
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%:  lecteur, lectrice, masculin feminin

\newcounter{MF}
\newcommand\stMF{\stepcounter{MF}}

\newcommand{\lec}{\stMF\ifodd\value{MF}lecteur \else 
lectrice \fi}
\newcommand{\lecz}{\stMF\ifodd\value{MF}lecteur\else lectrice\fi}

\newcommand{\lecs}{\stMF\ifodd\value{MF}lecteurs \else 
lectrices \fi}
\newcommand{\lecsz}{\stMF\ifodd\value{MF}lecteurs\else 
lectrices\fi}

\newcommand{\alec}{\stMF\ifodd\value{MF}au lecteur \else%
\`a la lectrice \fi}
\newcommand{\alecz}{\stMF\ifodd\value{MF}au lecteur\else%
\`a la lectrice\fi}

\newcommand{\dlec}{\stMF\ifodd\value{MF}du lecteur \else%
de la lectrice \fi}
\newcommand{\dlecz}{\stMF\ifodd\value{MF}du lecteur\else%
de la lectrice\fi}

\newcommand{\llec}{\stMF\ifodd\value{MF}le lecteur \else la lectrice \fi}
\newcommand{\llecz}{\stMF\ifodd\value{MF}le lecteur\else la lectrice\fi}

\newcommand{\Llec}{\stMF\ifodd\value{MF}Le lecteur \else La lectrice \fi}

% les suivants ne changent pas le genre
\newcommand{\lui}{\ifodd\value{MF}lui \else
elle \fi}
\newcommand{\luiz}{\ifodd\value{MF}lui\else
elle\fi}

\newcommand{\celui}{\ifodd\value{MF}celui \else
celle \fi}

\newcommand{\ceux}{\ifodd\value{MF}ceux \else
celles \fi}

\newcommand{\er}{\ifodd\value{MF}er \else
\`ere \fi}

\newcommand{\eux}{\ifodd\value{MF}eux \else
elles \fi}

\newcommand{\eUx}{\ifodd\value{MF}eux \else
euse \fi}

\newcommand{\leux}{\ifodd\value{MF}leux \else
leuse \fi}

\newcommand{\il}{\ifodd\value{MF}il \else
elle \fi}

\newcommand{\e}{\ifodd\value{MF} \else e \fi}
\newcommand{\ez}{\ifodd\value{MF}\else e\fi}

\newcommand{\n}{\ifodd\value{MF}n \else nne \fi}
\newcommand{\nz}{\ifodd\value{MF}n\else nne\fi}

%  \la* fait le ou la sans changer le genre,
%  \la fait le ou la en changeant de genre
\makeatletter
\newcommand{\la}{\@ifstar{\ifodd\value{MF}le\else
la\fi}{\stMF\ifodd\value{MF}le \else la \fi}}
\makeatother


%:  newtheorems  


%\theoremstyle{plain}
\newtheorem{theorem}{Th\'eor\`eme}[section]   
\newtheorem{thdef}[theorem]{Th\'eor\`eme et d\'efinition}
\newtheorem{plcc}[theorem]{Principe local-global concret}%[section]
\newtheorem{plca}[theorem]{Principe local-global abstrait$\mathbf{^*}$}
\newtheorem{prvq}[theorem]{Principe de recouvrement par quotients}
\newtheorem{prvqp}[theorem]{Principe de recouvrement par quotients principaux}
%\newtheorem{plcc}{Principe local-global concret}
%\newtheorem{plcd}{Principe local-global dynamique}
\newtheorem{proposition}[theorem]{Proposition}%[section]
\newtheorem{propdef}[theorem]{Proposition et d\'efinition}
\newtheorem{propnot}[theorem]{Proposition et notation}
\newtheorem{lemma}[theorem]{Lemme}
\newtheorem{corollary}[theorem]{Corollaire} 
\newtheorem{fact}[theorem]{Fait}
%
%%\theoremstyle{remark}
%\newtheorem{remark}[proposition]{Remarque}
%\newtheorem{remarks}[proposition]{Remarques}
%\newtheorem{comment}[proposition]{Commentaire}
%\newtheorem{comments}[proposition]{Commentaires}
%\newtheorem{example}[proposition]{Exemple}
%\newtheorem{examples}[proposition]{Exemples}
\newtheorem{definition}[theorem]{D\'efinition}
\newtheorem{definitions}[theorem]{D\'efinitions}
\newtheorem{definota}[theorem]{D\'efinition et notation}
%\newtheorem{convention}[proposition]{Convention}
\newtheorem{notation}[theorem]{Notation} 
%\newtheorem{notations}[proposition]{Notations} 
%\newtheorem{question}[proposition]{Question}
%\newtheorem{exercise}{Exercice}
%\newtheorem{problem}[exercise]{Probl\`eme}
%\newtheorem{algorithm}[proposition]{Algorithme}
%
%
%%\theoremstyle{plain}
\newtheorem{theoremc}[theorem]{Th\'{e}or\`{e}me\etoz}
%\newtheorem{lemmac}[proposition]{Lemme\etoz}
%\newtheorem{corollaryc}[proposition]{Corollaire\etoz}
%\newtheorem{proprietec}[proposition]{Propri\'{e}t\'{e}\etoz}
%\newtheorem{propositionc}[proposition]{Proposition\etoz}
%\newtheorem{factc}[proposition]{Fait\etoz}
%


\newcommand \rem{\rdb
\noi{\it Remarque. }}

\newcommand \REM[1]{\rdb
\noi{\it Remarque#1. }}

\newcommand \rems{\rdb
\noi{\it Remarques. }}

\newcommand \exl{\rdb
\noi{\bf Exemple. }}

\newcommand \EXL[1]{\rdb
\noi{\bf Exemple: #1. }}

\newcommand \exls{\rdb
\noi{\bf Exemples. }}

\newcommand \thref[1] {\thoz~\ref{#1}}
\newcommand \paref[1] {page~\pageref{#1}}


%%
\DeclareRobustCommand{\guig}{\mbox{{\usefont{U}{lasy}%
{\if b\expandafter\@car\f@series\@nil b\else m\fi}{n}%
\char40\kern-0.20em\char40}~}}
%%
\DeclareRobustCommand{\guid}{\mbox{~\usefont{U}{lasy}%
{\if b\expandafter\@car\f@series\@nil b\else m\fi}{n}%
\char41\kern-0.20em\char41}}
%%
\newcommand\gui[1]{\guig{#1}\guid}

\newcommand \facile{\begin{proof}
La \dem est laiss\'ee \alecz.
\end{proof}
}


\newcommand \num {{n$^{\mathrm{ o}}$}}
%\newcommand \num {{n\o}}

%:  Commentaires, remarques, exemples, problemes
\newcommand\comm{\rdb
\noi{\it Commentaire. }}

\newcommand\COM[1]{\rdb
\noi{\it Commentaire #1. }}

\newcommand\comms{\rdb
\noi{\it Commentaires. }}



\newcommand\Pb{\rdb
\noi{\bf Probl\`eme. }}

\newcommand\PB[1]{\rdb
\noi{\bf Probl\`eme #1. }}

%\newcommand \eoe {\hbox{}\nobreak\hfill
%\vrule width 1.4mm height 1.4mm depth 0mm \par \smallskip} 

\newcommand\eoq{\hbox{}\nobreak
\vrule width 1.4mm height 1.4mm depth 0mm}


%:   cad hdr spdg propeq ...
\newcommand \Cad {C'est-\`a-dire }
\newcommand \Cadz {C'est-\`a-dire}
\newcommand \recu {r\'ecur\-rence }
\newcommand \recuz {r\'ecur\-rence}
\newcommand \hdr {hypo\-th\`ese de \recu }
\newcommand \hdrz {hypo\-th\`ese de \recuz}
\newcommand \cad {c'est-\`a-dire }
\newcommand \cadz {c'est-\`a-dire}
\newcommand \cade {c'est-\`a-dire en\-co\-re }
\newcommand \ssi {si, et seu\-le\-ment si, }
\newcommand \ssiz {si, et seu\-le\-ment si,~}
\newcommand \cnes {con\-di\-tion n\'eces\-sai\-re et suf\-fi\-san\-te }
\newcommand \spdg {sans per\-te de g\'en\'e\-ra\-lit\'e }
\newcommand \Spdg {Sans per\-te de g\'en\'e\-ra\-lit\'e }

\newcommand \Propeq {Les pro\-pri\-\'e\-t\'es sui\-van\-tes sont 
\'equi\-va\-len\-tes.}
\newcommand \propeq {les pro\-pri\-\'e\-t\'es sui\-van\-tes sont 
\'equi\-va\-len\-tes.}


%:  Kev Alg etc...
\newcommand \Kev {$\gK$-\evc }
\newcommand \Kevs {$\gK$-\evcs }
\newcommand \Kevz {$\gK$-\evcz}
\newcommand \Kevsz {$\gK$-\evcsz}

\newcommand \Lev {$\gL$-\evc }
\newcommand \Levs {$\gL$-\evcs }
\newcommand \Levz {$\gL$-\evcz}
\newcommand \Levsz {$\gL$-\evcsz}

\newcommand \Qev {$\QQ$-\evc }
\newcommand \Qevs {$\QQ$-\evcs }
\newcommand \Qevz {$\QQ$-\evcz}
\newcommand \Qevsz {$\QQ$-\evcsz}

\newcommand \kev {$\gk$-\evc }
\newcommand \kevs {$\gk$-\evcs }
\newcommand \kevz {$\gk$-\evcz}
\newcommand \kevsz {$\gk$-\evcsz}

