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From : Marc Mertens <address@hidden>- To: address@hidden
- Subject: [TeXmacs] Problem using table of content and index
- Date: Thu, 26 Jun 2014 20:45:24 +0200
Hello,
I'm using the latest version of TeXmacs (version 1.99.2) (revision 8666)
and found some annoying bug. All my documents contains a 'Table of Content'
and a 'Index' which I used to use heavely when navigation through my document
during editing. However in the latest version of Texmacs when I click on the
page number in the 'Table of Content' or in the 'Index' Texmacs freezes. The
only thing that seems to work are the scrollbars. I have compiled TexMacs on
a
Open Suse 12.3 with or without the Guile2 support to no avail. As reference
I
include a small document that demonstrates the problem (although in my humble
opinion the problem is there for all documents using a 'Table of Content' or
a
'Index'.
Thanks a lot for any help and for a amazing product
Marc Mertens
<TeXmacs|1.99.2>
<style|<tuple|book|american>>
<\body>
<doc-data|<doc-title|Special Relativity>>
<\table-of-contents|toc>
<vspace*|1fn><with|font-series|bold|math-font-series|bold|1<space|2spc>Spaces>
<datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
<no-break><pageref|auto-1><vspace|0.5fn>
1.1<space|2spc>Translation groups <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
<no-break><pageref|auto-2>
<vspace*|1fn><with|font-series|bold|math-font-series|bold|Index>
<datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
<no-break><pageref|auto-6><vspace|0.5fn>
</table-of-contents>
<chapter|Spaces>
<section|Translation groups>
<\definition>
<label|translation group><index|translation group>Given a non empty set
<math|S> then a subset <math|\<cal-T\>\<subseteq\>S<rsup|S>> (the set of
functions <math|S\<rightarrow\>S>) is a
<with|font-series|bold|translation group of <math|S>> if an only
if<verbatim|>
<\enumerate>
<item><math|\<forall\>u,v\<in\>\<cal-T\>> we have that
<math|u\<circ\>v\<in\>\<cal-T\>>
<item><math|\<forall\>u,v\<in\>\<cal-T\>> we have <math|u\<circ\>
v=v\<circ\>u>
<item><math|\<forall\>x,y\<in\>S> there exists a
<math|u\<in\>\<cal-T\>> such that <math|u<around*|(|x|)>=y>
</enumerate>
</definition>
<\definition>
<label|fixed point><index|fixed point>Given a non empty set <math|S> and
<math|\<cal-T\>> a translation group on <math|S> then a
<math|v\<in\>\<cal-T\>> has a <with|font-series|bold|fixed point x> if
<math|x\<in\>S> and <math|x=v<around*|(|x|)>>.
</definition>
<\theorem>
<label|fixed point property>Given a non empty set <math|S> and
<math|\<cal-T\>> a translation group on <math|S> then if
<math|v\<in\>\<cal-T\>> has a<math|> fixed point then <math|v=1<rsub|S>>
(in other words <math|\<forall\>y\<in\>S> we have
<math|v<around*|(|y|)>=y>
</theorem>
<\proof>
Let <math|x\<in\>S> be the fixed point of <math|v> then
<math|v<around*|(|x|)>=x>, take now <math|y\<in\>S> then there exists a
<math|u\<in\>\<cal-T\>> (see <reference|translation group> (3)) such that
<math|y=u<around*|(|x|)>\<Rightarrow\>y=u<around*|(|v<around*|(|x|)>|)>=<around*|(|u\<circ\>
v|)><around*|(|x|)>=<around*|(|v\<circ\>u|)><around*|(|x|)>=v<around*|(|u<around*|(|x|)>|)>=v<around*|(|y|)>\<Rightarrow\>y=v<around*|(|y|)>>
</proof>
<\theorem>
<label|identity is in a transaltion group>Given a non empty set <math|S>
and <math|\<cal-T\>> a translation group on <math|S> then
<math|1<rsub|S>\<in\>\<cal-T\>>
</theorem>
<\proof>
As <math|S> is not empty there exists a <math|x\<in\>S> then by
<reference|translation group> (3) there exists a <math|u\<in\>\<cal-T\>>
such that <math|u<around*|(|x|)>=x> which by the previous theorem means
that <math|u=1<rsub|S>>
</proof>
<\theorem>
<label|uniqueness of translation>Given a non empty set <math|S> and
<math|\<cal-T\>> a translation group on <math|S> and <math|x,y\<in\>S>
then there exists only one <math|v\<in\>\<cal-T\>> such that
<math|y=v<around*|(|x|)>>
</theorem>
<\proof>
Existence is already guaranteed by the definition of a translation group,
So assume that there exists <math|v<rsub|1>,v<rsub|2>\<in\>\<cal-T\>>
such that <math|v<rsub|1><around*|(|x|)>=y=v<rsub|2><around*|(|x|)>>.
Take then <math|z\<in\>S> then there exists a <math|w\<in\>\<cal-S\>>
such that <math|w<around*|(|x|)>=z> so that
<math|v<rsub|1><around*|(|z|)>=v<rsub|1><around*|(|w<around*|(|x|)>|)>=<around*|(|v<rsub|1>\<circ\>w|)><around*|(|x|)>=<around*|(|w\<circ\>
v<rsub|1>|)><around*|(|x|)>=w<around*|(|v<rsub|1><around*|(|x|)>|)>=w<around*|(|y|)>=w<around*|(|v<rsub|2><around*|(|x|)>|)>=<around*|(|w\<circ\>
v<rsub|2>|)><around*|(|x|)>=<around*|(|v<rsub|2>\<circ\>w|)><around*|(|x|)>=v<rsub|2><around*|(|w<around*|(|x|)>|)>=v<rsub|2><around*|(|z|)>>
proving that as <math|z> was chosen arbitrary that
<math|v<rsub|1>=v<rsub|2>>.
</proof>
<\theorem>
<label|translation group exists of bijections>Given a non empty set
<math|S> and <math|\<cal-T\>> a translation group on <math|S> then
<math|\<forall\>u\<in\>\<cal-T\>> we have that <math|u> is a bijection.
So <math|u<rsup|-1>> exists, additionally we have that
\ <math|u<rsup|-1>\<in\>\<cal-T\>>
</theorem>
<\proof>
Let <math|u\<in\>\<cal-T\>> then we prove that is a bijection.
<\enumerate>
<item><dueto|injection>As <math|S\<neq\>\<emptyset\>> there exists a
<math|x\<in\>S> take then <math|y=u<around*|(|x|)>> then by
<reference|translation group> (3) there exists a
<math|v\<in\>\<cal-T\>> such that <math|v<around*|(|y|)>=x> and thus we
have <math|<around*|(|v\<circ\>u|)><around*|(|x|)>=v<around*|(|u<around*|(|x|)>|)>=v<around*|(|y|)>=x\<Rightarrow\><around*|(|v\<circ\>u|)>>
has a fixed point and thus by <reference|fixed point property> we have
that <math|<around*|(|v\<circ\>u|)>=1<rsub|S>>. So if
<math|u<around*|(|x<rsub|1>|)>=u<around*|(|x<rsub|2>|)>\<Rightarrow\>v<around*|(|u<around*|(|x<rsub|1>|)>|)>=v<around*|(|u<around*|(|x<rsub|2>|)>|)>\<Rightarrow\><around*|(|v\<circ\>u|)><around*|(|x<rsub|1>|)>=<around*|(|v\<circ\>u|)><around*|(|x<rsub|2>|)>\<Rightarrow\>1<rsub|S><around*|(|x<rsub|1>|)>=1<rsub|S><around*|(|x<rsub|2>|)>\<Rightarrow\>x<rsub|1>=x<rsub|2>>
proving injectivity.
<item><dueto|surjection>Let <math|y\<in\>S> take then
<math|x=u<around*|(|y|)>> using again <reference|translation group> (3)
there exists a <math|v\<in\>\<cal-T\>> such that
<math|v<around*|(|x|)>=y\<Rightarrow\>v<around*|(|u<around*|(|y|)>|)>=y\<Rightarrow\><around*|(|v\<circ\>u|)><around*|(|y|)>=y\<Rightarrowlim\><rsub|<with|mode|text|<reference|translation
group> (2)>><around*|(|u\<circ\> v|)><around*|(|y|)>=u<around*|(|v<around*|(|y|)>|)>>
proving surjectivity.
</enumerate>
To prove that <math|u<rsup|-1>\<in\>\<cal-T\>>, as
<math|S\<neq\>\<emptyset\>> there exists a <math|x\<in\>S> take \ then
<math|y=u<around*|(|x|)>> then there exists a <math|v\<in\>\<cal-T\>>
such that <math|v<around*|(|y|)>=x> so that
<math|<around*|(|v\<circ\>u|)><around*|(|x|)>=x> proving that
<math|v\<circ\>u> has a fixed point so that we must have that
<math|v\<circ\>u=1<rsub|S>> so that <math|u<rsup|-1>=1<rsub|S>\<circ\>u<rsup|-1>=<around*|(|v\<circ\>u|)>\<circ\>u<rsup|-1>=v\<circ\><around*|(|u\<circ\>u<rsup|-1>|)>=v\<circ\>1<rsub|S>=v\<in\>\<cal-T\>>
proving that <math|u<rsup|-1>\<in\>\<cal-T\>>.
</proof>
We show now that the name translation group is correct in the following
theorem
<\theorem>
Given a non empty set <math|S> and <math|\<cal-T\>> a translation group
on <math|S> then <math|<around*|\<langle\>|\<cal-T\>,\<circ\>|\<rangle\>>>
is a abelian group
</theorem>
<\proof>
We check if <math|\<circ\>:\<cal-T\>\<times\>\<cal-T\>\<rightarrow\>\<cal-T\>>
defined by <math|<around*|(|u,v|)>\<rightarrow\>u\<circ\>
v\<in\>\<cal-T\>> satisfies the group axioms
<\enumerate>
<item><dueto|associativity>This follows from the properties of
composition of functions
<item><dueto|neutral element>As <math|1<rsub|S>\<in\>\<cal-T\>> we have
for any <math|u\<in\>\<cal-T\>> that
<math|1<rsub|S>\<circ\>u=u=u\<circ\>1<rsub|S>>
<item><dueto|inverse element>Given <math|u\<in\>\<cal-T\>> we have by
<reference|translation group exists of bijections> that
<math|u<rsup|-1>\<in\>\<cal-T\>> and
<math|u\<circ\>u<rsup|-1>=1<rsub|S>=u<rsup|-1>\<circ\>u> proving that
<math|u<rsup|-1>> is the inverse of <math|u>
<item><dueto|commutativity>This follows from <reference|translation
group> (2)
</enumerate>
</proof>
<\notation>
Given a non empty set <math|S> and <math|\<cal-T\>> a translation group
on <math|S> then we use the following notations
<\enumerate>
<item>Elements of <math|S> are noted by normal letters <math|x,y,z
\<ldots\>\<ldots\>>
<item>Elements of <math|\<cal-T\>> are noted by
<math|<wide|u|\<vect\>>,<wide|v|\<vect\>>>,....
