}

\def\N{{\mathbb N}}
\def\Net{{\mathbb N}^*}
\def\Z{{\mathbb Z}}
\def\Q{{\mathbb Q}}
\def\R{{\mathbb R}}
\def\Ret{{\mathbb R}^*}
\def\Rpl{{\mathbb R}_+}
\def\Rplet{{\mathbb R}_+^*}
\def\Rmn{{\mathbb R}_-}
\def\Rmnet{{\mathbb R}_-^*}
\def\C{{\mathbb C}}
\def\K{{\mathbb K}}
\def\D{{\mathbb D}}

%__________ FLECHES ET LOGIQUE __________

\let\fc=\longmapsto              % pour "x donne f(x)"
\let\vers=\rightarrow            % pour "tend vers"
\let\flg=\leftarrow              % fleche simple courte vers la gauche
\let\rec=\longleftarrow          % symbole d'affectation
\let\Rec=\Longleftarrow          % fleche double longue vers la gauche
\let\Imp=\Rightarrow             % implique         (double barre)
\let\Ssi=\Leftrightarrow         % est equivalent a (double barre)

\let\ou=\vee                     % disjonction logique
\let\et=\wedge                   % conjonction logique
\let\non=\neg                    % negation logique
\let\imp=\Rightarrow         % implication logique
\let\ssi=\Longleftrightarrow     % equivalence logique
\let\eqv=\Longleftrightarrow

\let\qqs=\forall                 % quantificateur universel
\let\exi=\exists                 % quantificateur existentiel

\let\theor=\vdash                % dÃ©duction
\let\tauto=\models               % consÃ©quence logique
\def\vrai{\textit{vrai}}         % vrai
\def\faux{\textit{faux}}         % faux



%___________ COMBINATOIRE ET ENSEMBLES ______________

\def\cnp{\mathop{\raise -1pt\hbox{\large C}}\nolimits}
\def\card{\mathop{\rm Card}\nolimits}

\let\vide=\emptyset
\def\void{\not{\kern -1.55pt\rm o}}

\let\union=\cup
\let\inter=\cap
\let\moins=\setminus
\let\sse=\subset

\def\enspart#1{{\cal P}(#1)}

\def\rest#1#2{{#1}_{\left|\mathstrut #2\right.}}
\def\idl#1{{\goth #1}}

%__________ TOPO ____________________________

\def\min{\mathop{\rm Min}\limits}
\def\max{\mathop{\rm Max}\limits}
\def\sup{\mathop{\rm Sup}\limits}
\def\inf{\mathop{\rm Inf}\limits}

\let\nor=\Vert                 % norme
\def\nme#1{\nor #1\nor}
\def\trnme#1{|||#1|||}         % "triple" norme
\def\nmex#1{\left|\!\left|#1\right|\!\right|}
                               % norme extensible

\let\Inf=\infty                % infini
\def\plinf{{+\infty}}          % plus l'infini
\def\moinf{{-\infty}}          % moins l'infini

\def\Rpl{\R_{\sss +}}          % double R plus
\def\Rmo{\R_{\sss -}}          % double R moins

\def\intr#1{\,\buildrel{\kern 2pt\circ}\over{#1}} % intï¿½rieur

%_____________________ FONCTIONS _______________________

\def\tg{\mathop{\rm tg}\nolimits}
\def\arcsin{\mathop{\rm Arcsin}\nolimits}
\def\arccos{\mathop{\rm Arccos}\nolimits}
\def\arctg{\mathop{\rm Arctg}\nolimits}
\def\ch{\mathop{\rm ch}\nolimits}
\def\sh{\mathop{\rm sh}\nolimits}
\def\th{\mathop{\rm th}\nolimits}
\def\argsh{\mathop{\rm Argsh}\nolimits}
\def\argch{\mathop{\rm Argch}\nolimits}
\def\argth{\mathop{\rm Argth}\nolimits}
\def\log{\mathop{\rm ln}\nolimits}
\def\Log{\mathop{\rm Log}\nolimits}
\def\E{\mathop{\rm E}\nolimits}                % partie entiere
\def\e{\mathop{\rm e}\nolimits}                % exponentielle

%______________ ANALYSE _______________

\def\cl{{\cal C}}                             % classe d'une fonction

\def\intint{\int\!\!\!\!\int}                 % int. double
\def\intintint{\int\!\!\!\!\int\!\!\!\!\int}  % int. triple