\newcommand \lev {$\gl$-\evc }
\newcommand \levs {$\gl$-\evcs }
\newcommand \levz {$\gl$-\evcz}
\newcommand \levsz {$\gl$-\evcsz}


\newcommand \Alg {$\gA$-\alg}
\newcommand \Algs {$\gA$-\algs}
\newcommand \Algz {$\gA$-\algz}
\newcommand \Algsz {$\gA$-\algsz}

\newcommand \Blg {$\gB$-\alg}
\newcommand \Blgs {$\gB$-\algs}
\newcommand \Blgz {$\gB$-\algz}
\newcommand \Blgsz {$\gB$-\algsz}

\newcommand \Clg {$\gC$-\alg}
\newcommand \Clgs {$\gC$-\algs}
\newcommand \Clgz {$\gC$-\algz}
\newcommand \Clgsz {$\gC$-\algsz}

\newcommand \klg {$\gk$-\alg}
\newcommand \klgs {$\gk$-\algs}
\newcommand \klgz {$\gk$-\algz}
\newcommand \klgsz {$\gk$-\algsz}

\newcommand \llg {$\gl$-\alg}
\newcommand \llgs {$\gl$-\algs}
\newcommand \llgz {$\gl$-\algz}
\newcommand \llgsz {$\gl$-\algsz}

\newcommand \Klg {$\gK$-\alg}
\newcommand \Klgs {$\gK$-\algs}
\newcommand \Klgz {$\gK$-\algz}
\newcommand \Klgsz {$\gK$-\algsz}

\newcommand \Llg {$\gL$-\alg}
\newcommand \Llgs {$\gL$-\algs}
\newcommand \Llgz {$\gL$-\algz}
\newcommand \Llgsz {$\gL$-\algsz}

\newcommand \QQlg {$\QQ$-\alg}
\newcommand \QQlgs {$\QQ$-\algs}
\newcommand \QQlgz {$\QQ$-\algz}
\newcommand \QQlgsz {$\QQ$-\algsz}

\newcommand \Rlg {$\gR$-\alg}
\newcommand \Rlgs {$\gR$-\algs}
\newcommand \Rlgz {$\gR$-\algz}
\newcommand \Rlgsz {$\gR$-\algsz}

\newcommand \RRlg {$\RR$-\alg}
\newcommand \RRlgs {$\RR$-\algs}
\newcommand \RRlgz {$\RR$-\algz}
\newcommand \RRlgsz {$\RR$-\algsz}

\newcommand \ZZlg {$\ZZ$-\alg}
\newcommand \ZZlgs {$\ZZ$-\algs}
\newcommand \ZZlgz {$\ZZ$-\algz}
\newcommand \ZZlgsz {$\ZZ$-\algsz}


\newcommand \Amo {$\gA$-mo\-du\-le }
\newcommand \Amos {$\gA$-mo\-du\-les }
\newcommand \Amoz {$\gA$-mo\-du\-le}
\newcommand \Amosz {$\gA$-mo\-du\-les}

\newcommand \Bmo {$\gB$-mo\-du\-le }
\newcommand \Bmos {$\gB$-mo\-du\-les }
\newcommand \Bmoz {$\gB$-mo\-du\-le}
\newcommand \Bmosz {$\gB$-mo\-du\-les}

\newcommand \Cmo {$\gC$-mo\-du\-le }
\newcommand \Cmos {$\gC$-mo\-du\-les }
\newcommand \Cmoz {$\gC$-mo\-du\-le}
\newcommand \Cmosz {$\gC$-mo\-du\-les}

\newcommand \kmo {$\gk$-mo\-du\-le }
\newcommand \kmos {$\gk$-mo\-du\-les }
\newcommand \kmoz {$\gk$-mo\-du\-le}
\newcommand \kmosz {$\gk$-mo\-du\-les}

\newcommand \Kmo {$\gK$-mo\-du\-le }
\newcommand \Kmos {$\gK$-mo\-du\-les }
\newcommand \Kmoz {$\gK$-mo\-du\-le}
\newcommand \Kmosz {$\gK$-mo\-du\-les}

\newcommand \Lmo {$\gL$-mo\-du\-le }
\newcommand \Lmos {$\gL$-mo\-du\-les }
\newcommand \Lmoz {$\gL$-mo\-du\-le}
\newcommand \Lmosz {$\gL$-mo\-du\-les}

%\newcommand \lmo {$\gl$-mo\-du\-le }  % \lmo d\'ej\`a pris
%\newcommand \lmos {$\gl$-mo\-du\-les }
%\newcommand \lmoz {$\gl$-mo\-du\-le}
%\newcommand \lmosz {$\gl$-mo\-du\-les}

\newcommand \Ali {ap\-pli\-ca\-tion $\gA$-\lin }
\newcommand \Alis {ap\-pli\-ca\-tions $\gA$-\lins }
\newcommand \Aliz {ap\-pli\-ca\-tion $\gA$-\linz}
\newcommand \Alisz {ap\-pli\-ca\-tions $\gA$-\linsz}

\newcommand \Kli {ap\-pli\-ca\-tion $\gK$-\lin }
\newcommand \Klis {ap\-pli\-ca\-tions $\gK$-\lins }
\newcommand \Kliz {ap\-pli\-ca\-tion $\gK$-\linz}
\newcommand \Klisz {ap\-pli\-ca\-tions $\gK$-\linsz}

\newcommand \Bli {ap\-pli\-ca\-tion $\gB$-\lin }
\newcommand \Blis {ap\-pli\-ca\-tions $\gB$-\lins }
\newcommand \Bliz {ap\-pli\-ca\-tion $\gB$-\linz}
\newcommand \Blisz {ap\-pli\-ca\-tions $\gB$-\linsz}

\newcommand \Cli {ap\-pli\-ca\-tion $\gC$-\lin }
\newcommand \Clis {ap\-pli\-ca\-tions $\gC$-\lins }
\newcommand \Cliz {ap\-pli\-ca\-tion $\gC$-\linz}
\newcommand \Clisz {ap\-pli\-ca\-tions $\gC$-\linsz}

%:  abbr\'eviations a
\newcommand \ac{alg\'e\-bri\-quement clos }  
\newcommand \acz{alg\'e\-bri\-quement clos}  

\newcommand \acl {an\-neau \icl}
\newcommand \acls {an\-neaux \icl}
\newcommand \aclsz {an\-neaux \iclz}
\newcommand \aclz {an\-neau \iclz}

\newcommand \adk {an\-neau de Dedekind }
\newcommand \adks {an\-neaux de Dedekind }
\newcommand \adkz {an\-neau de Dedekind}
\newcommand \adksz {an\-neaux de Dedekind}

\newcommand \adp {an\-neau de Pr\"u\-fer }
\newcommand \adps {an\-neaux de Pr\"u\-fer }
\newcommand \adpsz {an\-neaux de Pr\"u\-fer}
\newcommand \adpz {an\-neau de Pr\"u\-fer}

\newcommand \adpc {\adp \coh }
\newcommand \adpcs {\adps \cohs }
\newcommand \adpcz {\adp \cohz}
\newcommand \adpcsz {\adps \cohsz}

\newcommand \adu {\alg de d\'ecom\-po\-sition univer\-selle }
\newcommand \adus {\algs de d\'ecom\-po\-sition univer\-selle }
\newcommand \aduz {\alg de d\'ecom\-po\-sition univer\-selle}
\newcommand \adusz {\algs de d\'ecom\-po\-sition univer\-selle}

\newcommand \adv {anneau de valuation }
\newcommand \advs {anneaux de valuation }
\newcommand \advsz {anneaux de valuation}
\newcommand \advz {anneau de valuation}

\newcommand \Advl {Anneau \dvla } 
\newcommand \Advls {Anneaux \dvlas } 
\newcommand \advl {anneau \dvla } 
\newcommand \advls {anneaux \dvlas } 
\newcommand \advlz {anneau \dvlaz } 
\newcommand \advlsz {anneaux \dvlasz } 

\newcommand \aG {\alg galoisienne }
\newcommand \aGs {\algs galoisiennes }
\newcommand \aGz {\alg galoisienne}
\newcommand \aGsz {\algs galoisiennes}

\newcommand \agB {\alg de Boole }
\newcommand \agBs {\algs de Boole }
\newcommand \agBz {\alg de Boole}
\newcommand \agBsz {\algs de Boole}


\newcommand \agH {\alg de Heyting }
\newcommand \agHs {\algs de Heyting }
\newcommand \agHz {\alg de Heyting}
\newcommand \agHsz {\algs de Heyting}


\newcommand \agq{alg\'e\-bri\-que }
\newcommand \agqs{alg\'e\-bri\-ques }
\newcommand \agqz{alg\'e\-bri\-que}
\newcommand \agqsz{alg\'e\-bri\-ques}

\newcommand \agqt{\agqz\-ment }

\newcommand \aKr {anneau de Krull }
\newcommand \aKrs {anneaux de Krull }
\newcommand \aKrz {anneau de Krull}
\newcommand \aKrsz {anneaux de Krull}


\newcommand \alg {alg\`e\-bre }
\newcommand \algs {alg\`e\-bres }
\newcommand \algz {alg\`e\-bre}
\newcommand \algsz {alg\`e\-bres}

\newcommand \algo{algo\-rithme }
\newcommand \algos{algo\-rithmes }
\newcommand \algoz{algo\-rithme}
\newcommand \algosz{algo\-rithmes}

\newcommand \algq{al\-go\-rith\-mi\-que }
\newcommand \algqs{al\-go\-rith\-mi\-ques }
\newcommand \algqz{al\-go\-rith\-mi\-que}
\newcommand \algqsz{al\-go\-rith\-mi\-ques}