<item><math|1<rsub|S>\<in\>\<cal-T\>> is noted \ as
<math|<wide|0|\<vect\>>>
<item>If <math|<wide|u|\<vect\>>,<wide|v|\<vect\>>\<in\>\<cal-T\> then
<wide|u|\<vect\>>\<circ\><wide|v|\<vect\>>> is noted by
<math|<wide|v|\<vect\>>+<wide|u|\<vect\>>>
<item>If <math|<wide|u|\<vect\>>\<in\>\<cal-T\>> then
<math|<wide|u|\<vect\>><rsup|-1>> is noted by <math|-<wide|u|\<vect\>>>
<item>If <math|<wide|u|\<vect\>>,<wide|v|\<vect\>>\<in\>\<cal-T\>> then
<math|<wide|v|\<vect\>>+<around*|(|-<wide|u|\<vect\>>|)>> is noted by
<math|<wide|v|\<vect\>>-<wide|u|\<vect\>>>
<item>If <math|x\<in\>S>, <math|<wide|u|\<vect\>>\<in\>\<cal-T\>> then
<math|<wide|u|\<vect\>><around*|(|x|)>> is noted by
<math|x+<wide|u|\<vect\>>>\
<item>If <math|x\<in\>S>, <math|<wide|u|\<vect\>>\<in\>\<cal-T\>> then
<math|x+<around*|(|-<wide|u|\<vect\>>|)>> is noted by
<math|<wide|x|\<vect\>>-<wide|u|\<vect\>>> (and is equal to
<math|<wide|u|\<vect\>><rsup|-1>*<around*|(|x|)>>
<item>If <math|x,y\<in\>S> then the unique <math|<wide|v|\<vect\>>>
such that <math|y=<wide|v|\<vect\>><around*|(|x|)>> (see
<reference|uniqueness of translation>) is noted by <math|y-x>\
</enumerate>
</notation>
The reason for this notation is that then some relations are expressed in a
very natural way as in the following theorem. Also because we have a
distinct notation for elements of <math|S> and <math|\<cal-T\>> we can use
the same operator <math|+> and <math|-> in the different ways without
having to introduce different\
<\theorem>
<label|properties of transalation groups>Given a non empty set <math|S>
and <math|\<cal-T\>> a translation group on <math|S> then we have
<\enumerate>
<item><math|<around*|\<langle\>|S,+|\<rangle\>>> is a abelian group
<item><math|\<forall\>x,y\<in\>S> we have <math|x+<around*|(|y-x|)>=y>
<item><math|\<forall\>x\<in\>S> and
<math|\<forall\><wide|u|\<vect\>>\<in\>\<cal-T\>> we have
<math|<around*|(|x+<wide|u|\<vect\>>|)>-x=<wide|u|\<vect\>>>
<item><math|\<forall\>x\<in\>S> we have <math|x-x=<wide|0|\<vect\>>>
<item><math|\<forall\><wide|u|\<vect\>>\<in\>\<cal-T\>> we have
<math|<wide|u|\<vect\>>-<wide|u|\<vect\>>=<wide|0|\<vect\>>>
<item><math|\<forall\>x\<in\>S> and
<math|\<forall\><wide|u|\<vect\>>,<wide|v|\<vect\>>\<in\>\<cal-T\>> we
have <math|x+<around*|(|<wide|u|\<vect\>>+<wide|v|\<vect\>>|)>=<around*|(|x+<wide|u|\<vect\>>|)>+<wide|v|\<vect\>>>
<item><math|\<forall\>x\<in\>S> and
<math|\<forall\><wide|u|\<vect\>>,<wide|v|\<vect\>>\<in\>\<cal-T\>> we
have <math|x+<wide|u|\<vect\>>=x+<wide|v|\<vect\>>\<Rightarrow\><wide|u|\<vect\>>=<wide|v|\<vect\>>>
<item><math|\<forall\>x,y\<in\>S> and
<math|\<forall\><wide|u|\<vect\>>\<in\>\<cal-T\>> we have
<math|x+<wide|u|\<vect\>>=y+<wide|u|\<vect\>>\<Rightarrow\>x=y>
<item><math|\<forall\>x,y\<in\>S> and
<math|\<forall\><wide|u|\<vect\>>\<in\>\<cal-T\>> we have
<math|<around*|(|x-y|)>+<wide|u|\<vect\>>=<around*|(|x+<wide|u|\<vect\>>|)>-y>
<item><math|\<forall\>x,y,z\<in\>S> we have
<math|<around*|(|x-y|)>+<around*|(|y-z|)>=x-z>
<item><math|\<forall\>x\<in\>S>, <math|<wide|u|\<vect\>>\<in\>\<cal-T\>>
we have <math|x-<around*|(|x+<wide|u|\<vect\>>|)>=-<wide|u|\<vect\>>>
</enumerate>
</theorem>
<\proof>
\ \
<\enumerate>
<item>This follows from the fact that
<math|<around*|\<langle\>|S,\<circ\>|\<rangle\>>> is a abelian group
and <math|<wide|u|\<vect\>>+<wide|v|\<vect\>>=<wide|v|\<vect\>>\<circ\><wide|u|\<vect\>>>
<item>By notation <math|> <math|<around*|(|y-x|)>=<wide|v|\<vect\>>\<in\>\<cal-T\>>
such that <math|<wide|v|\<vect\>><around*|(|x|)>=y> and as by notation
<math|y=<wide|v|\<vect\>><around*|(|x|)>=x+<wide|v|\<vect\>>=x+<around*|(|x-y|)>>
<item>By notation we have that <math|<around*|(|x+<wide|u|\<vect\>>|)>=<wide|u|\<vect\>><around*|(|x|)>>
so that by notation <math|<around*|(|x+<wide|u|\<vect\>>|)>-x=<wide|u|\<vect\>><around*|(|x|)>-x>
is the unique <math|<wide|v|\<vect\>>\<in\>\<cal-T\>> such that
<math|<wide|v|\<vect\>><around*|(|x|)>=<wide|u|\<vect\>><around*|(|x|)>>
which by <reference|uniqueness of translation> means that
<math|<wide|u|\<vect\>>=<wide|v|\<vect\>>> so that
<math|<around*|(|x+<wide|u|\<vect\>>|)>-x=<wide|u|\<vect\>>>
<item>As <math|1<rsub|S>=<wide|0|\<vect\>>> we have
<math|x+<wide|0|\<vect\>>=<wide|0|\<vect\>><around*|(|x|)>=1<rsub|S><around*|(|x|)>=x>
so that <math|x-x=<around*|(|x+<wide|0|\<vect\>>|)>-x\<equallim\><rsub|<around*|(|3|)>><wide|0|\<vect\>>>
<item><math|<wide|u|\<vect\>>-<wide|u|\<vect\>>=<wide|u|\<vect\>>+<around*|(|-<wide|u|\<vect\>>|)>=<wide|u|\<vect\>>+<wide|u|\<vect\>><rsup|-1>=<around*|(|<wide|u|\<vect\>><rsup|-1>|)>\<circ\><wide|u|\<vect\>>=1<rsub|S>=<wide|0|\<vect\>>>
<item><math|x+<around*|(|<wide|u|\<vect\>>+<wide|v|\<vect\>>|)>=<around*|(|<wide|u|\<vect\>>+<wide|v|\<vect\>>|)><around*|(|x|)>=<around*|(|<wide|v|\<vect\>>\<circ\><wide|u|\<vect\>>|)><around*|(|x|)>=<wide|v|\<vect\>><around*|(|<wide|u|\<vect\>><around*|(|x|)>|)>=<wide|v|\<vect\>><around*|(|x+<wide|u|\<vect\>>|)>=<around*|(|x+<wide|u|\<vect\>>|)>+<wide|v|\<vect\>>>
<item>If <math|x+<wide|u|\<vect\>>=x+<wide|v|\<vect\>>> \ then
<math|<wide|u|\<vect\>><around*|(|x|)>=<wide|v|\<vect\>><around*|(|x|)>>
so that by <reference|uniqueness of translation> we have
<math|<wide|u|\<vect\>>=<wide|v|\<vect\>>>
<item>If <math|x+<wide|u|\<vect\>>=y+<wide|u|\<vect\>>> then
<math|<wide|u|\<vect\>><around*|(|x|)>=<wide|u|\<vect\>><around*|(|y|)>\<Rightarrowlim\><rsub|<wide|u|\<vect\>>
is a bijection <around*|(|see <reference|translation group exists of
bijections>|)>>x=y>
<item>As <math|<around*|(|x-y|)>=<wide|v|\<vect\>>\<in\>\<cal-T\>> such
that <math|<wide|v|\<vect\>><around*|(|y|)>=x> we have that
<math|<around*|(|x-y|)>+<wide|u|\<vect\>>=<wide|v|\<vect\>>+<wide|u|\<vect\>>=<wide|u|\<vect\>>\<circ\><wide|v|\<vect\>>>
and thus <math|<around*|(|<around*|(|x-y|)>+<wide|u|\<vect\>>|)><around*|(|y|)>=<around*|(|<wide|u|\<vect\>>\<circ\><wide|v|\<vect\>>|)><around*|(|y|)>=<wide|u|\<vect\>><around*|(|<wide|v|\<vect\>><around*|(|y|)>|)>=<wide|u|\<vect\>><around*|(|x|)>=x+<wide|u|\<vect\>>>
meaning that <math|<around*|(|x-y|)>+<wide|u|\<vect\>>=<around*|(|x+<wide|u|\<vect\>>|)>-y>\
<item>We have by notation that <math|<around*|(|x-y|)>=<wide|u|\<vect\>>\<in\>\<cal-T\>>
such that <math|<wide|u|\<vect\>><around*|(|y|)>=x> and
<math|<around*|(|y-z|)>=<wide|v|\<vect\>>\<in\>\<cal-T\>> such that