\let\dep=\partial                             % d rond
\def\ddr{{\rm d\,}}                           % d droit
\def\dpar#1#2{\fr{\partial#1}{\partial#2}}    % d rond #1 sur d rond #2
\def\dtot#1#2{\fr{{\rm d}#1}{{\rm d}#2}}      % d droit #1 sur d droit #2

\def\ste#1#2#3{\left(#1_#2\right)_{#2\in #3}}
\def\sten#1#2{\ste{#1}{#2}\N}
\def\stenet#1#2{\ste{#1}{#2}{\N^*}}
\def\stenn#1{\sten{#1}n}
\def\stennet#1{\stenet{#1}n}

%______________ ALGEBRE Complexe _______________
\renewcommand{\Re}{\mathop{\mathrm{Re}}}
\renewcommand{\Im}{\mathop{\mathrm{Im}}}


%_______________ ALGEBRE LINEAIRE ______________________

\def\com{\mathop{\rm Com}\nolimits}
\def\det{\mathop{\rm D\acute e t}\nolimits}
\def\dim{\mathop{\rm dim}\nolimits}
\def\ker{\mathop{\rm Ker}\nolimits}
\def\id{\mathop{\rm id}\nolimits}
\def\tr{\mathop{\rm Tr}\limits}
\def\vect{\mathop{\rm Vect}\nolimits}
\def\rang{\mathop{\rm rang}\nolimits}
\def\diag{\mathop{\rm diag}\nolimits}
\def\sp{\mathop{\rm Sp}\nolimits}
\def\GL#1#2{{\cal GL}_{#1}(#2)}
\def\gl#1{{\cal GL}(#1)}
\def\mat#1#2{{\rm Mat}\,\left(#1\vir#2\right)}
\def\matb#1#2#3{{\rm Mat}\left(#1\vir#2\vir#3\right)}
\def\matbb#1{{\rm Mat}_{\cal B}\left(#1\right)}
\def\matcar#1#2{{\cal M}_{#1}\left(#2\right)}
\def\mn#1{\matcar n{#1}}
\def\matrect#1#2#3{{\cal M}_{#1,#2}\left(#3\right)}
\def\apl#1#2{{\cal L}\left(#1,#2\right)}
\def\aplr#1#2{{\cal L}\left(\R^#1,\R^#2\right)}
\def\endo#1{{\cal L}\left(#1\right)}
\def\endor#1{{\cal L}\left(\R^#1\right)}
\def\nul#1{0_{#1}}
\def\endonul#1{O_{{\cal L}(#1)}}
\def\evnul#1{\left\{\nul#1\right\}}
\def\fami#1#2{({#2}_1\,,\,{#2}_2\,,\,\ldots\,,\,{#2}_{#1})}
\def\famn#1{\fami{n}{#1}}

%_____ POUR MATRICES, points parallï¿½les ï¿½ diag 2 _____

\def\adots{\mathinner{\mkern2mu\raise1pt\hbox{.}
\mkern3mu\raise4pt\hbox{.}\mkern1mu\raise7pt\hbox{.}}}

%__________ GEOMETRIE _______________________________

\def\grad{\mathop{\rm grad}\nolimits}
\def\so{\mathop{\cal SO}\nolimits}
\def\og#1{{\cal O}(#1)}
\let\vc=\overrightarrow

\def\arc#1{\buildrel\frown\over{#1}}
\def\arcfl#1{\buildrel{\ds\bb y}\over{#1}}

\def\pscal#1#2{\langle #1\vir #2\rangle}

%__________ ARITHMETIQUE _______________________________

\def\mod{\mathop{\rm mod}\nolimits}
\def\dv{\mathop{\rm div}\nolimits}

%__________ Programmes _______________

\def\prol{\parindent=1cm\obeylines\tt }
\def\dec {\advance\parindent by 1cm}

\def\prog#1{\advance\parindent by #1 cm\obeylines\tt}
\def\ind#1{\advance\parindent by #1 cm}

%__________ en-dessus, au-dessous d'un symbole __________

\def\build#1_#2^#3{\mathrel{
\mathop{\kern 0pt#1}\limits_{#2}^{#3}}}

%____________________ DIVERS ____________________________

\def\bs{\char"5C}              % le backslash !!!

\def\vir{\,,\,} \def\cdotv{\raise 2pt\hbox{,}}

\def\jbar{\sur{\mathstrut\,j\,}}

\def\psaut{\vskip 2pt}
\def\saut{\vskip 5pt}
\def\gsaut{\vskip 8pt}
\def\Saut{\vskip 8pt}

\def\fin{\gsaut\centerline{\hbox to 3cm{\hrulefill}}}

\def\nopagenumbers{\def\folio{\hfil}}

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