\newcommand \ali {appli\-ca\-tion \lin }
\newcommand \alis {appli\-ca\-tions \lins }
\newcommand \aliz {appli\-ca\-tion \linz}
\newcommand \alisz {appli\-ca\-tions \linsz}

\newcommand \alo {an\-neau lo\-cal }
\newcommand \aloz {an\-neau lo\-cal}
\newcommand \alos {an\-neaux lo\-caux }
\newcommand \alosz {an\-neaux lo\-caux}

\newcommand \algb {an\-neau \lgb}
\newcommand \algbs {an\-neaux \lgbs}
\newcommand \algbz {an\-neau \lgbz}
\newcommand \algbsz {an\-neaux \lgbsz}

\newcommand \alrd {\alo \dcd }
\newcommand \alrds {\alos \dcds }
\newcommand \alrdz {\alo \dcdz}
\newcommand \alrdsz {\alos \dcdsz}

\newcommand \anar {anneau \ari}
\newcommand \anars {anneaux \aris}
\newcommand \anarsz {anneaux \arisz}
\newcommand \anarz {anneau \ariz}

\newcommand \anor {an\-neau nor\-mal }
\newcommand \anors {an\-neaux nor\-maux }
\newcommand \anorsz {an\-neaux nor\-maux}
\newcommand \anorz {an\-neau nor\-mal}

\newcommand \apf {\alg \pf}
\newcommand \apfs {\algs \pf}
\newcommand \apfz {\alg \pfz}
\newcommand \apfsz {\algs \pfz}

\newcommand \apG {\alg pr\'e\-galoisienne }
\newcommand \apGs {\algs pr\'e\-galoisiennes }
\newcommand \apGz {\alg pr\'e\-galoisienne}
\newcommand \apGsz {\algs pr\'e\-galoisiennes}

\newcommand \ari{arith\-m\'e\-tique }  
\newcommand \ariz{arith\-m\'e\-tique}  
\newcommand \aris{arith\-m\'e\-tiques }  
\newcommand \arisz{arith\-m\'e\-tiques}

\newcommand \atf {\alg \tf}
\newcommand \atfs {\algs \tf}
\newcommand \atfz {\alg \tfz}
\newcommand \atfsz {\algs \tfz}


\newcommand \auto {auto\-mor\-phis\-me }
\newcommand \autos {auto\-mor\-phis\-mes }
\newcommand \autoz {auto\-mor\-phis\-me}
\newcommand \autosz {auto\-mor\-phis\-mes}

%:  abbr\'eviations b c



\newcommand \bdg {base de Gr\"obner }
\newcommand \bdgs {bases de Gr\"obner }
\newcommand \bdgz {base de Gr\"obner}
\newcommand \bdgsz {bases de Gr\"obner}

\newcommand \bdp {base de \dcn partielle }
\newcommand \bdpz {base de \dcn partielle}
\newcommand \bdps {bases de \dcn partielle }
\newcommand \bdpsz {bases de \dcn partielle}

\newcommand \bdf {base de \fap }
\newcommand \bdfz {base de \fapz}

\newcommand \Bif {Borne inf\'e\-rieure } %
\newcommand \bif {borne inf\'e\-rieure } %
\newcommand \bifz {borne inf\'e\-rieure} %
\newcommand \bifs {bornes inf\'e\-rieures } %
\newcommand \bifsz {bornes inf\'e\-rieures} %

\newcommand \bsp {borne sup\'e\-rieure } %
\newcommand \bsps {borne sup\'e\-rieures } %
\newcommand \bspz {borne sup\'e\-rieure} %
\newcommand \bspsz {borne sup\'e\-rieures} %

\newcommand \cac{corps \ac }  
\newcommand \cacz{corps \acz}  

\newcommand \calf{calcul formel }  
\newcommand \calfz{calcul formel}  

\newcommand \cara{carac\-t\'e\-ris\-tique }  
\newcommand \caras{carac\-t\'e\-ris\-tiques }  
\newcommand \caraz{carac\-t\'e\-ris\-tique}  
\newcommand \carasz{carac\-t\'e\-ris\-tiques}

\newcommand \carn{carac\-t\'e\-ri\-sation }  
\newcommand \carns{carac\-t\'e\-ri\-sations }  

\newcommand \carar{carac\-t\'e\-riser }


\newcommand \carf{de carac\-t\`ere fini }  
\newcommand \carfz{de carac\-t\`ere fini}  

\newcommand \cdi{corps discret }
\newcommand \cdis{corps discrets }
\newcommand \cdiz{corps discret}
\newcommand \cdisz{corps discrets}
  
\newcommand \cdv{changement de variables }  
\newcommand \cdvs{changements de variables }  
\newcommand \cdvz{changement de variables}  
\newcommand \cdvsz{changements de variables}  

\newcommand \cli {cl\^o\-ture int\'e\-grale }
\newcommand \clis {cl\^o\-ture int\'e\-grale}

\newcommand \coe {coef\-fi\-cient }
\newcommand \coes {coef\-fi\-cients }
\newcommand \coez {coef\-fi\-cient}
\newcommand \coesz {coef\-fi\-cients}

\newcommand \coh {co\-h\'e\-rent }
\newcommand \cohs {co\-h\'e\-rents }
\newcommand \cohz {co\-h\'e\-rent}
\newcommand \cohsz {co\-h\'e\-rents}

\newcommand \cohc {co\-h\'e\-rence }
\newcommand \cohcz {co\-h\'e\-rence}


\newcommand \coli {combi\-nai\-son \lin }
\newcommand \colis {combi\-nai\-sons \lins }
\newcommand \coliz {combi\-nai\-son \linz}
\newcommand \colisz {combi\-nai\-sons \linsz}

\newcommand \com {co\-ma\-xi\-maux }
\newcommand \comz {co\-ma\-xi\-maux}
\newcommand \come {co\-ma\-xi\-ma\-les }
\newcommand \comez {co\-ma\-xi\-ma\-les}

\newcommand \cop {compl\'e\-men\-taire }
\newcommand \cops {compl\'e\-men\-taires }
\newcommand \copz {compl\'e\-men\-taire}
\newcommand \copsz {compl\'e\-men\-taires}

\newcommand \corg{\coh r\'egulier }
\newcommand \corgs{\cohs r\'eguliers }
\newcommand \corgz{\coh r\'egulier}
\newcommand \corgsz{\cohs r\'eguliers}


\newcommand \cpb {compa\-tible } 
\newcommand \cpbs {compa\-tibles } 
\newcommand \cpbz {compa\-tible} 
\newcommand \cpbsz {compa\-tibles} 

\newcommand \cpbt {compa\-tibi\-lit\'e } 
\newcommand \cpbtz {compa\-tibi\-lit\'e} 

%\newcommand \cro {croi\-s\'ee }
%\newcommand \cros {croi\-s\'ees }
%\newcommand \croz {croi\-s\'ee}
%\newcommand \crosz {croi\-s\'ees}

%:  abbr\'eviations d e


\newcommand \dcd {r\'e\-si\-duel\-le\-ment dis\-cret }
\newcommand \dcdz {r\'e\-si\-duel\-le\-ment dis\-cret}
\newcommand \dcds {r\'e\-si\-duel\-le\-ment dis\-crets }
\newcommand \dcdsz {r\'e\-si\-duel\-le\-ment dis\-crets}

\newcommand \dcn {d\'ecom\-po\-sition }
\newcommand \dcns {d\'ecom\-po\-sitions }
\newcommand \dcnz {d\'ecom\-po\-sition}
\newcommand \dcnsz {d\'ecom\-po\-sitions}

\newcommand \dcnb {\dcn born\'ee }
\newcommand \dcnbz {\dcn born\'ee}

\newcommand \dcnc {\dcn compl\`ete }
\newcommand \dcncz {\dcn compl\`ete}

\newcommand \dcnp {\dcn partielle }
\newcommand \dcnpz {\dcn partielle}

\newcommand \dcp {d\'ecom\-posa\-ble }
\newcommand \dcps {d\'ecom\-posa\-bles }
\newcommand \dcpz {d\'ecom\-posa\-ble}
\newcommand \dcpsz {d\'ecom\-posa\-bles}

\newcommand \ddk {dimension de~Krull }
\newcommand \ddkz {dimension de~Krull}
\newcommand \ddi {de dimension inf\'e\-rieure ou \'egale \`a~}

\newcommand \ddp {domaine de Pr\"u\-fer }
\newcommand \ddps {domaines de Pr\"u\-fer }
\newcommand \ddpz {domaine de Pr\"u\-fer}
\newcommand \ddpsz {domaines de Pr\"u\-fer}

\newcommand \Demo{D\'emon\-stra\-tion }     

\newcommand \demo{d\'emon\-stra\-tion }     
\newcommand \demoz{d\'emon\-stra\-tion}     
\newcommand \demos{d\'emon\-stra\-tions }     
\newcommand \demosz{d\'emon\-stra\-tions}     

\newcommand \dem{d\'emons\-tra\-tion }
\newcommand \demz{d\'emons\-tra\-tion}
\newcommand \dems{d\'emons\-tra\-tions }
\newcommand \demsz{d\'emons\-tra\-tions}


\newcommand \deno{d\'e\-no\-mi\-na\-teur }     
\newcommand \denos{d\'e\-no\-mi\-na\-teurs }   
\newcommand \denoz{d\'e\-no\-mi\-na\-teur}     
\newcommand \denosz{d\'e\-no\-mi\-na\-teurs}   