<math|<wide|v|\<vect\>><around*|(|z|)>=y> we have then that
<math|<around*|(|<around*|(|x-y|)>+<around*|(|y-z|)>|)><around*|(|z|)>=<around*|(|<wide|u|\<vect\>>+<wide|v|\<vect\>>|)><around*|(|z|)>=<around*|(|<wide|v|\<vect\>>\<circ\><wide|u|\<vect\>>|)><around*|(|z|)>=<around*|(|<wide|u|\<vect\>>\<circ\><wide|v|\<vect\>>|)><around*|(|z|)>=<wide|u|\<vect\>><around*|(|<wide|v|\<vect\>><around*|(|z|)>|)>=<wide|u|\<vect\>><around*|(|y|)>=x>
proving that <math|<around*|(|x-y|)>+<around*|(|y-z|)>=<around*|(|x-z|)>>
<item>Let <math|<wide|v|\<vect\>>=x-<around*|(|x+<wide|u|\<vect\>>|)>>
then <math|<wide|v|\<vect\>><around*|(|x+<wide|u|\<vect\>>|)>=x\<Rightarrow\><around*|(|<wide|v|\<vect\>><around*|(|<wide|u|\<vect\>><around*|(|x|)>|)>|)>=x\<Rightarrow\><around*|(|<wide|v|\<vect\>>\<circ\><wide|u|\<vect\>>|)><around*|(|x|)>=x=1<rsub|S><around*|(|x|)>\<Rightarrowlim\><rsub|<reference|uniqueness
of translation>><wide|v|\<vect\>>\<circ\><wide|u|\<vect\>>=1<rsub|S>\<Rightarrow\><wide|v|\<vect\>>=1<rsub|S>\<circ\><wide|u|\<vect\>><rsup|-1><rsub|>=<wide|u|\<vect\>><rsup|-1>=-<wide|u|\<vect\>>>
</enumerate>
</proof>
<\definition>
If <math|S> is a nonempty set, <math|\<cal-T\>> a translation group on
<math|S>, <math|x\<in\>S>, <math|<wide|v|\<vect\>>\<in\>\<cal-T\>>,
<math|G,H\<subseteq\>S> and <math|\<cal-U\>,\<cal-W\>\<subseteq\>\<cal-V\>>
then we define
<\enumerate>
<item><math|H+\<cal-U\>=<around*|{|x+<wide|u|\<vect\>>\|x\<in\>H\<wedge\><wide|u|\<vect\>>\<in\>\<cal-U\>|}>>
<item><math|H-\<cal-U\>=<around*|{|x-<wide|v|\<vect\>>\|x\<in\>H\<wedge\><wide|u|\<vect\>>\<in\>\<cal-U\>|}>>
<item><math|\<cal-U\>+\<cal-W\>=<around*|{|<wide|u|\<vect\>>+<wide|w|\<vect\>>\|<wide|u|\<vect\>>\<in\>\<cal-U\>\<wedge\><wide|w|\<vect\>>\<in\>\<cal-W\>|}>>
<item><math|\<cal-U\>-\<cal-W\>=<around*|{|<wide|u|\<vect\>>-<wide|w|\<vect\>>\|<wide|u|\<vect\>>\<in\>\<cal-U\>\<wedge\><wide|w|\<vect\>>\<in\>\<cal-W\>|}>>
<item><math|x+\<cal-U\>=<around*|{|x+<wide|u|\<vect\>>\|<wide|u|\<vect\>>\<in\>\<cal-U\>|}>>
<item><math|x-\<cal-U\>=<around*|{|x-<wide|u|\<vect\>>\|<wide|u|\<vect\>>\<in\>\<cal-U\>|}>>
<item><math|G+<wide|v|\<vect\>>=<around*|{|x+<wide|v|\<vect\>>\|x\<in\>G|}>>
<item><math|G-<wide|v|\<vect\>>=<around*|{|x-<wide|v|\<vect\>>\|x\<in\>G|}>>
<item><math|G-H=<around*|{|x-y\|x\<in\>G\<wedge\>y\<in\>H|}>>
<item><math|-\<cal-U\>=<around*|{|-<wide|u|\<vect\>>\|<wide|u|\<vect\>>\<in\>\<cal-U\>|}>>
</enumerate>
</definition>
\;
<\definition>
<label|flat space><index|flat space>A <with|font-series|bold|flat space>
<math|<around*|\<langle\>|S,\<cal-T\>,\<cdot\>|\<rangle\>>> is a non
empty set <math|S>, a <math|\<cal-T\>> translation group on <math|S>
together with a map <math|\<cdot\>:\<bbb-R\>\<times\>\<cal-T\>\<rightarrow\>\<cal-T\>>
mapping <math|<around*|(|\<alpha\>,<wide|u|\<vect\>>|)>\<rightarrow\>\<alpha\>\<cdot\><wide|u|\<vect\>>>
such that\
<\enumerate>
<item><math|\<forall\>\<alpha\>,\<beta\>\<in\>\<bbb-R\>>,
<math|\<forall\><wide|u|\<vect\>>\<in\>\<cal-T\>> we have
<math|\<alpha\>\<cdot\><around*|(|\<beta\>\<cdot\><wide|u|\<vect\>>|)>=<around*|(|\<alpha\>\<cdot\>\<beta\>|)>\<cdot\><wide|u|\<vect\>>>
<item><math|\<forall\>\<alpha\>,\<beta\>\<in\>\<bbb-R\>>,
<math|\<forall\><wide|u|\<vect\>>\<in\>\<cal-T\>> we have
<math|<around*|(|\<alpha\>+\<beta\>|)>\<cdot\><wide|u|\<vect\>>=\<alpha\>\<cdot\><wide|u|\<vect\>>+\<beta\>\<cdot\><wide|u|\<vect\>>>
<item><math|\<forall\>\<alpha\>\<in\>\<bbb-R\>>,
<math|\<forall\><wide|u|\<vect\>>,<wide|v|\<vect\>>\<in\>\<cal-T\>> we
have <math|\<alpha\>\<cdot\><around*|(|<wide|u|\<vect\>>+<wide|v|\<vect\>>|)>=\<alpha\>\<cdot\><wide|u|\<vect\>>+\<alpha\>\<cdot\><wide|v|\<vect\>>>
<item><math|\<forall\><wide|u|\<vect\>>\<in\>\<cal-T\>> we have
<math|1\<cdot\><wide|u|\<vect\>>=<wide|u|\<vect\>>>
</enumerate>
making essential <math|<around*|\<langle\>|\<cal-T\>,+,\<cdot\>|\<rangle\>>>
a vector space over <math|\<bbb-R\>>. If
<math|<around*|\<langle\>|\<cal-T\>,+,\<cdot\>|\<rangle\>>> is finite
dimensional then <math|dim<around*|(|<around*|\<langle\>|S,\<cal-T\>,\<cdot\>|\<rangle\>>|)>>
is by definition <math|dim<around*|(|<around*|\<langle\>|\<cal-T\>,+,\<cdot\>|\<rangle\>>|)>>
</definition>
<\definition>
<label|external translation space><index|external translation space>Let
<math|S> be a nonempty set, <math|<around*|\<langle\>|V,+,\<cdot\>|\<rangle\>>>
a vector space on <math|\<bbb-R\>> such that there is a mapping
<math|D:S\<times\>S\<rightarrow\>V> such that\
<\enumerate>
<item><math|\<forall\>x,y,z\<in\>S> we have
<math|D<around*|(|x,z|)>=D<around*|(|x,y|)>+D<around*|(|y,z|)>>
<item><math|\<forall\>x\<in\>S>, <math|\<forall\>v\<in\>V> there exists
exactly one <math|y\<in\>S> such that <math|D<around*|(|x,y|)>=v>
</enumerate>
then we call <math|V> a <with|font-series|bold|external translation
space> on <math|S.>
</definition>
The next theorem shows that given a external translation space for <math|S>
we can construct a translation space on S
\;
<\theorem>
Let <math|S> be a nonempty set, <math|<around*|\<langle\>|V,+,\<cdot\>|\<rangle\>>>
a vector space on <math|\<bbb-R\>> and assume that there is a mapping
<math|D:S\<times\>S\<rightarrow\>V> that satisfies\
<\enumerate>
<item><math|\<forall\>x,y,z\<in\>S> we have
<math|D<around*|(|x,z|)>=D<around*|(|x,y|)>+D<around*|(|y,z|)>>
<item><math|\<forall\>x\<in\>S>, <math|\<forall\>v\<in\>V> there exists
exact one <math|y\<in\>S> such that <math|D<around*|(|x,y|)>=v>
</enumerate>
in other words <math|V> is a external translation space for <math|S>.
Then there exists a unique flat space
<math|<around*|\<langle\>|S,\<cal-T\>,\<cdot\>|\<rangle\>>> such that
there is a unique <math|\<varphi\>:V\<rightarrow\>\<cal-T\>> with\
<\enumerate-roman>
<item><math|\<forall\>x,y\<in\>S> we have
<math|\<varphi\><around*|(|D<around*|(|x,y|)>|)>=x-y>\
<item><math|\<forall\>u\<in\>V> and
<math|\<forall\>\<alpha\>\<in\>\<bbb-R\>> we have
<math|\<varphi\><around*|(|\<alpha\>\<cdot\>u|)>=\<alpha\>\<cdot\>\<varphi\><around*|(|u|)>>
</enumerate-roman>
Finally the mapping <math|\<varphi\>> is a bijection and
<math|\<forall\>x,y\<in\>S> we have <math|\<varphi\><around*|(|x+y|)>=\<varphi\><around*|(|x|)>+\<varphi\><around*|(|y|)>>
(the mapping is linear). It follows that <math|V> and <math|\<cal-T\>>
are isomorph with each other, so we can identify <math|V> with
<math|\<cal-T\>>.
