COMMENT ⊗   VALID 00005 PAGES
C REC  PAGE   DESCRIPTION
C00001 00001
C00002 00002	% OP, DOTS, PARENTHESIS		READ MAXTEX for documentation.
C00007 00003	% ALIGNMENT (for equations), CHOP, etc.
C00009 00004	% THEOREMS, PROOFS, EQNOS
C00011 00005	% INITIALIZATIONS
C00012 ENDMK
C⊗;
% OP, DOTS, PARENTHESIS		READ MAXTEX for documentation.

	% Trascendental functions:
\def\log{\mathop{\char l\char o\char g}\limitswitch}
\def\ln{\mathop{\char l\char n}\limitswitch}
\def\sin{\mathop{\char s\char i\char n}\limitswitch}
\def\cos{\mathop{\char c\char o\char s}\limitswitch}
\def\tan{\mathop{\char t\char a\char n}\limitswitch}
\def\cot{\mathop{\char c\char o\char t}\limitswitch}
\def\sec{\mathop{\char s\char e\char c}\limitswitch}
\def\csc{\mathop{\char c\char s\char c}\limitswitch}
\def\exp{\mathop{\char e\char x\char p}\limitswitch}

	% Operators:
\def\modop{\<\,\mathbin{\char m\char o\char d}\<\,}
\def\mod#1{\penalty0\;(\char m\char o\char d\,\,#1)}
\def\eqv{\mathrel\char'421 }
\def\neqv{\mathrel{\not\eqv}}
\def\neq{\mathrel{\not=}}
\def\gets{\mathrel{:}{=}} % see ex. 18.7

	% Functions:
\def\det{\mathop{\char d\char e\char t}}
\def\gcd{\mathop{\char g\char c\char d}}
\def\choose{\comb()}

	% For analysts:
\def\lim{\mathop{\char l\char i\char m}}
\def\limsup{\mathop{\char l\char i\char m\,\char s\char u\char p}}
\def\liminf{\mathop{\char l\char i\char m\,\char i\char n\char f}}
\def\max{\mathop{\char m\char a\char x}}
\def\min{\mathop{\char m\char i\char n}}
\def\sup{\mathop{\char s\char u\char p}}
\def\inf{\mathop{\char i\char n\char f}}
\def\osc{\mathop{\char o\char s\char c}}
\def\diam{\mathop{\char d\char i\char a\char m}\limitswitch}
\def\dist{\mathop{\char d\char i\char s\char t}\limitswitch}

	% Sets:
\def\leftset{\mathopen{\{\,}}
\def\rightset{\mathclose{\,\}}}
\def\emptyset{\char'100 }
\def\psubset{\char'432 }			% (proper sub(sup)sets)
\def\psupset{\char'433 }
\def\charfn#1{\raise3pt\hbox{$\chi$}↓{#1}}	% Char. fun. of set S: $\charfn{S}$

	% Ellipses
\def\ldots{{.\≥.\≥.}}
\def\cdots{{\char'401\≥\char'401\≥\char'401}}
\def\ldotss{{.\≥.\≥.\≥}}
\def\cdotss{\cdots\≥}
\def\ldotsm{{\≥.\≥.\≥.\≥}}
\def\vdots{\vbox{\baselineskip 4pt\vskip 6pt\hbox{.}\hbox{.}\hbox{.}}}

	% Large parenthesis
\def\bigglp{\mathopen{\vcenter{\hbox{\:@\char'22}}}}
\def\biggrp{\mathclose{\vcenter{\hbox{\:@\char'23}}}}
\def\biggglp{\mathopen{\vcenter{\hbox{\:@\char'40}}}}
\def\bigggrp{\mathclose{\vcenter{\hbox{\:@\char'41}}}}

	% Aliases:
\def\scst{\scriptstyle}
\def\scscst{\scriptscriptstyle}
\def\txtst{\textstyle}
\def\dspst{\dispstyle}
\let\twobar=\|
% ALIGNMENT (for equations), CHOP, etc.

	% The following are explained in the manual:
\def\eqalign#1{\baselineskip15pt\lineskip3pt
	\vcenter{\halign{\hfill$\dispstyle{##}$⊗$\dispstyle{\null##}$\hfill\cr#1}}}
\def\eqalignno#1{\baselineskip15pt\lineskip3pt
  \vbox{\tabskip 0pt plus 1000pt minus 1000pt
	\halign to size{\hfill$\dispstyle{##}$\tabskip 0pt
	⊗$\dispstyle{\null##}$\hfill\tabskip 0 pt plus 1000pt minus 1000pt
	⊗\hfill$ ##$\tabskip 0pt\cr#1}}}
\def\twoline#1#2#3{\vbox{\hbox to size{$\quad\dispstyle{#1}$\hfill}
    \vskip#2\hbox to size{\hfill$\dispstyle{#3}\quad$}}}

\def\chop to#1pt#2{\hbox{\lower#1pt\null\vbox{
					\hbox{\lower100pt\hbox{\raise100pt
					\hbox{$\dispstyle{#2}$}}}\vskip-100pt}}}
					% (pretends that #2 is #1pt deep)

	% For a left brace (as in manual, p.90). E. g., Kronecker's delta:
	%  $$\delta↓{i,j}=\bracedef{1}{\hbox{if}\ i=j;}{0}{\hbox{otherwise.}}$$
\def\bracedef#1#2#3#4{\left\{
	  \vcenter{\halign{\lft{$## $}\qquad
		   ⊗\lft{$##$}\cr
		   #1⊗#2\cr #3⊗#4\cr}}\right.}

	% 2 by 2 matrices: \mattwo{a11}{a12}{a21}{a22}
\def\mattwo#1#2#3#4{\left(\vcenter{\halign{\¬
		\ctr{$##$}\quad⊗\ctr{$##$}\quad\cr
		#1⊗#2\cr
		#3⊗#4\cr}}\right)}
% THEOREMS, PROOFS, EQNOS
   
	% To typeset a theorem, type: \thm Not all numbers are prime.\Pf Suppose...
	% The numbering is given automatically (starting at 1). When chapters are
	%   used, numbering is in form "2.8" (chap.thm: see DOCMAC for details).
	% Statement is italicized (\Pf resets it to \rm); so, type \rm at the end.
\def\thm{\stmt{\Theorem}}
\def\lem{\stmt{\Lemma}}
\def\cor{\stmt{\Corollary}}
\def\prop{\stmt{\Proposition}}
\def\rem{\stmt{\Remark}}
	% To typeset other things: \stmt{Algorithm}
\def\stmt#1{\addq1{\varstmt}\yyskip\noindent
	    {\bf#1\ \consnum{\varstmt}.\quad}\stmtfnt}
\def\stmtfnt{\ti{}}

\def\proofbegin{\par\penalty 25\vskip 6pt plus 12pt minus 4pt
   		\noindent{\sl\Proof\/}:\xskip\rm}
\def\Pf{\proofbegin}
	% Guess what:
\def\QED{\penalty 100000\quad\penalty 100000
	 \hbox{\hskip1pt\vrule width4pt height 6pt depth 1.5pt}\par\yyskip}

	% The following are similar to \eqno and \leqno, except these number
	%   automatically.
\def\neqno{\addq1{\vareqno}\eqno(\hbox{\rm\consnum{\vareqno}})}
\def\nleqno{\addq1{\vareqno}\leqno(\hbox{\rm\consnum{\vareqno}})}
% INITIALIZATIONS

\inivar{\varstmt}1	% start at theorem 1
\inivar{\vareqno}1	% start at eq. no. 1