\newcommand \deter {d\'eter\-mi\-nant }  
\newcommand \deters {d\'eter\-mi\-nants }  
\newcommand \deterz {d\'eter\-mi\-nant}  
\newcommand \detersz {d\'eter\-mi\-nants}  
  
\newcommand \dfn{d\'efi\-nition }  
\newcommand \dfns{d\'efi\-nitions }  
\newcommand \dfnz{d\'efi\-nition}  
\newcommand \dfnsz{d\'efi\-nitions}  

  
\newcommand \dil{diff\'e\-rentiel }  
\newcommand \dils{diff\'e\-rentiels }  
\newcommand \dile{diff\'e\-ren\-tielle }  
\newcommand \diles{diff\'e\-ren\-tielles }  
\newcommand \dilz{diff\'e\-rentiel}  
\newcommand \dilsz{diff\'e\-rentiels}  
\newcommand \dilez{diff\'e\-ren\-tielle}  
\newcommand \dilesz{diff\'e\-ren\-tielles}  

\newcommand \dip{diviseur principal }
\newcommand \dips{diviseurs principaux }
\newcommand \dipz{diviseur principal}
\newcommand \dipsz{diviseurs principaux}

\newcommand \discri{discri\-minant }  
\newcommand \discris{discri\-minants }  
\newcommand \discriz{discri\-minant}  
\newcommand \discrisz{discri\-minants} 

\newcommand \divle {dimension divisorielle } 
 
%\newcommand \dvp{diviseur principal }

\newcommand \dit{distri\-bu\-ti\-vit\'e }
\newcommand \ditz{distri\-bu\-ti\-vit\'e}

\newcommand \dlg{d'\'elar\-gis\-sement }  
\newcommand \dlgz{d'\'elar\-gis\-sement}  

\newcommand \dok {domaine de Dedekind }
\newcommand \doks {domaines de Dedekind }
\newcommand \dokz {domaine de Dedekind}
\newcommand \doksz {domaines de Dedekind}

\newcommand \dvla {\`a diviseurs }% {diviso\-riel } %anneaux
\newcommand \dvlaz {\`a diviseurs}% %anneaux
\newcommand \dvlas {\`a diviseurs }% %anneaux
\newcommand \dvlasz {\`a diviseurs}% %anneaux

\newcommand \dvld {\dvlt d\'ecom\-pos\'e } %
\newcommand \dvlds {\dvlt d\'ecom\-pos\'es } %
\newcommand \dvldz {\dvlt d\'ecom\-pos\'e} %
\newcommand \dvldsz {\dvlt d\'ecom\-pos\'es} %

\newcommand \dvlg {diviso\-riel } %groupes
\newcommand \dvlgz {diviso\-riel} %groupes
\newcommand \dvlgs {diviso\-riels } %groupes
\newcommand \dvlgsz {diviso\-riels} %groupes

\newcommand \dvli {\dvlt inver\-sible } %ideal
\newcommand \dvlis {\dvlt inver\-sibles } %ideal
\newcommand \dvliz {\dvlt inver\-sible} %ideal
\newcommand \dvlisz {\dvlt inver\-sibles} %ideal

\newcommand \dvlt {diviso\-riel\-lement } %
\newcommand \dvltz {diviso\-riel\-lement} %

\newcommand \dvz {di\-viseur de z\'e\-ro }
\newcommand \dvzs {di\-viseurs de z\'e\-ro }
\newcommand \dvzz {di\-viseur de z\'e\-ro}
\newcommand \dvzsz {di\-viseurs de z\'e\-ro}

\newcommand \dve {divi\-si\-bi\-lit\'e }
\newcommand \dvez {divi\-si\-bi\-lit\'e}

\newcommand \dvee {\`a \dve explicite }
\newcommand \dveez {\`a \dve explicite}

\newcommand \dvr {diviseur }
\newcommand \dvrs {diviseurs }
\newcommand \dvrz {diviseur}
\newcommand \dvrsz {diviseurs}

\newcommand \dvrp {diviseur prin\-cipal }
\newcommand \dvrps {diviseurs prin\-cipaux }
\newcommand \dvrpz {diviseur prin\-cipal}
\newcommand \dvrpsz {diviseurs prin\-cipaux}

%:  e

\newcommand \Eds {Exten\-sion des sca\-laires }
\newcommand \Edsz {Exten\-sion des sca\-laires}
\newcommand \edss {exten\-sions des sca\-laires }
\newcommand \edssz {exten\-sions des sca\-laires}
\newcommand \eds {exten\-sion des sca\-laires }
\newcommand \edsz {exten\-sion des sca\-laires}

\newcommand \eco {\elts \com}
\newcommand \ecoz {\elts \comz}

\newcommand \egmt {\'ega\-le\-ment }
\newcommand \egmtz {\'ega\-le\-ment}

\newcommand \egt {\'ega\-li\-t\'e }
\newcommand \egts {\'ega\-li\-t\'es }
\newcommand \egtz {\'ega\-li\-t\'e}
\newcommand \egtsz {\'ega\-li\-t\'es}

\newcommand \eli{\'elimi\-nation }  
\newcommand \eliz{\'elimi\-nation}  

\newcommand \elr{\'el\'e\-men\-tai\-re }  
\newcommand \elrs{\'el\'e\-men\-tai\-res }  
\newcommand \elrz{\'el\'e\-men\-tai\-re}  
\newcommand \elrsz{\'el\'e\-men\-tai\-res}

\newcommand \elrt{\'el\'e\-men\-tai\-rement }  
\newcommand \elrtz{\'el\'e\-men\-tai\-rement}  


\newcommand \elt{\'el\'e\-ment }  
\newcommand \elts{\'el\'e\-ments }  
\newcommand \eltz{\'el\'e\-ment}  
\newcommand \eltsz{\'el\'e\-ments}  

\def \endo {en\-do\-mor\-phisme }
\def \endos {en\-do\-mor\-phismes }
\def \endoz {en\-do\-mor\-phisme}
\def \endosz {en\-do\-mor\-phismes}


\def \entrel {rela\-tion impli\-ca\-tive }
\def \entrelz {rela\-tion impli\-ca\-tive}
\def \entrels {rela\-tions impli\-ca\-tives }
\def \entrelsz {rela\-tions impli\-ca\-tives}


\newcommand\evc{es\-pa\-ce vec\-to\-riel } 
\newcommand\evcs{es\-pa\-ces vec\-to\-riels } 
\newcommand\evcz{es\-pa\-ce vec\-to\-riel} 
\newcommand\evcsz{es\-pa\-ces vec\-to\-riels} 

\newcommand \eqv {\'equi\-valent }  
\newcommand \eqve {\'equi\-valente }  
\newcommand \eqvs {\'equi\-valents }  
\newcommand \eqves {\'equi\-valentes }  
\newcommand \eqvz {\'equi\-valent}  
\newcommand \eqvez {\'equi\-valente}  
\newcommand \eqvsz {\'equi\-valents}  
\newcommand \eqvesz {\'equi\-valentes}  

\newcommand \eqvc {\'equi\-va\-lence }  
\newcommand \eqvcs {\'equi\-va\-lences }  
\newcommand \eqvcz {\'equi\-va\-lence}  
\newcommand \eqvcsz {\'equi\-va\-lences}  


%:  abbr\'eviations f g
\newcommand \fab {\fcn born\'ee }
\newcommand \fabs {\fcns born\'ees }
\newcommand \fabz {\fcn born\'ee}
\newcommand \fabsz {\fcns born\'ees}

\newcommand \fat {\fcn totale }
\newcommand \fats {\fcn totales }
\newcommand \fatz {\fcn totale}
\newcommand \fatsz {\fcn totales}

\newcommand \fap {\fcn partielle }
\newcommand \faps {\fcns partielles }
\newcommand \fapz {\fcn partielle}
\newcommand \fapsz {\fcns partielles}

\newcommand \fip {filtre pre\-mier }
\newcommand \fips {filtres pre\-miers }
\newcommand \fipz {filtre pre\-mier}
\newcommand \fipsz {filtres pre\-miers}

\newcommand \fipma {\fip maxi\-mal }
\newcommand \fipmas {\fips maxi\-maux }
\newcommand \fipmaz {\fip maxi\-mal}
\newcommand \fipmazs {\fips maxi\-maux}

\newcommand \fcn {factorisation }
\newcommand \fcns {factorisations }
\newcommand \fcnz {factorisation}
\newcommand \fcnsz {factorisations}


\newcommand \fdi {for\-te\-ment dis\-cret }
\newcommand \fdis {for\-te\-ment dis\-crets }
\newcommand \fdisz {for\-te\-ment dis\-crets}
\newcommand \fdiz {for\-te\-ment dis\-cret}


\newcommand\gmt{g\'eo\-m\'etrie }  
\newcommand\gmts{g\'eo\-m\'etries }  
\newcommand\gmtz{g\'eo\-m\'etrie }  
\newcommand\gmtsz{g\'eo\-m\'etries}  


\newcommand\gaq{\gmt \agq}  
\newcommand\gaqz{\gmt \agqz}  

\newcommand\gmq{g\'eo\-m\'e\-trique }  
\newcommand\gmqs{g\'eo\-m\'e\-triques }  
\newcommand\gmqz{g\'eo\-m\'e\-trique}  
\newcommand\gmqsz{g\'eo\-m\'e\-triques} 

\newcommand\gmqt{g\'eo\-m\'e\-tri\-quement }  
\newcommand\gmqtz{g\'eo\-m\'e\-tri\-quement}  
 