</theorem>
<\proof>
Given a <math|v\<in\>V> then using (2) for every <math|x> there exists
exactly one <math|y> such that <math|D<around*|(|x,y|)>=v>, which defines
a function <math|\<psi\><rsub|v>:S\<rightarrow\>S>. Also if there is
another function <math|\<varphi\><rsub|v>> such
<math|D<around*|(|x,\<varphi\><rsub|v><around*|(|x|)>|)>=v> then by (2)
<math|\<forall\>x\<in\>S> we have <math|\<varphi\><rsub|v><around*|(|x|)>=\<psi\><rsub|v><around*|(|x|)>\<Rightarrow\>\<varphi\>=\<psi\>>).
So there exists a exactly one function
<math|\<psi\><rsub|v>:S\<rightarrow\>S> such that
<\equation>
<label|eq 1.1>\<forall\>v\<in\>V,\<forall\>x\<in\>S we have
D<around*|(|x,\<psi\><rsub|v><around*|(|x|)>|)>=v
</equation>
We define then
<\equation>
<label|eq 1.2>\<cal-T\>=<around*|{|\<psi\><rsub|v>\|v\<in\>V|}>
</equation>
If <math|u,v\<in\>V> then <math|u+v\<in\>V> and using (1) we have
<math|\<forall\>x\<in\>S> that <math|D<around*|(|x,<around*|(|\<psi\><rsub|u>\<circ\>\<psi\><rsub|v>|)><around*|(|x|)>|)>=D<around*|(|x,\<psi\><rsub|u><around*|(|\<psi\><rsub|v><around*|(|x|)>|)>|)>=D<around*|(|x,\<psi\><rsub|v><around*|(|x|)>|)>+D<around*|(|\<psi\><rsub|v><around*|(|x|)>,\<psi\><rsub|u><around*|(|\<psi\><rsub|v><around*|(|x|)>|)>|)>\<equallim\><rsub|<text|<reference|eq
1.1>>>v+u\<equallim\><rsub|<text|<reference|eq
1.1>>>D<around*|(|x,\<psi\><rsub|v+u><around*|(|x|)>|)>> so that
<math|D<around*|(|x,<around*|(|\<psi\><rsub|u>\<circ\>\<psi\><rsub|v><around*|(|x|)>|)>|)>=D<around*|(|x,\<psi\><rsub|v+u><around*|(|x|)>|)>>
which by (2) means that <math|<around*|(|\<psi\><rsub|u>\<circ\>\<psi\><rsub|v>|)><around*|(|x|)>=\<psi\><rsub|v+u><around*|(|x|)>>
proving that
<\equation>
<label|eq 1.3>\<forall\>u,v\<in\>V we have
\<psi\><rsub|u>\<circ\>\<psi\><rsub|v>=\<psi\><rsub|v+u>\<in\>\<cal-T\>
</equation>
As in <math|V> we have also <math|u+v=v+u> we have automatically
<\equation>
<label|eq 1.4>\<forall\>u,v\<in\>V we have
\<psi\><rsub|u>\<circ\>\<psi\><rsub|v>=\<psi\><rsub|v>\<circ\>\<psi\><rsub|u>
</equation>
If <math|x,y\<in\>S> then by (2) there exists a <math|v\<in\>V> such that
<math|D<around*|(|x,y|)>=v\<equallim\><rsub|<reference|eq
1.1>>D<around*|(|x,\<psi\><rsub|v><around*|(|x|)>|)>\<Rightarrowlim\><rsub|<around*|(|2|)>>y=\<psi\><rsub|v><around*|(|x|)>>
so that
<\equation>
<label|eq 1.5>\<forall\>x,y\<in\>S we have
\<exists\>\<psi\><rsub|v>\<in\>\<cal-T\> such that
\<psi\><rsub|v><around*|(|x|)>=y
</equation>
Using the definition of a translation group (see <reference|translation
group>) and <reference|eq 1.2>,<reference|eq 1.3>,<reference|eq
1.4>,<reference|eq 1.4> we have that <math|\<cal-T\>> is indeed a
translation group on <math|S>. Define now
<math|\<cdot\>:\<bbb-R\>\<times\>\<cal-T\>\<rightarrow\>\<cal-T\>> by
<math|<around*|(|\<alpha\>,\<psi\><rsub|u>|)>\<rightarrow\>\<alpha\>\<cdot\>\<psi\><rsub|v>\<equallim\><rsub|defined
by>\<psi\><rsub|\<alpha\>\<cdot\>v>> then we have:
<\enumerate>
<item>If <math|\<alpha\>,\<beta\>\<in\>\<bbb-R\>> and
<math|<wide|u|\<vect\>>\<in\>\<cal-T\>> then there exists a
<math|v\<in\>V> such that <math|<wide|u|\<vect\>>=\<psi\><rsub|v>> and
then <math|<around*|(|\<alpha\>\<cdot\>\<beta\>|)>\<cdot\><wide|u|\<vect\>>=<around*|(|\<alpha\>\<cdot\>\<beta\>|)>\<cdot\>\<psi\><rsub|v>=\<psi\><rsub|<around*|(|\<alpha\>\<cdot\>\<beta\>|)>\<cdot\>v>=\<psi\><rsub|\<alpha\>\<cdot\><around*|(|\<beta\>\<cdot\>v|)>>=\<alpha\>\<cdot\>\<psi\><rsub|\<beta\>\<cdot\>v>=\<alpha\>\<cdot\><around*|(|\<beta\>\<cdot\>\<psi\><rsub|v>|)>=\<alpha\>\<cdot\><around*|(|\<beta\>\<cdot\><wide|u|\<vect\>>|)>>
<item>If <math|\<alpha\>,\<beta\>\<in\>\<bbb-R\>> and
<math|<wide|u|\<vect\>>\<in\>\<cal-T\>> then there exists a
<math|v\<in\>V> such that <math|<wide|u|\<vect\>>=\<psi\><rsub|v>> so
that <math|<around*|(|\<alpha\>+\<beta\>|)>\<cdot\><wide|u|\<vect\>>=<around*|(|\<alpha\>+\<beta\>|)>\<cdot\>\<psi\><rsub|v>=\<psi\><rsub|<around*|(|\<alpha\>+\<beta\>|)>\<cdot\>v>=\<psi\><rsub|\<alpha\>\<cdot\>v+\<beta\>\<cdot\>v>\<equallim\><rsub|<reference|eq
1.3>>\<psi\><rsub|\<beta\>\<cdot\>v>\<circ\>\<psi\><rsub|\<alpha\>\<cdot\>v>=<around*|(|\<beta\>\<cdot\>\<psi\><rsub|v>|)>\<circ\><around*|(|\<alpha\>\<cdot\>\<psi\><rsub|v>|)>=\<alpha\>\<cdot\><wide|u|\<vect\>>+\<beta\>\<cdot\><wide|u|\<vect\>>>
<item>If <math|\<alpha\>\<in\>\<bbb-R\>> and
<math|<wide|u|\<vect\>>,<wide|v|\<vect\>>\<in\>\<cal-T\>> then there
exists a <math|u<rprime|'>,v<rprime|'>\<in\>V> such that
<math|<wide|u|\<vect\>>=\<psi\><rsub|u<rprime|'>>>,<math|<wide|v|\<vect\>>=\<psi\><rsub|v<rprime|'>>>.
Then we have <math|\<alpha\>\<cdot\><around*|(|<wide|u|\<vect\>>+<wide|v|\<vect\>>|)>=\<alpha\>\<cdot\><around*|(|\<psi\><rsub|u<rprime|'>>+\<psi\><rsub|v<rprime|'>>|)>=\<alpha\>\<cdot\><around*|(|\<psi\><rsub|v<rprime|'>>\<circ\>\<psi\><rsub|u<rprime|'>>|)>\<equallim\><rsub|<reference|eq
1.3>>\<alpha\>\<cdot\>\<psi\><rsub|u<rprime|'>+v<rprime|'>>=\<psi\><rsub|\<alpha\>\<cdot\><around*|(|u<rprime|'>+v<rprime|'>|)>>=\<psi\><rsub|\<alpha\>\<cdot\>u<rprime|'>+\<alpha\>\<cdot\>v<rprime|'>>\<equallim\><rsub|<reference|eq
1.3>>\<psi\><rsub|\<alpha\>\<cdot\>v<rprime|'>>\<circ\>\<psi\><rsub|\<alpha\>\<cdot\>u<rprime|'>>=<around*|(|\<alpha\>\<cdot\>\<psi\><rsub|v<rprime|'>>|)>\<circ\><around*|(|\<alpha\>\<cdot\>\<psi\><rsub|u<rprime|'>>|)>=<around*|(|\<alpha\>\<cdot\><wide|v|\<vect\>>|)>\<circ\><around*|(|\<alpha\>\<cdot\><wide|u|\<vect\>>|)>=\<alpha\>\<cdot\><wide|u|\<vect\>>+\<alpha\>\<cdot\><wide|v|\<vect\>>>
<item>If <math|<wide|u|\<vect\>>\<in\>\<cal-T\>> then there exists a
<math|v\<in\>V> such that <math|<wide|u|\<vect\>>=\<psi\><rsub|v><rsub|>>
then <math|1\<cdot\><wide|u|\<vect\>>=1\<cdot\>\<psi\><rsub|v>=\<psi\><rsub|1\<cdot\>v>=\<psi\><rsub|v>=<wide|u|\<vect\>>>
</enumerate>
Using the four above statements we have then that
<math|<around*|\<langle\>|S,\<cal-T\>,\<cdot\>|\<rangle\>>> is a flat
space.\
Define now <math|\<varphi\>:V\<rightarrow\>\<cal-T\>> by
<math|v\<rightarrow\>\<varphi\><around*|(|v|)>=\<psi\><rsub|v>> (which is
a function because of <reference|eq 1.1>) then we have that
<math|\<varphi\>> is a bijection because:
<\enumerate>
<item><dueto|injectivity>If <math|v,u\<in\>V> and
<math|\<varphi\><around*|(|u|)>=\<varphi\><around*|(|v|)>> then
<math|v=D<around*|(|x,\<psi\><rsub|v><around*|(|x|)>|)>=D<around*|(|x,\<varphi\><around*|(|v|)><around*|(|x|)>|)>=D<around*|(|x,\<varphi\><around*|(|u|)><around*|(|x|)>|)>=D<around*|(|x,\<psi\><rsub|u><around*|(|x|)>|)>=u\<Rightarrow\>u=v>
proving that <math|\<varphi\>> is injective.
