<?xml version="1.0"?>
<div class="add">

<b>set </b><font color="Maroon" title="c2">D</font> =  <a href="anproj_1.html#K4" title="ANPROJ_1:func.4">ProjectivePoints</a> <font color="Maroon" title="c1">V</font>;<br/>
<b>set </b><font color="Maroon" title="c3">XXX</font> = <span class="p1"><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="anproj_1.html#K4" title="ANPROJ_1:func.4">ProjectivePoints</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p2">(<span class="default"><a href="anproj_1.html#K4" title="ANPROJ_1:func.4">ProjectivePoints</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p2">(<span class="default"><a href="anproj_1.html#K4" title="ANPROJ_1:func.4">ProjectivePoints</a> <font color="Maroon" title="c1">V</font></span>)</span></span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font>, <font color="Olive" title="b3">r</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font> st <br/>( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><span class="p2">(<span class="default"><a href="anproj_1.html#K3" title="ANPROJ_1:func.3">Dir</a> <font color="Olive" title="b1">p</font></span>)</span>,<span class="p2">(<span class="default"><a href="anproj_1.html#K3" title="ANPROJ_1:func.3">Dir</a> <font color="Olive" title="b2">q</font></span>)</span>,<span class="p2">(<span class="default"><a href="anproj_1.html#K3" title="ANPROJ_1:func.3">Dir</a> <font color="Olive" title="b3">r</font></span>)</span></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span> &amp; <font color="Olive" title="b1">p</font> <a href="anproj_1.html#NR1" title="ANPROJ_1:NR.1">is_Prop_Vect</a>  &amp; <font color="Olive" title="b2">q</font> <a href="anproj_1.html#NR1" title="ANPROJ_1:NR.1">is_Prop_Vect</a>  &amp; <font color="Olive" title="b3">r</font> <a href="anproj_1.html#NR1" title="ANPROJ_1:NR.1">is_Prop_Vect</a>  &amp; <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font>,<font color="Olive" title="b3">r</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  );<br/>
<b>consider </b><font color="Maroon" title="c4">R</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E2:44_1_1"/><i><font color="Green" title="E32">A1</font></i>: 
for <font color="Olive" title="b1">xyz</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">xyz</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">R</font> iff ( <font color="Olive" title="b1">xyz</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="anproj_1.html#K4" title="ANPROJ_1:func.4">ProjectivePoints</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p2">(<span class="default"><a href="anproj_1.html#K4" title="ANPROJ_1:func.4">ProjectivePoints</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p2">(<span class="default"><a href="anproj_1.html#K4" title="ANPROJ_1:func.4">ProjectivePoints</a> <font color="Maroon" title="c1">V</font></span>)</span></span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">xyz</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1" title="XBOOLE_0:sch.1">XBOOLE_0:sch 1</a>();<br/></i>
<a NAME="E3:44_1_1"/>
<font color="Maroon" title="c4">R</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="anproj_1.html#K4" title="ANPROJ_1:func.4">ProjectivePoints</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p2">(<span class="default"><a href="anproj_1.html#K4" title="ANPROJ_1:func.4">ProjectivePoints</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p2">(<span class="default"><a href="anproj_1.html#K4" title="ANPROJ_1:func.4">ProjectivePoints</a> <font color="Maroon" title="c1">V</font></span>)</span></span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:44_1_1_1"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">R</font> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="anproj_1.html#K4" title="ANPROJ_1:func.4">ProjectivePoints</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p2">(<span class="default"><a href="anproj_1.html#K4" title="ANPROJ_1:func.4">ProjectivePoints</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p2">(<span class="default"><a href="anproj_1.html#K4" title="ANPROJ_1:func.4">ProjectivePoints</a> <font color="Maroon" title="c1">V</font></span>)</span></span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span>
 
<b>by </b><i><a class="txt" href="anproj_1.html#E2:44_1_1"><i><font color="Green" title="E32">A1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:44_1_1_1"/>