\newcommand\gne{g\'e\-n\'e\-ra\-li\-s\'e }  
\newcommand\gnee{g\'e\-n\'e\-ra\-li\-s\'ee }  
\newcommand\gnes{g\'e\-n\'e\-ra\-li\-s\'es }  
\newcommand\gnees{g\'e\-n\'e\-ra\-li\-s\'ees }  
\newcommand\gnez{g\'e\-n\'e\-ra\-li\-s\'e}  
\newcommand\gneez{g\'e\-n\'e\-ra\-li\-s\'ee}  
\newcommand\gnesz{g\'e\-n\'e\-ra\-li\-s\'es}  
\newcommand\gneesz{g\'e\-n\'e\-ra\-li\-s\'ees}  

\newcommand\gnl{g\'e\-n\'e\-ral }  
\newcommand\gnle{g\'e\-n\'e\-ra\-le }  
\newcommand\gnls{g\'e\-n\'e\-raux }  
\newcommand\gnles{g\'e\-n\'e\-ra\-les }  
\newcommand\gnlz{g\'e\-n\'e\-ral}  
\newcommand\gnlez{g\'e\-n\'e\-ra\-le}  
\newcommand\gnlsz{g\'e\-n\'e\-raux}  
\newcommand\gnlesz{g\'e\-n\'e\-ra\-les}  

\newcommand\gnlt{g\'e\-n\'e\-ra\-le\-ment }  
\newcommand\gnltz{g\'e\-n\'e\-ra\-le\-ment}  

\newcommand\gnn{g\'e\-n\'e\-ra\-li\-sa\-tion }  
\newcommand\gnns{g\'e\-n\'e\-ra\-li\-sa\-tions }  
\newcommand\gnnz{g\'e\-n\'e\-ra\-li\-sa\-tion}  
\newcommand\gnnsz{g\'e\-n\'e\-ra\-li\-sa\-tions}

\newcommand\gnr{g\'e\-n\'e\-ra\-li\-ser }  

\newcommand \gns{g\'en\'e\-ra\-lise }

\newcommand \gnt{g\'en\'e\-ra\-lit\'e }
\newcommand \gnts{g\'en\'e\-ra\-lit\'es }
\newcommand \gntz{g\'en\'e\-ra\-lit\'e}
\newcommand \gntsz{g\'en\'e\-ra\-lit\'es}

\newcommand \grl{groupe r\'eti\-cul\'e }
\newcommand \grls{groupes r\'eti\-cul\'es }
\newcommand \grlz{groupe r\'eti\-cul\'e}
\newcommand \grlsz{groupes r\'eti\-cul\'es}

\newcommand\gtr{g\'e\-n\'e\-ra\-teur }  
\newcommand\gtrs{g\'e\-n\'e\-ra\-teurs }  
\newcommand\gtrz{g\'e\-n\'e\-ra\-teur}  
\newcommand\gtrsz{g\'e\-n\'e\-ra\-teurs}  

%:  abbr\'eviations h i

\newcommand \homo {ho\-mo\-mor\-phisme }
\newcommand \homos {ho\-mo\-mor\-phismes }
\newcommand \homosz {ho\-mo\-mor\-phismes}
\newcommand \homoz {ho\-mo\-mor\-phisme}

\newcommand \hmg {homo\-g\`ene }
\newcommand \hmgs {homo\-g\`enes }
\newcommand \hmgz {homo\-g\`ene}
\newcommand \hmgsz {homo\-g\`enes}

\newcommand \icl {int\'e\-gra\-le\-ment clos }
\newcommand \icle {int\'e\-gra\-le\-ment close }
\newcommand \iclez {int\'e\-gra\-le\-ment close}
\newcommand \iclz {int\'e\-gra\-le\-ment clos}

\newcommand \id {id\'eal }
\newcommand \ids {id\'eaux }
\newcommand \idz {id\'eal}
\newcommand \idsz {id\'eaux}

\newcommand \ida {\idt \agq }
\newcommand \idas {\idts \agqs }
\newcommand \idasz {\idts \agqsz}
\newcommand \idaz {\idt \agqz}

\newcommand \idc  {\idt de Cramer }
\newcommand \idcs {\idts de Cramer }
\newcommand \idcsz {\idts de Cramer}
\newcommand \idcz {\idt de Cramer}

\newcommand \idd {id\'eal d\'eter\-minan\-tiel }
\newcommand \idds {id\'eaux d\'eter\-minan\-tiels }
\newcommand \iddz {id\'eal d\'eter\-minan\-tiel}
\newcommand \iddsz {id\'eaux d\'eter\-minan\-tiels}

\newcommand \idema {\id maxi\-mal }
\newcommand \idemas {\ids maxi\-maux }
\newcommand \idemaz {\id maxi\-mal}
\newcommand \idemasz {\ids maxi\-maux}


\newcommand \idep {\id pre\-mier }
\newcommand \idepz {\id pre\-mier}
\newcommand \ideps {\ids pre\-miers }
\newcommand \idepsz {\ids pre\-miers}

\newcommand \idemi {\idep minimal }
\newcommand \idemis {\ideps minimaux }
\newcommand \idemiz {\idep minimal}
\newcommand \idemisz {\ideps minimaux}

\newcommand \idf {id\'eal de Fitting }
\newcommand \idfs {id\'eaux de Fitting }
\newcommand \idfz {id\'eal de Fitting}
\newcommand \idfsz {id\'eaux de Fitting}

\newcommand \idif {id\'eal \dvlg fini }
\newcommand \idifs {id\'eaux \dvlgs finis }
\newcommand \idifz {id\'eal \dvlg fini}
\newcommand \idifsz {id\'eaux \dvlgs finis}

\newcommand \idli {\id \dvli } %ideal
\newcommand \idlis {\ids \dvlis } %ideal
\newcommand \idliz {\id \dvliz } %ideal
\newcommand \idlisz {\ids \dvlisz } %ideal

\newcommand \idm {idem\-po\-tent }
\newcommand \idms {idem\-po\-tents }
\newcommand \idmz {idem\-po\-tent}
\newcommand \idmsz {idem\-po\-tents}

\newcommand \idme {idem\-po\-tente }
\newcommand \idmes {idem\-po\-tentes }
\newcommand \idmez {idem\-po\-tente}
\newcommand \idmesz {idem\-po\-tentes}

\newcommand \idp {id\'e\-al prin\-ci\-pal }
\newcommand \idps {id\'e\-aux prin\-ci\-paux }
\newcommand \idpsz {id\'e\-aux prin\-ci\-paux}
\newcommand \idpz {id\'e\-al prin\-ci\-pal}

\newcommand \idt {iden\-ti\-t\'e }
\newcommand \idts {iden\-ti\-t\'es }
\newcommand \idtz {iden\-ti\-t\'e}
\newcommand \idtsz {iden\-ti\-t\'es}

\newcommand \idtr {in\-d\'e\-ter\-mi\-n\'ee }
\newcommand \idtrs {in\-d\'e\-ter\-mi\-n\'ees }
\newcommand \idtrz {in\-d\'e\-ter\-mi\-n\'ee}
\newcommand \idtrsz {in\-d\'e\-ter\-mi\-n\'ees}

\newcommand \ifr {id\'eal frac\-tion\-nai\-re }
\newcommand \ifrs {id\'eaux frac\-tion\-nai\-res }
\newcommand \ifrz {id\'eal frac\-tion\-nai\-re}
\newcommand \ifrsz {id\'eaux frac\-tion\-nai\-res}

\newcommand \imd {imm\'e\-diat }
\newcommand \imde {imm\'e\-diate }
\newcommand \imds {imm\'e\-diats }
\newcommand \imdes {imm\'e\-diates }
\newcommand \imdz {imm\'e\-diat}
\newcommand \imdez {imm\'e\-diate}
\newcommand \imdsz {imm\'e\-diats}
\newcommand \imdesz {imm\'e\-diates}

\newcommand \imdt {imm\'e\-dia\-te\-ment }
\newcommand \imdtz {imm\'e\-dia\-te\-ment}


\newcommand \ing {in\-ver\-se \gne }
\newcommand \ings {in\-ver\-ses \gnes }
\newcommand \ingz {in\-ver\-se \gnez}
\newcommand \ingsz {in\-ver\-ses \gnesz}

\newcommand \iMP {in\-ver\-se de Moo\-re-Pen\-ro\-se }
\newcommand \iMPz {in\-ver\-se de Moo\-re-Pen\-ro\-se}
\newcommand \iMPs {in\-ver\-ses de Moo\-re-Pen\-ro\-se }
\newcommand \iMPsz {in\-ver\-ses de Moo\-re-Pen\-ro\-se}

\newcommand \ipp {\idep poten\-tiel }
\newcommand \ipps {\ideps poten\-tiels }
\newcommand \ippz {\idep poten\-tiel}
\newcommand \ippsz {\ideps poten\-tiels}

\newcommand \ird {irr\'e\-duc\-tible }
\newcommand \irds {irr\'e\-duc\-tibles }
\newcommand \irdz {irr\'e\-duc\-tible}
\newcommand \irdsz {irr\'e\-duc\-tibles}

\newcommand \iso {iso\-mor\-phisme }
\newcommand \isos {iso\-mor\-phismes }
\newcommand \isosz {iso\-mor\-phismes}
\newcommand \isoz {iso\-mor\-phisme}

\newcommand \isoc {iso\-morphe }
\newcommand \isocs {iso\-morphes }
\newcommand \isocz {iso\-morphe}
\newcommand \isocsz {iso\-morphes}

\newcommand \itf {id\'e\-al \tf}
\newcommand \itfs {id\'e\-aux \tf}
\newcommand \itfz {id\'e\-al \tfz}
\newcommand \itfsz {id\'e\-aux \tfz}