<item><dueto|surjectivity>If <math|<wide|u|\<vect\>>\<in\>\<cal-T\>>
then there exists a <math|v\<in\>S> such that
<math|\<varphi\><around*|(|v|)>=\<psi\><rsub|v>=<wide|u|\<vect\>>>
</enumerate>
Next we prove (i,ii and linearity)\
<\enumerate-roman>
<item>If <math|x,y\<in\>S> then <math|y-x=<wide|u|\<vect\>>\<in\>\<cal-T\>>
where <math|<wide|u|\<vect\>><around*|(|x|)>=y> so there exists a
<math|v\<in\>V> such that <math|<wide|u|\<vect\>>=\<varphi\><around*|(|v|)>=\<psi\><rsub|v>>
and thus <math|D<around*|(|x,y|)>=D<around*|(|x,<wide|u|\<vect\>><around*|(|x|)>|)>=D<around*|(|x,\<psi\><rsub|v><around*|(|x|)>|)>=v>
so that <math|\<varphi\><around*|(|D<around*|(|x,y|)>|)>=\<varphi\><around*|(|v|)>=<wide|u|\<vect\>>>
<item>If <math|v\<in\>V> and <math|\<alpha\>\<in\>\<bbb-R\>> then
<math|\<varphi\><around*|(|\<alpha\>\<cdot\>v|)>=\<psi\><rsub|\<alpha\>\<cdot\>v>=\<alpha\>\<cdot\>\<psi\><rsub|v>=\<alpha\>\<cdot\>\<varphi\><around*|(|v|)>>
<item>If <math|u,v\<in\>V> then <math|\<varphi\><around*|(|u+v|)>=\<psi\><rsub|u+v>\<equallim\><rsub|<reference|eq
1.3>>\<psi\><rsub|v>\<circ\>\<psi\><rsub|u>=\<varphi\><around*|(|v|)>\<circ\>\<varphi\><around*|(|u|)>=\<varphi\><around*|(|u|)>+\<varphi\><around*|(|v|)>>
</enumerate-roman>
Finally assume that there is another translation group
<math|\<cal-T\><rprime|'>> such that <math|<around*|\<langle\>|S,\<cal-T\><rprime|'>,\<ast\>|\<rangle\>>>
is a flat space and that there is a bijection
<math|\<varphi\><rprime|'>:V\<rightarrow\>\<cal-T\><rprime|'>\<subseteq\>S<rsup|S>>
that satisfies (i,ii). As <math|S> is not empty there exists a
<math|x<rsub|0>\<in\>S> then given a <math|<wide|u|\<vect\>>\<in\>\<cal-T\><rprime|'>>
it follows that
<\equation>
<label|eq 1.6>\<varphi\><rprime|'><around*|(|D<around*|(|x<rsub|0>,x<rsub|0>+<wide|u|\<vect\>>|)>|)>\<equallim\><rsub|<around*|(|i|)>><around*|(|x<rsub|0>+<wide|u|\<vect\>>|)>-x<rsub|0>\<equallim\><rsub|<reference|properties
of transalation groups>><wide|u|\<vect\>>
</equation>
as <math|<wide|u|\<vect\>>> is choosen arbitrary we have proven that
<\equation>
<label|eq 1.7>\<varphi\><rprime|'>:V\<rightarrow\>\<cal-T\><rprime|'>
is a surjection
</equation>
If <math|v\<in\>V> take then <math|\<varphi\><around*|(|v|)>=\<psi\><rsub|v>\<in\>\<cal-T\>>
then <math|v\<equallim\><rsub|<reference|eq
1.1>>D<around*|(|x,\<psi\><rsub|v><around*|(|x|)>|)>=D<around*|(|x,x+\<psi\><rsub|v>|)>\<Rightarrowlim\><rsub|<around*|(|i|)>>\<varphi\><rprime|'><around*|(|v|)>=\<varphi\><rprime|'><around*|(|D<around*|(|x,x+\<psi\><rsub|v>|)>|)>=<around*|(|x+\<psi\><rsub|v>|)>-x=\<psi\><rsub|v>=\<varphi\><around*|(|v|)>>
proving that as <math|v> is choosen arbitrary that
<\equation>
<label|eq 1.8>\<varphi\>=\<varphi\><rprime|'>
</equation>
So as <math|\<varphi\>> is injective we have that
<math|\<varphi\><rprime|'>> is injective proving that
<math|\<varphi\><rprime|'>> is a bijection. Also we have that
<math|\<cal-T\>=\<varphi\><around*|(|V|)>=\<varphi\><rprime|'><around*|(|V|)>=\<cal-T\><rprime|'>>
proving that
<\equation>
<label|eq 1.9>\<cal-T\>=\<cal-T\><rprime|'>
</equation>
</proof>
<\definition>
Let <math|<around*|\<langle\>|S,\<cal-T\>,\<cdot\>|\<rangle\>>> a flat
space <math|\<cal-T\>>, <math|\<cal-U\>> a subspace of <math|\<cal-T\>>
\ and <math|\<emptyset\>\<neq\>H\<subseteq\>S> then we say that <math|H>
<with|font-series|bold|is flat in> S <with|font-series|bold|with
direction space ><math|\<cal-U\>> if <math|H+\<cal-U\>=H> and
<math|H-H=\<cal-U\>>
</definition>
<\theorem>
Let <math|<around*|\<langle\>|S,\<cal-T\>,\<cdot\>|\<rangle\>>> a flat
space and <math|\<emptyset\>\<neq\>H\<subseteq\>S> is flat in S with
direction space <math|\<cal-U\>> and direction space
<math|\<cal-U\><rprime|'>> then <math|\<cal-U\>=\<cal-U\><rprime|'>>
</theorem>
<\proof>
<math|\<cal-U\>=H-H-\<cal-U\><rprime|'>>\
</proof>
Because of the above the following is well defined
<\definition>
Let <math|<around*|\<langle\>|S,\<cal-T\>,\<cdot\>|\<rangle\>>> be a flat
space and <math|\<cal-U\>> a subspace of <math|\<cal-T\>> such that
<math|H> [where <math|\<emptyset\>\<neq\>H\<subseteq\>S>] is flat in
<math|\<cal-U\>> then if <math|\<cal-U\>> is finite dimensional we say
that <math|H> is finite dimensional and
<math|dim<around*|(|H|)>=dim<around*|(|\<cal-U\>|)>>. If
<math|dim<around*|(|H|)>=1> then we say that <math|H> is a straight line.
</definition>
The following theorem proves that this correspond with the classic
definition of a straight line
<\theorem>
Let <math|<around*|\<langle\>|S,\<cal-T\>,\<cdot\>|\<rangle\>>> be a flat
space and <math|\<emptyset\>\<neq\>H\<subseteq\>S> then <math|H> is a
straight line if an only if <math|\<exists\>x<rsub|0>\<in\>S> and
<math|<wide|0|\<vect\>>\<neq\><wide|e|\<vect\>>\<in\>\<cal-T\>> such that
<math|H=<around*|{|x<rsub|0>+\<alpha\>\<cdot\><wide|e|\<vect\>>\|\<alpha\>\<in\>\<bbb-R\>|}>>
</theorem>
<\proof>
\
<\enumerate>
<item><dueto|<math|\<Rightarrow\>>>If <math|H> is a straight line then
<math|>there exists a one dimensional subspace such of <math|\<cal-T\>>
such that <math|H+\<cal-U\>=H> and <math|H-H=\<cal-U\>>. As
<math|\<cal-U\>> is one dimensional there exists a
<math|<wide|e|\<vect\>>\<neq\><wide|0|\<vect\>>> such that
<math|\<cal-U\>=<around*|{|\<alpha\>\<cdot\><wide|e|\<vect\>>\|\<alpha\>\<in\>\<bbb-R\>|}>>.
Also as <math|S\<neq\>\<emptyset\>> there exists a
<math|x<rsub|0>\<in\>S>. So if <math|y\<in\>H> then
<math|y-x<rsub|0>\<in\>H-H=\<cal-U\>> or
<math|y-x<rsub|0>=\<alpha\>\<cdot\><wide|e|\<vect\>>\<Rightarrow\>y=<around*|(|y-x<rsub|0>|)>+x<rsub|0>=\<alpha\>\<cdot\><wide|e|\<vect\>>+x<rsub|0>>.
<item><dueto|<math|\<Leftarrow\>>>If
<math|H=<around*|{|x<rsub|0>+\<alpha\>\<cdot\><wide|e|\<vect\>>\|\<alpha\>\<in\>\<bbb-R\>|}>>
where <math|x<rsub|0>\<in\>S and <wide|0|\<vect\>>\<neq\><wide|e|\<vect\>>\<in\>\<cal-T\>>
then <math|\<cal-U\>=<around*|{|\<alpha\>\<cdot\><wide|e|\<vect\>>\|\<alpha\>\<in\>\<bbb-R\>|}>>
\ is a one dimensional space. If <math|<wide|u|\<vect\>>\<in\>H-H>
there exists <math|\<alpha\>,\<beta\>\<in\>\<bbb-R\>> such that
<math|<wide|u|\<vect\>>=<around*|(|x<rsub|0>+\<alpha\>\<cdot\><wide|e|\<vect\>>|)>-<around*|(|x<rsub|0>+\<beta\>\<cdot\><wide|e|\<vect\>>|)>\<equallim\><rsub|<reference|properties
of transalation groups> <around*|(|9|)>><around*|(|x<rsub|0>-<around*|(|x<rsub|0>+\<beta\>\<cdot\><wide|e|\<vect\>>|)>|)>+\<alpha\>\<cdot\><wide|e|\<vect\>>\<equallim\><rsub|<text|<reference|properties
of transalation groups> (10>>-<around*|(|\<beta\>\<cdot\><wide|e|\<vect\>>|)>+\<alpha\>\<cdot\><wide|e|\<vect\>>=<around*|(|\<beta\>-\<alpha\>|)>\<cdot\><wide|e|\<vect\>>\<in\>\<cal-U\>>,
if <math|<wide|u|\<vect\>>\<in\>\<cal-U\>> then there exists a
<math|\<alpha\>\<in\>\<bbb-R\>> with
<math|<wide|u|\<vect\>>=\<alpha\>\<cdot\><wide|e|\<vect\>>=<around*|(|x<rsub|0>+\<alpha\>\<cdot\><wide|e|\<vect\>>|)>-x<rsub|0>=<around*|(|x<rsub|0>\<upl\>\<alpha\>\<cdot\><wide|u|\<vect\>>|)>-<around*|(|x<rsub|0>+<wide|0|\<vect\>>|)>=<around*|(|x<rsub|0>+\<alpha\>\<cdot\><wide|u|\<vect\>>|)>-<around*|(|x<rsub|0>+0\<cdot\><wide|u|\<vect\>>|)>\<in\>H-H>.