<font color="Maroon" title="c4">R</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="anproj_1.html#K4" title="ANPROJ_1:func.4">ProjectivePoints</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p2">(<span class="default"><a href="anproj_1.html#K4" title="ANPROJ_1:func.4">ProjectivePoints</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p2">(<span class="default"><a href="anproj_1.html#K4" title="ANPROJ_1:func.4">ProjectivePoints</a> <font color="Maroon" title="c1">V</font></span>)</span></span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c5">R'</font> = <font color="Maroon" title="c4">R</font> as    <a href="collsp.html#M1" title="COLLSP:mode.1">Relation3</a> of  <a href="anproj_1.html#K4" title="ANPROJ_1:func.4">ProjectivePoints</a> <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="ref" href="collsp.html#D1" title="COLLSP:def.1">COLLSP:def 1</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c5">R'</font>
;<br/>

<b>let </b><font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c7">y</font>, <font color="Maroon" title="c8">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<div><i><font color="Green" title="E33">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:44_1_1_2"/>
<span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c8">z</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">R'</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c9">p</font>, <font color="Maroon" title="c10">q</font>, <font color="Maroon" title="c11">r</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E3:44_1_1_2"/><i><font color="Green" title="E34">A3</font></i>: 
( <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c8">z</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><span class="p2">(<span class="default"><a href="anproj_1.html#K3" title="ANPROJ_1:func.3">Dir</a> <font color="Maroon" title="c9">p</font></span>)</span>,<span class="p2">(<span class="default"><a href="anproj_1.html#K3" title="ANPROJ_1:func.3">Dir</a> <font color="Maroon" title="c10">q</font></span>)</span>,<span class="p2">(<span class="default"><a href="anproj_1.html#K3" title="ANPROJ_1:func.3">Dir</a> <font color="Maroon" title="c11">r</font></span>)</span></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span> &amp; <font color="Maroon" title="c9">p</font> <a href="anproj_1.html#NR1" title="ANPROJ_1:NR.1">is_Prop_Vect</a>  &amp; <font color="Maroon" title="c10">q</font> <a href="anproj_1.html#NR1" title="ANPROJ_1:NR.1">is_Prop_Vect</a>  &amp; <font color="Maroon" title="c11">r</font> <a href="anproj_1.html#NR1" title="ANPROJ_1:NR.1">is_Prop_Vect</a>  &amp; <font color="Maroon" title="c9">p</font>,<font color="Maroon" title="c10">q</font>,<font color="Maroon" title="c11">r</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  )
 <b>by </b><i><a class="txt" href="anproj_1.html#E2:44_1_1"><i><font color="Green" title="E32">A1</font></i></a></i>;<br/><a NAME="E4:44_1_1_2"/>
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K3" title="ANPROJ_1:func.3">Dir</a> <font color="Maroon" title="c9">p</font> &amp; <font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K3" title="ANPROJ_1:func.3">Dir</a> <font color="Maroon" title="c10">q</font> &amp; <font color="Maroon" title="c8">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K3" title="ANPROJ_1:func.3">Dir</a> <font color="Maroon" title="c11">r</font> )
 <b>by </b><i><a class="txt" href="anproj_1.html#E3:44_1_1_2"><i><font color="Green" title="E34">A3</font></i></a>, <a class="ref" href="mcart_1.html#T28" title="MCART_1:th.28">MCART_1:28</a></i>;<br/><b>hence </b><a NAME="E5:44_1_1_2"/>
 ex <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font>, <font color="Olive" title="b3">r</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font> st <br/>( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K3" title="ANPROJ_1:func.3">Dir</a> <font color="Olive" title="b1">p</font> &amp; <font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K3" title="ANPROJ_1:func.3">Dir</a> <font color="Olive" title="b2">q</font> &amp; <font color="Maroon" title="c8">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K3" title="ANPROJ_1:func.3">Dir</a> <font color="Olive" title="b3">r</font> &amp; <font color="Olive" title="b1">p</font> <a href="anproj_1.html#NR1" title="ANPROJ_1:NR.1">is_Prop_Vect</a>  &amp; <font color="Olive" title="b2">q</font> <a href="anproj_1.html#NR1" title="ANPROJ_1:NR.1">is_Prop_Vect</a>  &amp; <font color="Olive" title="b3">r</font> <a href="anproj_1.html#NR1" title="ANPROJ_1:NR.1">is_Prop_Vect</a>  &amp; <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font>,<font color="Olive" title="b3">r</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  )
 <b>by </b><i><a class="txt" href="anproj_1.html#E3:44_1_1_2"><i><font color="Green" title="E34">A3</font></i></a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_1_3"><b>now </b></a><div class="add"><b>given </b><font color="Maroon" title="c9">p</font>, <font color="Maroon" title="c10">q</font>, <font color="Maroon" title="c11">r</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font><b> such that </b><a NAME="E2:44_1_1_3"/><i><font color="Green" title="E34">A4</font></i>: 