\newcommand \iv {inversible }
\newcommand \ivs {inversibles }
\newcommand \ivz {inversible}
\newcommand \ivsz {inversibles}

\newcommand \ivdg {inverse divisoriel } %groupes
\newcommand \ivdgs {inverses divisoriels } %groupes
\newcommand \ivdgz {inverse divisoriel} %groupes
\newcommand \ivdgsz {inverses divisoriels} %groupes

\newcommand \ivde {inverse divisorielle } %anneaux, liste 
\newcommand \ivdes {inverses divisorielles } %anneaux, liste 
\newcommand \ivdez {inverse divisorielle} %anneaux, liste 
\newcommand \ivdesz {inverses divisorielles} %anneaux, liste 

\newcommand \ivda {inverse divisoriel } %anneaux, ideaux
\newcommand \ivdas {inverses divisoriels } %anneaux
\newcommand \ivdaz {inverse divisoriel} %anneaux
\newcommand \ivdasz {inverses divisoriels} %anneaux

%:  abbr\'eviations l m

\newcommand \lgb {local-global }
\newcommand \lgbe {locale-globale }
\newcommand \lgbs {local-globals }
\newcommand \lgbz {local-global}
\newcommand \lgbez {locale-globale}
\newcommand \lgbsz {local-globals}

\newcommand \lin {lin\'e\-aire }
\newcommand \lins {lin\'e\-aires }
\newcommand \linz {lin\'e\-aire}
\newcommand \linsz {lin\'e\-aires}

\newcommand \lint {lin\'e\-ai\-rement }

\newcommand \lmo {\lot mono\-g\`ene }
\newcommand \lmos {\lot mono\-g\`enes }
\newcommand \lmoz {\lot mono\-g\`ene}
\newcommand \lmosz {\lot mono\-g\`enes}

\newcommand \lnl {\lot \nl}
\newcommand \lnls {\lot \nls}
\newcommand \lnlz {\lot \nlz}
\newcommand \lnlsz {\lot \nlsz}

\newcommand \lot {loca\-le\-ment }
\newcommand \lotz {loca\-le\-ment}

\newcommand \lon {loca\-li\-sa\-tion }
\newcommand \lons {loca\-li\-sa\-tions }
\newcommand \lonz {loca\-li\-sa\-tion}
\newcommand \lonsz {loca\-li\-sa\-tions}

\newcommand \lop {\lot prin\-ci\-pal }
\newcommand \lops {\lot prin\-ci\-paux }
\newcommand \lopsz {\lot prin\-ci\-paux}
\newcommand \lopz {\lot prin\-ci\-pal}

\newcommand \lsdz {\lot \sdz}
\newcommand \lsdzz {\lot \sdzz}

\newcommand \mdi {mo\-dule des \diles }
\newcommand \mdiz {mo\-dule des \dilesz}

\newcommand \mlm {mo\-dule \lmo}
\newcommand \mlms {mo\-dules \lmos}
\newcommand \mlmz {mo\-dule \lmoz}
\newcommand \mlmsz {mo\-dules \lmosz}

\newcommand \mlmo {ma\-tri\-ce de loca\-li\-sa\-tion
mono\-g\`ene }
\newcommand \mlmos {ma\-tri\-ces de loca\-li\-sa\-tion
mono\-g\`ene }
\newcommand \mlmoz {ma\-tri\-ce de loca\-li\-sa\-tion
mono\-g\`ene}
\newcommand \mlmosz {ma\-tri\-ces de loca\-li\-sa\-tion
mono\-g\`ene}

\newcommand \mlp {ma\-tri\-ce de loca\-li\-sa\-tion
prin\-ci\-pa\-le }
\newcommand \mlpz {ma\-tri\-ce de loca\-li\-sa\-tion
prin\-ci\-pa\-le}
\newcommand \mlps {ma\-tri\-ces de loca\-li\-sa\-tion
prin\-ci\-pa\-le }
\newcommand \mlpsz {ma\-tri\-ces de loca\-li\-sa\-tion
prin\-ci\-pa\-le}

\newcommand \mo {mo\-no\"{\i}de }
\newcommand \mos {mo\-no\"{\i}des }
\newcommand \mosz {mo\-no\"{\i}des}
\newcommand \moz {mo\-no\"{\i}de}

\newcommand \moco {\mos\com}
\newcommand \mocoz {\mos\comz}

\newcommand \molo {morphisme de \lon }
\newcommand \molos {morphismes de \lon }
\newcommand \moloz {morphisme de \lonz }
\newcommand \molosz {morphismes de \lonz }

\newcommand \mom {mo\-n\^o\-me }
\newcommand \moms {mo\-n\^o\-mes }
\newcommand \momz {mo\-n\^o\-me}
\newcommand \momsz {mo\-n\^o\-mes}


\newcommand \moquo {morphisme de passage au quotient }
\newcommand \moquoz {morphisme de passage au quotient}
\newcommand \moquos {morphismes de passage au quotient }
\newcommand \moquosz {morphismes de passage au quotient}


\newcommand \mpf {mo\-dule \pf}
\newcommand \mpfs {mo\-dules \pf}
\newcommand \mpfz {mo\-dule \pfz}
\newcommand \mpfsz {mo\-dules \pfz}

\newcommand \mpl {mo\-dule plat }
\newcommand \mpls {mo\-dules plats }
\newcommand \mplz {mo\-dule plat}
\newcommand \mplsz {mo\-dules plats}

\newcommand \mpn {ma\-trice de \pn }
\newcommand \mpns {ma\-trices de \pn }
\newcommand \mpnz {ma\-trice de \pnz}
\newcommand \mpnsz {ma\-trices de \pnz}

\newcommand \mprn {ma\-trice de \prn }
\newcommand \mprns {ma\-trices de \prn }
\newcommand \mprnz {ma\-trice de \prnz}
\newcommand \mprnsz {ma\-trices de \prnz}

\newcommand \mptf {mo\-dule \ptf}
\newcommand \mptfs {mo\-dules \ptfs}
\newcommand \mptfz {mo\-dule \ptfz}
\newcommand \mptfsz {mo\-dules \ptfsz}

\newcommand \mrc {mo\-dule \prc }
\newcommand \mrcz {mo\-dule \prcz}
\newcommand \mrcs {mo\-dules \prcs }
\newcommand \mrcsz {mo\-dules \prcsz}

\newcommand \mtf {mo\-du\-le \tf}
\newcommand \mtfs {mo\-du\-les \tf}
\newcommand \mtfz {mo\-du\-le \tfz}
\newcommand \mtfsz {mo\-du\-les \tfz}

%:  abbr\'eviations n 

\newcommand \ncr{n\'e\-ces\-sai\-re }       
\newcommand \ncrs{n\'e\-ces\-sai\-res }       
\newcommand \ncrz{n\'e\-ces\-sai\-re}       
\newcommand \ncrsz{n\'e\-ces\-sai\-res}       

\newcommand \ncrt{n\'e\-ces\-sai\-re\-ment }       
\newcommand \ncrtz{n\'e\-ces\-sai\-re\-ment}       

\newcommand \ndz {r\'egu\-lier }
\newcommand \ndzs {r\'egu\-liers }
\newcommand \ndzz {r\'egu\-lier}
\newcommand \ndzsz {r\'egu\-liers}

\newcommand \ndze {r\'egu\-li\`ere }
\newcommand \ndzes {r\'egu\-li\`eres }
\newcommand \ndzez {r\'egu\-li\`ere}
\newcommand \ndzesz {r\'egu\-li\`eres}

\newcommand \nl {sim\-ple }
\newcommand \nlz {sim\-ple}
\newcommand \nls {sim\-ples }
\newcommand \nlsz {sim\-ples}

\newcommand \noco {\noe \coh }
\newcommand \nocos {\noes \cohs }
\newcommand \nocoz {\noe \cohz }
\newcommand \nocosz {\noes \cohsz }

\newcommand \Noe {Noether }
\newcommand \Noez {Noether}

\newcommand \noe {noeth\'e\-rien }
\newcommand \noes {noeth\'e\-riens }
\newcommand \noee {noeth\'e\-rienne }
\newcommand \noees {noeth\'e\-riennes }
\newcommand \noez {noeth\'e\-rien}
\newcommand \noesz {noeth\'e\-riens}
\newcommand \noeez {noeth\'e\-rienne}
\newcommand \noeesz {noeth\'e\-riennes}

\newcommand \noet {noeth\'e\-ria\-nit\'e }
\newcommand \noetz {noeth\'e\-ria\-nit\'e}



\newcommand \nst {Null\-stellen\-satz }
\newcommand \nstz {Null\-stellen\-satz}
\newcommand \nsts {Null\-stellen\-s\"atze }
\newcommand \nstsz {Null\-stellen\-s\"atze}


\newcommand \op{op\'e\-ra\-tion }  
\newcommand \ops{op\'e\-ra\-tions }
\newcommand \opz{op\'e\-ra\-tion}  
\newcommand \opsz{op\'e\-ra\-tions}
\newcommand \opari{\op\ari}  
\newcommand \oparis{\ops\aris}  
\newcommand \opariz{\op\ariz}  
\newcommand \oparisz{\ops\arisz}  
\newcommand \oparisv{\ops\arisv}  



\newcommand \oqc {ouvert \qc}
\newcommand \oqcs {ouverts \qcs}
\newcommand \oqcz {ouvert \qcz}
\newcommand \oqcsz {ouverts \qcsz}