This proves that
<\equation>
<label|eq 1.10>H-H=\<cal-U\>.
</equation>
If <math|x\<in\>H+\<cal-U\>> then there exists a
<math|\<alpha\>,\<beta\>\<in\>\<bbb-R\>> such that
<math|x=<around*|(|x<rsub|0>+\<alpha\>\<cdot\><wide|e|\<vect\>>|)>+\<beta\>\<cdot\><wide|e|\<vect\>>\<equallim\><rsub|<text|<reference|properties
of transalation groups> (6)>>x<rsub|0>+<around*|(|\<alpha\>\<cdot\><wide|e|\<vect\>>+\<beta\>\<cdot\><wide|e|\<vect\>>|)>=x<rsub|0>+<around*|(|\<alpha\>+\<beta\>|)>\<cdot\><wide|e|\<vect\>>\<in\>H>,
if <math|x\<in\>H> then <math|\<exists\>\<alpha\>\<in\>\<bbb-R\>> such
that <math|x=x<rsub|0>+\<alpha\>\<cdot\><wide|e|\<vect\>>=<around*|(|x<rsub|0>+<wide|0|\<vect\>>|)>+\<alpha\>\<cdot\><wide|e|\<vect\>>=<around*|(|x<rsub|0>+0\<cdot\><wide|e|\<vect\>>|)>+\<alpha\>\<cdot\><wide|e|\<vect\>>\<in\>H+\<cal-U\>>.
This proves
<\equation>
<label|eq 1.11>H+\<cal-U\>=H
</equation>
From <reference|eq 1.10>,<reference|eq 1.11> it follows that <math|H>
is a straight line.
</enumerate>
</proof>
<\definition>
<label|line segment><index|line segment>Let
<math|<around*|\<langle\>|S,\<cal-T\>,\<cdot\>|\<rangle\>>> be a flat
space then if <math|x,y\<in\>S> the <with|font-series|bold|the line
segment from x to y> noted by <math|<around*|[|x,y|]>> is defined by
<math|<around*|[|x,y|]>=<around*|{|x+\<alpha\>\<cdot\><around*|(|y-x|)>\|\<alpha\>\<in\><around*|[|0,1|]>\<subseteq\>\<bbb-R\>|}>>.
We define also <math|<around*|[|x,y|[>=<around*|[|x,y|]>\\<around*|{|y|}>>,
<math|<around*|]|x,y|]>=<around*|[|x,y|]>\\<around*|{|x|}>>,
<math|<around*|]|x,y|[>=<around*|[|x,y|]>\\<around*|{|x,y|}>>.
</definition>
\;
\;
<section|Events>
<\definition>
<label|eventworld><index|eventworld>A <with|font-series|bold|eventworld>
is a pair <math|<around*|\<langle\>|E,\<prec\>|\<rangle\>>> where
<math|E> is a set and <math|\<prec\>\<subseteq\>E\<times\>E> is a
relation on <math|E> which satisfies the following:
<\enumerate>
<item><dueto|reflexivity><math|\<forall\>x\<in\>E> we have
<math|x\<prec\>x>
<item><dueto|transitivity><math|\<forall\>x,y,z\<in\>E> is
<math|x\<prec\>y\<wedge\>y\<prec\>z\<Rightarrow\>x\<prec\>z>
<item><math|\<forall\>x,y\<in\>E> there exists a <math|z\<in\>E> such
that <math|z\<prec\>x\<wedge\>z\<prec\>y>
</enumerate>
We call <math|\<prec\>> a <with|font-series|bold|precedence relation> on
<math|E> the set of <with|font-series|bold|events>. If
<math|x\<prec\>y\<wedge\>y\<prec\>z> then we say that <math|y> is
<with|font-series|bold|intermediate between x and z>, we note this in
short by <math|x\<prec\>y\<prec\>z>.
</definition>
<\definition>
<label|history of a event><index|history of a event>Let
<math|<around*|\<langle\>|E,\<prec\>|\<rangle\>>> be a eventworld and
<math|e\<in\>E> then the <with|font-series|bold|history of e> noted by
<math|Hist<around*|(|e|)>> is defined by
<math|Hist<around*|(|e|)>=<around*|{|x\<in\>E\|x\<prec\>e|}>> (it is the
set of events preceding <math|e>)
</definition>
<\theorem>
<label|property 1 of eventworlds>If <math|<around*|\<langle\>|E,\<prec\>|\<rangle\>>>
is a eventworld then we have\
<\enumerate>
<item>If <math|x,y\<in\>E> then <math|Hist<around*|(|x|)><big|cap>Hist<around*|(|y|)>\<neq\>\<emptyset\>>
(the history of two events have some event in common, this disallows e
eventworld with parallel histories)
<item><math|x\<prec\>y> if and only if
<math|Hist<around*|(|x|)>\<subseteq\>Hist<around*|(|y|)>>
</enumerate>
</theorem>
<\proof>
\
<\enumerate>
<item>If <math|x,y\<in\>E> then by the definition of a eventworld (see
<reference|eventworld> (3)) there exists a <math|z\<in\>E> such that
<math|z\<prec\>x\<wedge\>z\<prec\>y\<Rightarrow\>z\<in\>Hist<around*|(|x|)>\<wedge\>z\<in\>Hist<around*|(|y|)>\<Rightarrow\>z\<in\>Hist<around*|(|x|)><big|cap>Hist<around*|(|y|)>>.
<item> To prove the equivalence note that:
<\enumerate>
<item><dueto|<math|\<Rightarrow\>>>If <math|x\<prec\>y> and if
<math|z\<in\>Hist<around*|(|x|)>\<Rightarrow\>z\<prec\>x\<Rightarrowlim\><rsub|x\<prec\>z>z\<prec\>y\<Rightarrow\>z\<in\>Hist<around*|(|y|)>\<Rightarrow\>Hist<around*|(|x|)>\<prec\>Hist<around*|(|y|)>>
<item><dueto|<math|\<Leftarrow\>>>Suppose
<math|Hist<around*|(|x|)>\<prec\>Hist<around*|(|y|)>> then as
<math|x\<prec\>x\<Rightarrow\>x\<in\>Hist<around*|(|x|)>\<Rightarrow\>x\<in\>Hist<around*|(|y|)>\<Rightarrow\>x\<prec\>y>
</enumerate>
</enumerate>
</proof>
<\definition>
<label|simultaneity relation><index|simultaneity relation>Let
<math|<around*|\<langle\>|E,\<prec\>|\<rangle\>>> be a eventworld then
<math|x,y\<in\>E> are <with|font-series|bold|simultaneou>s noted by
<math|x\<sim\>y> if <math|x\<prec\>y\<prec\>x>. This defines a relation
<math|\<sim\>\<subseteq\>E\<times\>E> called the
<with|font-series|bold|simultaneity relation on E>
</definition>
<\theorem>
If <math|<around*|\<langle\>|E,\<prec\>|\<rangle\>>> is a eventworld then
<math|\<sim\>> is a equivalence relation.
</theorem>
<\proof>
\
<\enumerate>
<item><dueto|reflexitivity>If <math|x\<in\>E> then
<math|x\<prec\>x\<wedge\>x\<prec\>x\<Rightarrow\>x\<sim\>x>
<item><dueto|symmetry>If <math|x\<sim\>y> then
<math|x\<prec\>y\<wedge\>y\<prec\>x\<Rightarrow\>y\<prec\>x\<wedge\>x\<prec\>y\<Rightarrow\>x\<sim\>y>
<item><dueto|transitivity>If <math|x\<sim\>y\<wedge\>y\<sim\>z> then
<math|x\<prec\>y\<wedge\>y\<prec\>x\<wedge\>y\<prec\>z\<wedge\>z\<prec\>y\<Rightarrow\>x\<prec\>z\<wedge\>z\<prec\>x\<Rightarrow\>x\<sim\>z>
</enumerate>
</proof>
<\theorem>
Let <math|<around*|\<langle\>|E,\<prec\>|\<rangle\>>> be a eventworld and
<math|x,y\<in\>E> then <math|x\<sim\>y> if and only if
<math|Hist<around*|(|x|)>=Hist<around*|(|y|)>>. Two events <math|x,y> are
simultaneous if <math|z> is in the history of <math|x> inf and only if
<math|z> is in the history of <math|y>\
</theorem>
<\proof>
\
<\enumerate>
<item>If <math|x\<sim\>y> then <math|x\<prec\>y> and <math|y\<prec\>x>
then by <reference|property 1 of eventworlds>
<math|Hist<around*|(|x|)>\<subseteq\>Hist<around*|(|y|)>\<wedge\>Hist<around*|(|y|)>\<subseteq\>Hist<around*|(|x|)>\<Rightarrow\>Hist<around*|(|x|)>=Hist<around*|(|y|)>>
<item>If <math|Hist<around*|(|x|)>=Hist<around*|(|y|)>\<Rightarrow\>Hist<around*|(|x|)>\<subseteq\>Hist<around*|(|y|)>\<wedge\>Hist<around*|(|y|)>\<subseteq\>Hist<around*|(|x|)>\<Rightarrowlim\><rsub|<text|<reference|property
1 of eventworlds>>>x\<prec\>y\<wedge\>y\<prec\>x\<Rightarrow\>x\<sim\>y>
</enumerate>
</proof>
<\definition>
<label|strict precedence><index|strict precedence>If
<math|<around*|\<langle\>|E,\<prec\>|\<rangle\>>> is a eventworld and
<math|x,y\<in\>E> then <math|x\<precdot\>y> if and only if
<math|x\<prec\>y\<wedge\>not<around*|(|x\<sim\>y|)>>. We call the
relation <math|\<precdot\>> a <with|font-series|bold|strict precedence>.
