
\newtheorem{lause}{Lause}[section]
\newtheorem{seuraus}[lause]{Seuraus}
\newtheorem{lemma}[lause]{Lemma}
\newtheorem{prop}[lause]{Propositio}
\newtheorem{ax}{Axiom}
\newtheorem{vaite}[lause]{V\"aite} 
\renewcommand{\proofname}{Tod}

\theoremstyle{definition}
\newtheorem{maarit}{M\"a\"aritelm\"a}[section]
\newtheorem{esim}{Esimerkki}[section] 
\newtheorem{huom}{Huomautus}[section] 
\newtheorem{teht}{Teht\"av\"a}[section] 

\theoremstyle{remark}
\newtheorem{notation}{Notation}
\renewcommand{\thenotation}{}  % to make the notation environment unnumbered
\renewcommand{\newline}{\null\hfill\break} 

\numberwithin{equation}{section}

\newcommand{\thmref}[1]{Lause~\ref{#1}}
\newcommand{\secref}[1]{\S\ref{#1}}
\newcommand{\lemref}[1]{Lemma~\ref{#1}}

\renewcommand{\labelenumi}{\arabic{enumi}.}
\newenvironment{Tehtavia}{\pagebreak[2]
\textbf{Tehtäviä \arabic{section}.\arabic{subsection}.}
\begin{enumerate}\setlength{\itemsep}{10pt}}{\end{enumerate}}

\font\eightrm=cmr8
\font\fiverm=cmr5
\newcommand{\stext}[1]{\text{\eightrm #1 }}

%       Math definitions

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\newcommand{\F}{{\mathcal F}}
\renewcommand{\H}{{\mathcal H}}
\renewcommand{\S}{{\mathcal S}}
\newcommand{\I}{{\mathcal I}}
\newcommand{\K}{{\mathcal K}}
\mathchardef\lscr="7160
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\newcommand{\Rat}{{\mathcal R}}
\newcommand{\U}{{\mathcal U}}
\newcommand{\X}{{\mathcal X}}
\newcommand{\Y}{{\mathcal Y}}
%      \interval is used to provide better spacing after a [ that
%      is used as a closing delimiter.
\newcommand{\interval}[1]{\mathinner{#1}}

%%%
\newcommand{\hf}{\null\hfill\null}
\newcommand{\dps}{\displaystyle}
\newcommand{\txs}{\textstyle}
\renewcommand{\ss}{\scriptstyle}
\newcommand{\sss}{\scriptscriptstyle}
\renewcommand{\hat}{\widehat}
\renewcommand{\Re}{\operatorname{Re}}
\renewcommand{\Im}{\operatorname{Im}}
\newcommand{\rank}{\operatorname{rank}}
\newcommand{\dist}{\operatorname{dist}}
\newcommand{\Off}{\operatorname{Off}}

\def\identically{\equiv}
\def\bdots{{.^{\,\dps.^{\,\dps.}}}}
\def\svdots{\vbox{\baselineskip=2pt\lineskiplimit=0pt
        \kern2pt \hbox{$\ss .$}\hbox{$\ss .$}\hbox{$\ss .$}}}
\def\sdots{{\ss{\dots}}}
\def\sddots{\mathinner{\raise4pt\vbox{\hbox{$\ss .$}}
    \raise2pt\hbox{$\ss .$}\hbox{$\ss .$}}}
\def\towith#1{\ {\mathop{\to}\limits_{#1}}\ }
\def\eqsby#1{\ {\mathop{=}\limits^{#1}}\ }
\def\eqs#1{\ {\mathop{=}\limits_{#1}}\ }
\def\ttowith#1{\ {\mathop{\longrightarrow}\limits_{#1}}\ }
\def\toappl#1{\ {\mathop{\to}\limits^{#1}}\ }
\def\ttoappl#1{\ {\mathop{\longrightarrow}\limits^{#1}}\ }
\def\otappl#1{\ {\mathop{\leftarrow}\limits^{#1}}\ }
\def\ottappl#1{\ {\mathop{\longleftarrow}\limits^{#1}}\ }

\newcommand{\mikroteht}{{$[\mu$-tehtävä$]$}}
\newcommand{\cont}{{\mathcal C}}

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\newcommand{\stiff}[1]{\hbox{$ #1 $}}
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\newcommand{\tabs}[1]{| #1 |}
\newcommand{\floor}[1]{\left\lfloor #1 \right\rfloor}
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\newcommand{\tdop}[1]{\tfrac{d\ }{d #1}}
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\newcommand{\tdops}[1]{\tfrac{d^2\ }{d #1^2}}
\newcommand{\pd}[2]{\frac{\partial #1 }{\partial #2}}
\newcommand{\tpd}[2]{\tfrac{\partial #1 }{\partial #2}}
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\newcommand{\tpdops}[1]{\tfrac{\partial^2\ }{\partial #1^2}}
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\newcommand{\tinprod}[1]{{\txs\langle #1 \rangle}}
\newcommand{\restr}[2]{ {#1}_{\big|_{\textstyle{{#2}}}}}
\newcommand{\trestr}[2]{ {#1}_{|_{#2}}}
\newcommand{\subst}[2]{ {\Big/_{\hskip-0.2cm #1}^{\,#2}\,}}
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\newcommand{\binomial}[2]{ \left( \ontop{#1}{#2}\right)}
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\hfill\epsfxsize=#3truecm\epsffile{#4}\break}

\newcommand{\equivalent}{\Longleftrightarrow}
\def\ProofBox{\hbox{\enspace{\vrule height 5pt depth 0pt width 0.5pt}\rlap
{\vrule height 5pt depth -4.5pt width 3pt}{\vrule
height 0.5pt depth 0pt width 3pt}{\vrule
height 5pt depth 0pt width 0.5pt}}}
\def\endPf{\null\hfill\ProofBox}

\def\primesum{\vphantom{\sum^n}'}

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\newcommand{\mat}[1]{{\boldsymbol{#1}}}
\def\b{\vec{b}} 
\def\bc{\vec{c}} 
\def\e{\vec{e}} 
\def\f{\vec{f}}
\def\g{\vec{g}} 
\def\h{\vec{h}} 
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\def\n{\vec{n}}
\def\p{\vec{p}} 
\def\q{\vec{q}} 
\def\r{\vec{r}}
\def\s{\vec{s}}
\def\t{\vec{t}}
\def\u{\vec{u}} 
\def\v{\vec{v}}
\def\w{\vec{w}} 
\def\x{\vec{x}}
\def\y{\vec{y}} 
\def\z{\vec{z}}
\def\bphi{\vec{\phi}}
\def\bpsi{\vec{\psi}}
\def\bP{\vec {P}} 
\def\bQ{\vec {Q}}
\def\A{\mat{A}} 
\def\B{\mat{B}} 
\def\BB{\mat{B}} 
\def\BC{\mat{C}} 
\def\BD{\mat{D}} 
\def\BE{\mat {E}} 
\def\BF{\mat {F}} 
\def\G{\mat{G}} 
\def\H{\mat{H}} 
\def\BI{\mat{I}} 
\def\BJ{\mat {J}}
\def\J{\mat {J}}
\def\BP{\mat {P}} 
\def\BQ{\mat {Q}} 
\def\BR{\mat {R}}
\def\BS{\mat {S}} 
\def\BU{\mat {U}} 
\def\V{\mat {V}} 
\def\BY{\mat{Y}}
\def\bbK{\mathbb{K}}
\def\whocares{\#}