\newcommand \ort{or\-tho\-go\-nal }  
\newcommand \orte{or\-tho\-go\-na\-le }  
\newcommand \orts{or\-tho\-go\-naux }  
\newcommand \ortes{or\-tho\-go\-na\-les }  
\newcommand \ortz{or\-tho\-go\-nal}  
\newcommand \ortez{or\-tho\-go\-na\-le}  
\newcommand \ortsz{or\-tho\-go\-naux}  
\newcommand \ortesz{or\-tho\-go\-na\-les}  

%:  abbr\'eviations p

\newcommand \pa {couple satur\'e }
\newcommand \pas {couples satur\'es }
\newcommand \paz {couple satur\'e}
\newcommand \pasz {couples satur\'es}
 
\newcommand \paral{pa\-ral\-l\`e\-le }  
\newcommand \parals{pa\-ral\-l\`e\-les }  
\newcommand \paralz{pa\-ral\-l\`e\-le}  
\newcommand \paralsz{pa\-ral\-l\`e\-les}

\newcommand \paralm{pa\-ral\-l\`e\-le\-ment }   


\newcommand \pb{pro\-bl\`e\-me }  
\newcommand \pbs{pro\-bl\`e\-mes }  
\newcommand \pbz{pro\-bl\`e\-me}  
\newcommand \pbsz{pro\-bl\`e\-mes} 

\newcommand \pf {de \pn finie }
\newcommand \pfz {de \pn finie}

\newcommand \plc {r\'esi\-duel\-lement \zed }
\newcommand \plcs {r\'esi\-duel\-lement \zeds }
\newcommand \plcz {r\'esi\-duel\-lement \zedz}
\newcommand \plcsz {r\'esi\-duel\-lement \zedsz}

\newcommand \Plg {Prin\-cipe \lgb }
\newcommand \plg {prin\-cipe \lgb }
\newcommand \plgs {prin\-cipes \lgbs }
\newcommand \plgz {prin\-cipe \lgbz}
\newcommand \plgsz {prin\-cipes \lgbsz}

\newcommand \plga {\plg abs\-trait }
\newcommand \plgas {\plgs abs\-traits }
\newcommand \plgaz {\plg abs\-trait}
\newcommand \plgasz {\plgs abs\-traits}

\newcommand \Plgc {\Plg con\-cret }
\newcommand \plgc {\plg con\-cret }
\newcommand \plgcz {\plg con\-cret}
\newcommand \plgcs {\plgs con\-crets }
\newcommand \plgcsz {\plgs con\-crets}

\newcommand \pn {pr\'esen\-ta\-tion }
\newcommand \pns {pr\'esen\-ta\-tions }
\newcommand \pnz {pr\'esen\-ta\-tion}
\newcommand \pnsz {pr\'esen\-ta\-tions}


\newcommand \pog {\pol \hmg }
\newcommand \pogs {\pols \hmgs }
\newcommand \pogz {\pol \hmgz}
\newcommand \pogsz {\pols \hmgsz}

\newcommand \Pol {Poly\-n\^ome }
\newcommand \Pols {Poly\-n\^omes }

\newcommand \pol {poly\-n\^ome }
\newcommand \pols {poly\-n\^omes }
\newcommand \polz {poly\-n\^ome}
\newcommand \polsz {poly\-n\^omes}


\newcommand \poll{poly\-nomial }  
\newcommand \polls{poly\-nomiaux }  
\newcommand \pollsz{poly\-nomiaux}  
\newcommand \pollz{poly\-nomial}  
\newcommand \polle{poly\-no\-miale }  
\newcommand \polles{poly\-no\-miales }  
\newcommand \pollesz{poly\-no\-miales}  
\newcommand \pollez{poly\-no\-miale}  

\newcommand \pollt{poly\-no\-mia\-le\-ment }  
\newcommand \polltz{poly\-no\-mia\-le\-ment}  



\newcommand \polfon {\pol fon\-da\-men\-tal }
\newcommand \polfonz {\pol fon\-da\-men\-tal}
\newcommand \polmu {\pol rang }
\newcommand \polmus {\pols rang }
\newcommand \polmuz{\pol rang}
\newcommand \polcar {\pol ca\-rac\-t\'e\-ris\-ti\-que }
\newcommand \polcarz {\pol ca\-rac\-t\'e\-ris\-ti\-que}
\newcommand \polmin {\pol mini\-mal }
\newcommand \polminz {\pol mini\-mal}

\newcommand \prc {\pro de rang constant }
\newcommand \prcs {\pros de rang constant }
\newcommand \prcz {\pro de rang constant}
\newcommand \prcsz {\pros de rang constant}

\newcommand \prcc {prin\-ci\-pe de \rcc }
\newcommand \prca {prin\-ci\-pe de \rca }
\newcommand \prce {prin\-ci\-pe de \rce }

\newcommand \prmt {pr\'eci\-s\'e\-ment }
\newcommand \prmtz {pr\'eci\-s\'e\-ment}
\newcommand \Prmt {Pr\'eci\-s\'e\-ment }
\newcommand \Prmtz {Pr\'eci\-s\'e\-ment}

\newcommand \prn {pro\-jec\-tion }
\newcommand \prns {pro\-jec\-tions }
\newcommand \prnz {pro\-jec\-tion}
\newcommand \prnsz {pro\-jec\-tions}

\newcommand \pro {pro\-jec\-tif }
\newcommand \pros {pro\-jec\-tifs }
\newcommand \proz {pro\-jec\-tif}
\newcommand \prosz {pro\-jec\-tifs}

\newcommand \Prof {Profondeur }
\newcommand \prof {profondeur }
\newcommand \profz {profondeur}
\newcommand \profs {profondeurs }
\newcommand \profsz {profondeurs}

\newcommand \prr {pro\-jec\-teur }
\newcommand \prrs {pro\-jec\-teurs }
\newcommand \prrz {pro\-jec\-teur}
\newcommand \prrsz {pro\-jec\-teurs}

\newcommand \Prt {Pro\-pri\-\'et\'e }
\newcommand \Prts {Pro\-pri\-\'et\'es }
\newcommand \prt {pro\-pri\-\'et\'e }
\newcommand \prts {pro\-pri\-\'et\'es }
\newcommand \prtz {pro\-pri\-\'et\'e}
\newcommand \prtsz {pro\-pri\-\'et\'es}


\newcommand \ptf {\pro \tf }
\newcommand \ptfz {\pro \tfz}
\newcommand \ptfs {\pros \tf }
\newcommand \ptfsz {\pros \tfz}


\newcommand \qc {quasi-compact }
\newcommand \qcs {quasi-compacts }
\newcommand \qcz {quasi-compact}
\newcommand \qcsz {quasi-compacts}


\newcommand \qi {qua\-si in\-t\`e\-gre }
\newcommand \qis {qua\-si in\-t\`e\-gres }
\newcommand \qisz {qua\-si in\-t\`e\-gres}
\newcommand \qiz {qua\-si in\-t\`e\-gre}


\newcommand \qnl {quasi-\nl}
\newcommand \qnls {quasi-\nls}
\newcommand \qnlz {quasi-\nlz}
\newcommand \qnlsz {quasi-\nlsz}

%:  abbr\'eviations r

\newcommand \rcc {\rcm con\-cret }
\newcommand \rca {\rcm abs\-trait }
\newcommand \rce {\rcc des \'e\-ga\-li\-t\'es }

\newcommand \rcm {recol\-le\-ment }
\newcommand \rcmz {recol\-le\-ment}
\newcommand \rcms {recol\-le\-ments }
\newcommand \rcmsz {recol\-le\-ments}

\newcommand \rcv {recou\-vrement } 
\newcommand \rcvs {recou\-vrements } 
\newcommand \rcvz {recou\-vrement} 
\newcommand \rcvsz {recou\-vrements} 

%\newcommand \rcvq {recou\-vrement par quotients } 
%\newcommand \rcvqs {recou\-vrements par quotients } 
%\newcommand \rcvqz {recou\-vrement par quotients} 
%\newcommand \rcvqsz {recou\-vrements par quotients} 

\newcommand \rde {rela\-tion de d\'epen\-dance }
\newcommand \rdes {rela\-tions de d\'epen\-dance }
\newcommand \rdesz {rela\-tions de d\'epen\-dance}
\newcommand \rdez {rela\-tion de d\'epen\-dance}

\newcommand \rdi {\rde int\'e\-grale }
\newcommand \rdis {\rdes int\'e\-grales }
\newcommand \rdiz {\rde int\'e\-grale}
\newcommand \rdisz {\rdes int\'e\-grales}

\newcommand \rdl {\rde \lin }
\newcommand \rdls {\rdes \lins }
\newcommand \rdlsz {\rde \linsz}
\newcommand \rdlz {\rdes \linz}

\newcommand \rmq {\rcm de quotients } % recollement de quotients
\newcommand \rvq {\rcv par quotients } % recouvrement par quotients
\newcommand \rmqs {\rcms de quotients } % 
\newcommand \rvqs {\rcvs par quotients } % 
\newcommand \rmqz {\rcm de quotients} % s
\newcommand \rvqz {\rcv par quotients} % 
\newcommand \rmqsz {\rcms de quotients} % 
\newcommand \rvqsz {\rcvs par quotients} % 

\newcommand \rpf {r\'eduite-de-\pnz-finie }
\newcommand \rpfs {r\'eduites-de-\pnz-finie }
\newcommand \rpfz {r\'eduite-de-\pnz-finie}
\newcommand \rpfsz {r\'eduites-de-\pnz-finie}

\newcommand \scf {sch\'ema fini\-taire }
\newcommand \scfs {sch\'emas fini\-taires }
\newcommand \scfz {sch\'ema fini\-taire}
\newcommand \scfsz {sch\'emas fini\-taires}