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K3" title="ANPROJ_1:func.3">Dir</a> <font color="Maroon" title="c9">p</font> &amp; <font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K3" title="ANPROJ_1:func.3">Dir</a> <font color="Maroon" title="c10">q</font> &amp; <font color="Maroon" title="c8">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K3" title="ANPROJ_1:func.3">Dir</a> <font color="Maroon" title="c11">r</font> &amp; <font color="Maroon" title="c9">p</font> <a href="anproj_1.html#NR1" title="ANPROJ_1:NR.1">is_Prop_Vect</a>  &amp; <font color="Maroon" title="c10">q</font> <a href="anproj_1.html#NR1" title="ANPROJ_1:NR.1">is_Prop_Vect</a>  &amp; <font color="Maroon" title="c11">r</font> <a href="anproj_1.html#NR1" title="ANPROJ_1:NR.1">is_Prop_Vect</a>  &amp; <font color="Maroon" title="c9">p</font>,<font color="Maroon" title="c10">q</font>,<font color="Maroon" title="c11">r</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  )
 ;<br/><b>set </b><font color="Maroon" title="c12">xyz</font> = <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c8">z</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span>;<br/><a NAME="E3:44_1_1_3"/>
(  <a href="anproj_1.html#K3" title="ANPROJ_1:func.3">Dir</a> <font color="Maroon" title="c9">p</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="anproj_1.html#K4" title="ANPROJ_1:func.4">ProjectivePoints</a> <font color="Maroon" title="c1">V</font> &amp;  <a href="anproj_1.html#K3" title="ANPROJ_1:func.3">Dir</a> <font color="Maroon" title="c10">q</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="anproj_1.html#K4" title="ANPROJ_1:func.4">ProjectivePoints</a> <font color="Maroon" title="c1">V</font> &amp;  <a href="anproj_1.html#K3" title="ANPROJ_1:func.3">Dir</a> <font color="Maroon" title="c11">r</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="anproj_1.html#K4" title="ANPROJ_1:func.4">ProjectivePoints</a> <font color="Maroon" title="c1">V</font> )
 <b>by </b><i><a class="txt" href="anproj_1.html#E2:44_1_1_3"><i><font color="Green" title="E34">A4</font></i></a>, <a class="ref" href="anproj_1.html#T34" target="_self" title="ANPROJ_1:th.34">Th34</a></i>;<br/><a NAME="E4:44_1_1_3"/><b>then </b>
<span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c8">z</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="anproj_1.html#K4" title="ANPROJ_1:func.4">ProjectivePoints</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p2">(<span class="default"><a href="anproj_1.html#K4" title="ANPROJ_1:func.4">ProjectivePoints</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p2">(<span class="default"><a href="anproj_1.html#K4" title="ANPROJ_1:func.4">ProjectivePoints</a> <font color="Maroon" title="c1">V</font></span>)</span></span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span>
 <b>by </b><i><a class="txt" href="anproj_1.html#E2:44_1_1_3"><i><font color="Green" title="E34">A4</font></i></a>, <a class="ref" href="mcart_1.html#T73" title="MCART_1:th.73">MCART_1:73</a></i>;<br/><b>hence </b><a NAME="E5:44_1_1_3"/>
<span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c8">z</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">R'</font>
 <b>by </b><i><a class="txt" href="anproj_1.html#E2:44_1_1"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="anproj_1.html#E2:44_1_1_3"><i><font color="Green" title="E34">A4</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:44_1_1"/>
( <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c8">z</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">R'</font> iff  ex <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font>, <font color="Olive" title="b3">r</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font> st <br/>( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K3" title="ANPROJ_1:func.3">Dir</a> <font color="Olive" title="b1">p</font> &amp; <font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K3" title="ANPROJ_1:func.3">Dir</a> <font color="Olive" title="b2">q</font> &amp; <font color="Maroon" title="c8">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K3" title="ANPROJ_1:func.3">Dir</a> <font color="Olive" title="b3">r</font> &amp; <font color="Olive" title="b1">p</font> <a href="anproj_1.html#NR1" title="ANPROJ_1:NR.1">is_Prop_Vect</a>  &amp; <font color="Olive" title="b2">q</font> <a href="anproj_1.html#NR1" title="ANPROJ_1:NR.1">is_Prop_Vect</a>  &amp; <font color="Olive" title="b3">r</font> <a href="anproj_1.html#NR1" title="ANPROJ_1:NR.1">is_Prop_Vect</a>  &amp; <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font>,<font color="Olive" title="b3">r</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  ) )
 <b>by </b><i><a class="txt" href="anproj_1.html#E5:44_1_1"><i><font color="Green" title="E33">A2</font></i></a></i>;<br/>


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