</definition>
<\definition>
Given a set <math|E> and <math|R\<subseteq\>E\<times\>E> a relation then
<math|R> is <with|font-series|bold|strictly antisymmetric> if
<math|<around*|(|x,y|)>\<in\>R\<Rightarrow\><around*|(|y,x|)>\<nin\>R> or
in other ways <math|xRy\<Rightarrow\>not<around*|(|yRx|)>>\
</definition>
<\theorem>
If <math|<around*|\<langle\>|E,\<prec\>|\<rangle\>>> is a eventworld then
<math|\<precdot\>> is strictly antisymmetric and transitive
</theorem>
<\proof>
\
<\enumerate>
<item><dueto|strictly antisymmetry>If <math|x\<precdot\>y> then
<math|x\<prec\>y\<wedge\>not<around*|(|x\<sim\>y|)>> so
<math|x\<prec\>y\<wedge\>not<around*|(|x\<prec\>y\<wedge\>y\<prec\>x|)>\<Rightarrow\>x\<prec\>y\<wedge\><around*|(|not<around*|(|x\<prec\>y|)>\<vee\>not<around*|(|y\<prec\>x|)>|)>=x\<prec\>y\<wedge\>not<around*|(|y\<prec\>x|)>>
<item><dueto|transitivity>Let <math|x\<precdot\>y\<wedge\>y\<precdot\>z>
then <math|x\<prec\>y\<wedge\>not<around*|(|x\<sim\>y|)>\<wedge\>y\<prec\>z\<wedge\>not<around*|(|y\<sim\>z|)>>
so that by transitivity of <math|\<prec\>> we have <math|x\<prec\>z>.
Assume now that <math|x\<sim\>z> then we would have that
<math|x\<prec\>z> and <math|z\<prec\>x\<Rightarrowlim\><rsub|y\<prec\>z>y\<prec\>x\<Rightarrow\>x\<sim\>y>
contradicting <math|not<around*|(|x\<sim\>y|)>>.\
</enumerate>
</proof>
<\definition>
Let <math|<around*|\<langle\>|E,\<prec\>|\<rangle\>>> be a eventworld
then if <math|x\<in\>E> we define
<\enumerate>
<item><math|Past<around*|(|x|)>=<around*|{|y\<in\>E\|y\<lessdot\>x|}>\<subseteq\>Hist<around*|(|x|)>>
(past of <math|x>)
<item><math|Pres<around*|(|x|)>=<around*|{|y\<in\>E\|x\<sim\>y|}>\<subseteq\>Hist<around*|(|x|)>>
(present of <math|x>)
<item><math|Fut<around*|(|x|)>=<around*|{|y\<in\>E\|x\<precdot\>y|}>>
(future of <math|x>)
</enumerate>
</definition>
\;
<\theorem>
Let <math|<around*|\<langle\>|E,\<prec\>|\<rangle\>>> be a eventworld and
<math|x\<in\>E> then
<\enumerate>
<item><math|Past<around*|(|x|)><big|cap>Pres<around*|(|x|)>=\<emptyset\>>
<item><math|Pres<around*|(|x|)><big|cap>Fut<around*|(|x|)>=\<emptyset\>>
<item><math|Fut<around*|(|x|)><big|cap>Past<around*|(|x|)>=\<emptyset\>>
</enumerate>
</theorem>
<\proof>
\
<\enumerate>
<item>If <math|z\<in\>Past<around*|(|x|)><big|cap>Pres<around*|(|x|)>>
then <math|z\<precdot\>x\<wedge\>z\<sim\>x\<Rightarrow\>z\<prec\>z\<wedge\>not<around*|(|z\<sim\>x|)>\<wedge\>z\<sim\>x>
a contradiction so <math|Past<around*|(|x|)><big|cap>Pres<around*|(|x|)>=\<emptyset\>>
<item>If <math|z\<in\>Pres<around*|(|x|)><big|cap>Fut<around*|(|x|)>>
then <math|z\<sim\>x\<wedge\>x\<precdot\>z\<Rightarrow\>z\<sim\>x\<wedge\>x\<prec\>z\<wedge\>not<around*|(|x\<sim\>z|)>\<Rightarrow\>z\<sim\>x\<wedge\>not<around*|(|z\<sim\>x|)>>
a contradiction so that <math|Pres<around*|(|x|)><big|cap>Fut<around*|(|x|)>=\<emptyset\>>
<item>If <math|z\<in\>Past<around*|(|x|)><big|cap>Fut<around*|(|x|)>>
then <math|z\<precdot\>x\<wedge\>x\<precdot\>z\<Rightarrow\>z\<prec\>x\<wedge\>not<around*|(|z\<sim\>x|)>\<wedge\>x\<prec\>z\<wedge\>not<around*|(|x\<sim\>z|)>\<Rightarrow\>z\<sim\>x\<wedge\>not<around*|(|z\<sim\>x|)>>
a contradiction so that <math|Past<around*|(|x|)><big|cap>Fut<around*|(|x|)>=\<emptyset\>>
</enumerate>
</proof>
<\definition>
Let <math|E> be a set and <math|R\<subseteq\>E\<times\>E> be a relation
on <math|E>, <math|x,y\<in\>E> then <math|<around*|[|x,y|]><rsub|R>=<around*|{|z\<in\>E\|xRz\<wedge\>zRy|}>>
</definition>
Let's recap some definitions concerning relations
<\definition>
Let <math|E> be a set with a relation <math|R\<subseteq\>E\<times\>E>
then if <math|F\<subseteq\>R> we define <math|R<rsub|\|F>> by
<math|R<rsub|\|F>=R<big|cap><around*|(|E\<times\>E|)>>
</definition>
<\definition>
<label|pre-order><index|pre-order>Let <math|E> be a set then a relation
<math|R\<subseteq\>E\<times\>E> is a pre-order if it satisfies
<\enumerate>
<item><dueto|reflexitivity>If <math|x\<in\>E> then <math|xRx>
<item><dueto|transitivity>If <math|x,y,z\<in\>E> then
<math|xRy\<wedge\>yRz\<Rightarrow\>xRz>
</enumerate>
</definition>
<\definition>
<label|order><index|order>Let <math|E> be a set then a relation
<math|R\<subseteq\>E\<times\>E> is a <with|font-series|bold|order>
relation if it is a pre-order and satisfies also\
<\enumerate>
<item><dueto|anti-symmetry><math|\<forall\>x,y\<in\>E> if
<math|xRy\<wedge\>yRx\<Rightarrow\>x=y>
</enumerate>
</definition>
<\definition>
<label|total relation><index|total relation>Let <math|E> be a set and
<math|R\<subseteq\>E\<times\>E> a relation then <math|R>
<with|font-series|bold|is total> if <math|\<forall\>x,y\<in\>E> we have
either <math|xRy>, <math|yRx> or <math|x=y>.
</definition>
<\definition>
<label|R-total><index|R-total set>Let <math|E> be a set with a pre-order
<math|R\<subseteq\>E\<times\>E> then a subset <math|F\<subseteq\>E> is
<with|font-series|bold|R-total> iff <math|R<rsub|\|F>> is total.
</definition>
<\definition>
<label|r-totally ordered><index|R-totally ordered set>If <math|E> is a
set and <math|R\<subseteq\>E\<times\>E> a pre-order then
<math|F\<subseteq\>E> is <with|font-series|bold|R-totally ordered> if
<math|R<rsub|\|F>> is a total order relation
</definition>
<\definition>
<label|maximally totally ordered><index|maximally totally ordred set>If
<math|E> is a set and <math|R\<subseteq\>E\<times\>E> a pre-order then
<math|F\<subseteq\>E> is <with|font-series|bold|maximally totally ordered
with respect to R> if <math|F> is R-totally ordered and if for every
<math|G> that is R-totally ordered and satisfies
<math|F\<subseteq\>G\<subseteq\>E> we have that <math|F=G>.
</definition>
<\definition>
If E is a set and <math|R\<subseteq\>E\<times\>E> a pre-order then
<math|F\<subseteq\>E> is <with|font-series|bold|locally maximally totally
ordered with respect to R> if <math|\<forall\>x,y\<in\>F> we have that
<math|F<big|cap><around*|[|x,y|]><rsub|R>> is maximally totally ordered
with respect to <math|R<rsub|\|<around*|[|x,y|]><rsub|R>>>
</definition>
<\definition>
If <math|E> is a set with a pre-order <math|R\<subseteq\>E\<times\>E> and
<math|F\<subseteq\>E> then
<\enumerate>
<item><math|x\<in\>F> is a greatest element of <math|F> if
<math|\<forall\>y\<in\>F> we have <math|yRx>
<item><math|x\<in\>F> is lowest element of <math|F> if
<math|\<forall\>y\<in\>F> we have <math|xRy>
</enumerate>
</definition>
<\theorem>
<label|minimum><label|maximum>If <math|E> is a set with a pre-order
<math|R\<subseteq\>E\<times\>E> and <math|F\<subseteq\>E> \ locally
maximally totally ordered with respect to <math|R> then <math|F> can have
only one greatest element and only one lowest element. If the lowest
element exists then it is called the <with|font-series|bold|minimum> of
<math|F>, if the gretest element exists then it is called the
<with|font-series|bold|maximum> of <math|F>
</theorem>
<\proof>
\ If <math|m<rsub|1>,m<rsub|2>> are two greatest (lowest) elements of
<math|F> then as <math|m<rsub|1>,m<rsub|2>\<in\>F> we have that
<math|m<rsub|1>Rm<rsub|2>> and <math|m<rsub|2>Rm<rsub|1>>. Because F is
maximally totally ordered we have that
<math|F<big|cap><around*|[|m<rsub|1>,m<rsub|2>|]><rsub|R>> is maximally
totally ordered and thus totally ordered. Because of reflexitivity we
have <math|m<rsub|1>Rm<rsub|1>> and <math|m<rsub|2>Rm<rsub|2>> this
together with <math|m<rsub|1>Rm<rsub|2>> gives
<math|m<rsub|1>Rm<rsub|1>\<wedge\>m<rsub|1>Rm<rsub|2>> and
<math|m<rsub|1>Rm<rsub|2>\<wedge\>m<rsub|2>Rm<rsub|2>> so that
<math|m<rsub|1>,m<rsub|2>\<in\><around*|[|m<rsub|1>,m<rsub|2>|]>\<Rightarrow\>m<rsub|1>,m<rsub|2>\<in\><around*|[|m<rsub|1>,m<rsub|2>|]><big|cap>F>
and then from the anti-symmetry property in a ordered set we have that
from <math|m<rsub|1>Rm<rsub|2>\<wedge\>m<rsub|2>Rm<rsub|1>> it follows
that <math|m<rsub|1>=m<rsub|2>>
</proof>
<\note>
If <math|<around*|\<langle\>|E,\<prec\>|\<rangle\>>> is a eventworld then
<math|\<prec\>> is a pre-order (this follows from the definition of a
pre-order and a eventworld)
</note>
<\definition>
<index|<math|\<preceqdot\>>>Let <math|<around*|\<langle\>|E,\<prec\>|\<rangle\>>>
be a eventworld and <math|x,y\<in\>E> then <math|x\<preceqdot\>y> if and
only if <math|x\<precdot\>y\<vee\>x=y>
</definition>
<\definition>
Let <math|<around*|\<langle\>|E,\<prec\>|\<rangle\>>> be a eventworld
then a <math|F\<subseteq\>E> is called a
<with|font-series|bold|worldpath> if\
<\enumerate>
<item><math|F> is <math|\<preceqdot\>>-total
<item>F is locally maximally totally ordered with respect to
<math|\<preceqdot\>>
</enumerate>
A worldpath <math|F> is a <with|font-series|bold|worldline> if it is
maximally totally ordered with respect to <math|\<preceqdot\>>
</definition>
<\definition>
Let <math|<around*|\<langle\>|E,\<prec\>|\<rangle\>>> be a eventworld and
<math|F\<subseteq\>E> then if <math|F> has a minimum (maximum) with
respect to <math|\<prec\>> then this minimum (maximum) is called the
<with|font-series|bold|beginning> (<with|font-series|bold|ending>) of
<math|F> and is noted by <math|beg<around*|(|F|)>>
(<math|end<around*|(|F|)>>. If <math|F> has both a beginning and ending
denoted by x and y then we say that <math|F> is a worldpath from <math|x>
to <math|y>.