\newcommand \scl {sch\'ema \elr}
\newcommand \scls {sch\'emas \elrs}
\newcommand \sclz {sch\'ema \elrz}
\newcommand \sclsz {sch\'emas \elrsz}

\newcommand \sdo {\sdr \orte }
\newcommand \sdos {\sdrs \ortes }
\newcommand \sdoz {\sdr \ortez }
\newcommand \sdosz {\sdrs \ortesz }

\newcommand \sdr {somme directe }
\newcommand \sdrs {sommes directes }
\newcommand \sdrz {somme directe}
\newcommand \sdrsz {sommes directes}

\newcommand \sdz {sans \dvz}
\newcommand \sdzz {sans \dvzz}

\newcommand \sfio {sys\-t\`eme fondamental d'\idms ortho\-gonaux }
\newcommand \sfios {sys\-t\`emes fondamentaux d'\idms ortho\-gonaux }
\newcommand \sfioz {sys\-t\`eme fondamental d'\idms ortho\-gonaux}
\newcommand \sfiosz {sys\-t\`emes fondamentaux d'\idms ortho\-gonaux}

\newcommand \sgr {\sys \gtr }
\newcommand \sgrs {\syss \gtrs }
\newcommand \sgrz {\sys \gtrz}
\newcommand \sgrsz {\syss \gtrsz}

\newcommand \slgb {stricte\-ment \lgb }
\newcommand \slgbs {stricte\-ment \lgbs }
\newcommand \slgbz {stricte\-ment \lgbz}
\newcommand \slgbsz {stricte\-ment \lgbsz}

\newcommand \sli {\sys \lin }
\newcommand \slis {\syss \lins }
\newcommand \slisz {\syss \linsz}
\newcommand \sliz {\sys \linz}

\newcommand \smq {sym\'e\-trique }
\newcommand \smqs {sym\'e\-triques }
\newcommand \smqz {sym\'e\-trique}
\newcommand \smqsz {sym\'e\-triques}

\newcommand \spb {s\'epa\-rable }  % algebres
\newcommand \spbs {s\'epa\-rables }
\newcommand \spbz {s\'epa\-rable}
\newcommand \spbsz {s\'epa\-rables}

\newcommand \spl {s\'epa\-rable }  % polynomes
\newcommand \spls {s\'epa\-rables }
\newcommand \splz {s\'epa\-rable}
\newcommand \splsz {s\'epa\-rables}

\newcommand \spt{s\'epa\-ra\-bi\-lit\'e }
\newcommand \sptz{s\'epa\-ra\-bi\-lit\'e}

\newcommand \srg {suite r\'egu\-li\`ere }
\newcommand \srgs {suites r\'egu\-li\`eres }
\newcommand \srgz {suite r\'egu\-li\`ere}
\newcommand \srgsz {suites r\'egu\-li\`eres}

\newcommand \stf {stricte\-ment fini }
\newcommand \stfs {stricte\-ment finis }
\newcommand \stfz {stricte\-ment fini}
\newcommand \stfsz {stricte\-ment finis}
\newcommand \stfe {stricte\-ment finie }
\newcommand \stfes {stricte\-ment finies }
\newcommand \stfez {stricte\-ment finie}
\newcommand \stfesz {stricte\-ment finies}

\newcommand \stl {stable\-ment libre }
\newcommand \stls {stable\-ment libres }
\newcommand \stlz {stable\-ment libre}
\newcommand \stlsz {stable\-ment libres}

\newcommand \sul {suppl\'e\-men\-taire }
\newcommand \suls {suppl\'e\-men\-taires }
\newcommand \sulz {suppl\'e\-men\-taire}
\newcommand \sulsz {suppl\'e\-men\-taires}

\newcommand \Sut {Support }
\newcommand \Suts {Supports }
\newcommand \sut {support }
\newcommand \suts {supports }
\newcommand \sutz {support}
\newcommand \sutsz {supports}

\newcommand \syc {\sys de coordon\-n\'ees }
\newcommand \sycs {\syss de coordon\-n\'ees }
\newcommand \sycz {\sys de coordon\-n\'ees}
\newcommand \sycsz {\syss de coordon\-n\'ees}

\newcommand \syp {\sys \poll }
\newcommand \syps {\syss \polls }
\newcommand \sypz {\sys \pollz}
\newcommand \sypsz {\syss \pollsz}

\newcommand \sys {sys\-t\`eme }
\newcommand \syss {sys\-t\`emes }
\newcommand \sysz {sys\-t\`eme}
\newcommand \syssz {sys\-t\`emes}

%:  abbr\'eviations t

\newcommand \tf {de ty\-pe fi\-ni }
\newcommand \tfz {de ty\-pe fi\-ni} 

\newcommand \Tho {Th\'eo\-r\`eme }
\newcommand \tho {th\'eo\-r\`eme }
\newcommand \thos {th\'eo\-r\`emes }
\newcommand \thoz {th\'eo\-r\`eme}
\newcommand \thosz {th\'eo\-r\`emes}


\newcommand \thoc {th\'eo\-r\`eme$\mathbf{^*}$~}


\newcommand \trdi {treil\-lis dis\-tri\-bu\-tif }
\newcommand \trdis {treil\-lis dis\-tri\-bu\-tifs }
\newcommand \trdiz {treil\-lis dis\-tri\-bu\-tif}
\newcommand \trdisz {treil\-lis dis\-tri\-bu\-tifs}

\newcommand \trel {trans\-for\-mation \elr}
\newcommand \trels {trans\-for\-mations \elrs}
\newcommand \trelz {trans\-for\-mation \elrz}
\newcommand \trelsz {trans\-for\-mations \elrsz}

\newcommand \umd {unimo\-du\-laire }
\newcommand \umds {unimo\-du\-laires }
\newcommand \umdz {unimo\-du\-laire}
\newcommand \umdsz {unimo\-du\-laires}

\newcommand \unt {uni\-taire }
\newcommand \unts {uni\-taires }
\newcommand \untz {uni\-taire}
\newcommand \untsz {uni\-taires}

\newcommand \vgq {\vrt \agq}
\newcommand \vgqs {\vrts \agqs}
\newcommand \vgqz {\vrt \agqz}
\newcommand \vgqsz {\vrts \agqsz}

\newcommand \vrt {vari\'e\-t\'e }
\newcommand \vrts {vari\'e\-t\'es }
\newcommand \vrtz {vari\'e\-t\'e}
\newcommand \vrtsz {vari\'e\-t\'es}

\newcommand \vfn {v\'eri\-fi\-cation }
\newcommand \vfns {v\'eri\-fi\-cations }
\newcommand \vfnz {v\'eri\-fi\-cation}
\newcommand \vfnsz {v\'eri\-fi\-cations}


\def \vmd {vec\-teur \umd}
\def \vmds {vec\-teurs \umds}
\def \vmdz {vec\-teur \umdz}
\def \vmdsz {vec\-teurs \umdsz}



\newcommand \zed {z\'{e}\-ro-di\-men\-sion\-nel }
\newcommand \zedz {z\'{e}\-ro-di\-men\-sion\-nel}
\newcommand \zede {z\'{e}\-ro-di\-men\-sion\-nel\-le }
\newcommand \zedez {z\'{e}\-ro-di\-men\-sion\-nel\-le}
\newcommand \zeds {z\'{e}\-ro-di\-men\-sion\-nels }
\newcommand \zedsz {z\'{e}\-ro-di\-men\-sion\-nels}
\newcommand \zedes {z\'{e}\-ro-di\-men\-sion\-nel\-les }
\newcommand \zedesz {z\'{e}\-ro-di\-men\-sion\-nel\-les}

\newcommand \zedr {\zed r\'eduit }
\newcommand \zedrs {\zeds r\'eduits }
\newcommand \zedrz {\zed r\'eduit}
\newcommand \zedrsz {\zeds r\'eduits}


\newcommand \zmt {\tho de Zariski-Grothen\-dieck }
\newcommand \zmtz {\tho de Zariski-Grothen\-dieck}



%:  ------- maths constructives

\newcommand \cof {cons\-truc\-tif }
\newcommand \cofs {cons\-truc\-tifs }
\newcommand \cofz {cons\-truc\-tif}
\newcommand \cofsz {cons\-truc\-tifs}

\newcommand \cov {cons\-truc\-ti\-ve }
\newcommand \covz {cons\-truc\-ti\-ve}
\newcommand \covsz {cons\-truc\-ti\-ves}
\newcommand \covs {cons\-truc\-ti\-ves }

\newcommand \coma {\maths\covs}
\newcommand \comaz {\maths\covsz}
\newcommand \clama {\maths clas\-si\-ques }
\newcommand \clamaz {\maths clas\-si\-ques}

\renewcommand \cot {cons\-truc\-ti\-vement }
\newcommand \cotz {cons\-truc\-ti\-vement}

\newcommand \LLPO{{\sf LLPO}}
\newcommand \LPO{{\sf LPO}}

\newcommand \matn {math\'e\-ma\-ti\-cien }
\newcommand \matne {math\'e\-ma\-ti\-cienne }
\newcommand \matns {math\'e\-ma\-ti\-ciens }
\newcommand \matnes {math\'e\-ma\-ti\-ciennes }

\newcommand \maths {math\'e\-ma\-tiques }
\newcommand \mathsz {math\'e\-ma\-tiques}
\newcommand \mathe {math\'e\-ma\-tique }
\newcommand \mathz {math\'e\-ma\-tique}

\newcommand \prco {preuve \cov}
\newcommand \prcos {preuves \covs}
\newcommand \prcoz {preuve \covz}
\newcommand \prcosz {preuve \covsz}