</definition>
\;
<\theorem>
Let <math|<around*|\<langle\>|E,\<prec\>|\<rangle\>>> be a eventworld and
<math|F\<subseteq\>E> then <math|\<precdot\><rsub|\|F>> is total if and
only if <math|\<prec\><rsub|\|F>> is a total order
</theorem>
<\proof>
\
<\enumerate>
<item><dueto|<math|\<Rightarrow\>>>If <math|\<precdot\><rsub|\|F>> is
total then we have if <math|x,y\<in\>F> that either
<math|x\<precdot\>y>, <math|y\<precdot\>> or <math|x=y> then we have
<\enumerate>
<item><math|<around*|(|x\<prec\>y\<wedge\>not<around*|(|x\<sim\>y|)>|)>>,
<math|<around*|(|y\<prec\>x\<wedge\>not<around*|(|y\<sim\>x|)>|)>> or
<math|x=y> so that either <math|x\<prec\>y>, <math|y\<prec\>x> or
<math|x=y> proving that <math|\<prec\><rsub|F>> is total
<item>If <math|x\<prec\>y\<wedge\>y\<prec\>x> then <math|x\<sim\>y>
so we can not have that <math|x\<precdot\>y> or <math|y\<precdot\>x>
so that we must have <math|x=y> proving that <math|\<prec\><rsub|F>>
is anti symmetric and thus a order
</enumerate>
<item><dueto|<math|\<Leftarrow\>>>If <math|\<prec\><rsub|\|F>> is a
total order then if <math|x,y\<in\>F> then we have <math|x\<prec\>y>,
<math|y\<prec\>x> or <math|x=y> we have then to consider the following
cases\
<\enumerate>
<item><dueto|<math|not<around*|(|x\<sim\>y|)>>>then
<math|x\<precdot\>y>, <math|y\<precdot\>x> or <math|x=y>
<item><dueto|<math|x\<sim\>y>>then <math|x\<prec\>y> and
<math|y\<prec\>x\<Rightarrowlim\><rsub|anti= symmetry>x=y>
</enumerate>
so in all cases we have that <math|x\<precdot\>y>, <math|y\<precdot\>>
or <math|x=y>
</enumerate>
</proof>
\;
<\the-index|idx>
<index-1|fixed point|<pageref|auto-4>>
<index-1|flat space|<pageref|auto-5>>
<index-1|translation group|<pageref|auto-3>>
</the-index>
</body>
<\initial>
<\collection>
<associate|page-medium|paper>
</collection>
</initial>
<\references>
<\collection>
<associate|1.1|<tuple|iii|?>>
<associate|1.2|<tuple|1.1|?>>
<associate|1.3|<tuple|1.2|7>>
<associate|R-total|<tuple|1.35|12>>
<associate|auto-1|<tuple|1|5>>
<associate|auto-10|<tuple|1.20|10>>
<associate|auto-11|<tuple|1.22|10>>
<associate|auto-12|<tuple|1.25|11>>
<associate|auto-13|<tuple|1.32|12>>
<associate|auto-14|<tuple|1.33|12>>
<associate|auto-15|<tuple|1.34|12>>
<associate|auto-16|<tuple|1.35|12>>
<associate|auto-17|<tuple|1.36|12>>
<associate|auto-18|<tuple|1.37|12>>
<associate|auto-19|<tuple|1.42|13>>
<associate|auto-2|<tuple|1.1|5>>
<associate|auto-20|<tuple|b|15>>
<associate|auto-21|<tuple|1.42|?|special relativity 006.tm>>
<associate|auto-22|<tuple|1.44|?|special relativity 006.tm>>
<associate|auto-3|<tuple|1.1|5>>
<associate|auto-4|<tuple|1.2|5>>
<associate|auto-5|<tuple|1.11|7>>
<associate|auto-6|<tuple|1.12|7>>
<associate|auto-7|<tuple|1.18|10>>
<associate|auto-8|<tuple|1.2|10>>
<associate|auto-9|<tuple|1.19|10>>
<associate|eq 1.1|<tuple|1.1|8>>
<associate|eq 1.10|<tuple|1.10|9>>
<associate|eq 1.11|<tuple|1.11|10>>
<associate|eq 1.2|<tuple|1.2|8>>
<associate|eq 1.3|<tuple|1.3|8>>
<associate|eq 1.4|<tuple|1.4|8>>
<associate|eq 1.5|<tuple|1.5|8>>
<associate|eq 1.6|<tuple|1.6|9>>
<associate|eq 1.7|<tuple|1.7|9>>
<associate|eq 1.8|<tuple|1.8|9>>
<associate|eq 1.9|<tuple|1.9|9>>
<associate|eventworld|<tuple|1.19|10>>
<associate|external translation space|<tuple|1.12|7>>
<associate|fixed point|<tuple|1.2|5>>
<associate|fixed point property|<tuple|1.3|5>>
<associate|flat space|<tuple|1.11|7>>
<associate|history of a event|<tuple|1.20|10>>
<associate|identity is in a transaltion group|<tuple|1.4|5>>
<associate|line segment|<tuple|1.18|10>>
<associate|maximally totally ordered|<tuple|1.37|12>>
<associate|maximum|<tuple|1.40|12>>
<associate|minimum|<tuple|1.40|12>>
<associate|minimum or maximum|<tuple|1.40|?|special relativity 006.tm>>
<associate|order|<tuple|1.33|12>>
<associate|pre-order|<tuple|1.32|12>>
<associate|properties of transalation groups|<tuple|1.9|6>>
<associate|property 1 of eventworlds|<tuple|1.21|10>>
<associate|r-totally ordered|<tuple|1.36|12>>
<associate|simultaneity relation|<tuple|1.22|10>>
<associate|strict precedence|<tuple|1.25|11>>
<associate|total relation|<tuple|1.34|12>>
<associate|translation group|<tuple|1.1|5>>
<associate|translation group exists of bijections|<tuple|1.6|5>>
<associate|uniqueness of translation|<tuple|1.5|5>>
</collection>
</references>
<\auxiliary>
<\collection>
<\associate|idx>
<tuple|<tuple|translation group>|<pageref|auto-3>>
<tuple|<tuple|fixed point>|<pageref|auto-4>>
<tuple|<tuple|flat space>|<pageref|auto-5>>
<tuple|<tuple|external translation space>|<pageref|auto-6>>
<tuple|<tuple|line segment>|<pageref|auto-7>>
<tuple|<tuple|eventworld>|<pageref|auto-9>>
<tuple|<tuple|history of a event>|<pageref|auto-10>>
<tuple|<tuple|simultaneity relation>|<pageref|auto-11>>
<tuple|<tuple|strict precedence>|<pageref|auto-12>>
<tuple|<tuple|pre-order>|<pageref|auto-13>>
<tuple|<tuple|order>|<pageref|auto-14>>
<tuple|<tuple|total relation>|<pageref|auto-15>>
<tuple|<tuple|R-total set>|<pageref|auto-16>>
<tuple|<tuple|R-totally ordered set>|<pageref|auto-17>>
<tuple|<tuple|maximally totally ordred set>|<pageref|auto-18>>
<tuple|<tuple|<with|mode|<quote|math>|\<preceqdot\>>>|<pageref|auto-19>>
</associate>
<\associate|toc>
<vspace*|1fn><with|font-series|<quote|bold>|math-font-series|<quote|bold>|1<space|2spc>Spaces>
<datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
<no-break><pageref|auto-1><vspace|0.5fn>
1.1<space|2spc>Translation groups <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
<no-break><pageref|auto-2>
1.2<space|2spc>Events <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
<no-break><pageref|auto-8>
<vspace*|1fn><with|font-series|<quote|bold>|math-font-series|<quote|bold>|Index>
<datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
<no-break><pageref|auto-20><vspace|0.5fn>
</associate>
</collection>
</auxiliary>
- [TeXmacs] Problem using table of content and index, Marc Mertens, 06/26/2014
- Re: [TeXmacs] Problem using table of content and index, Sam Liddicott, 06/26/2014
- Re: [TeXmacs] Problem using table of content and index, François Poulain, 06/27/2014
- Re: [TeXmacs] Problem using table of content and index, François Poulain, 06/27/2014
- Re: [TeXmacs] Problem using table of content and index, Marc Mertens, 06/27/2014
- Re: [TeXmacs] Problem using table of content and index, François Poulain, 06/27/2014
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