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

<b>let </b><font color="Maroon" title="c1">AS</font> be   <a href="diraf.html#NM1" title="DIRAF:NM.1">AffinSpace</a>;<br/>

<b>assume </b><a NAME="E1:73"/><i><font color="Green" title="E57">A1</font></i>: 
 <a href="afproj.html#K13" title="AFPROJ:func.13">IncProjSp_of</a> <font color="Maroon" title="c1">AS</font> is <a href="incproj.html#V11" title="INCPROJ:attr.11">Pappian</a>
 ;<br/>

<b>set </b><font color="Maroon" title="c2">XX</font> =  <a href="afproj.html#K13" title="AFPROJ:func.13">IncProjSp_of</a> <font color="Maroon" title="c1">AS</font>;<br/>
<a NAME="E2:73"/>
for <font color="Olive" title="b1">M</font>, <font color="Olive" title="b2">N</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font><br/> for <font color="Olive" title="b3">o</font>, <font color="Olive" title="b4">a</font>, <font color="Olive" title="b5">b</font>, <font color="Olive" title="b6">c</font>, <font color="Olive" title="b7">a'</font>, <font color="Olive" title="b8">b'</font>, <font color="Olive" title="b9">c'</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">AS</font>  st <font color="Olive" title="b1">M</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <font color="Olive" title="b2">N</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <font color="Olive" title="b1">M</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">N</font> &amp; <font color="Olive" title="b3">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">M</font> &amp; <font color="Olive" title="b3">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">N</font> &amp; <font color="Olive" title="b3">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b4">a</font> &amp; <font color="Olive" title="b3">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b7">a'</font> &amp; <font color="Olive" title="b3">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b5">b</font> &amp; <font color="Olive" title="b3">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b8">b'</font> &amp; <font color="Olive" title="b3">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b6">c</font> &amp; <font color="Olive" title="b3">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b9">c'</font> &amp; <font color="Olive" title="b4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">M</font> &amp; <font color="Olive" title="b5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">M</font> &amp; <font color="Olive" title="b6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">M</font> &amp; <font color="Olive" title="b7">a'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">N</font> &amp; <font color="Olive" title="b8">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">N</font> &amp; <font color="Olive" title="b9">c'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">N</font> &amp; <font color="Olive" title="b4">a</font>,<font color="Olive" title="b8">b'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b5">b</font>,<font color="Olive" title="b7">a'</font> &amp; <font color="Olive" title="b5">b</font>,<font color="Olive" title="b9">c'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b6">c</font>,<font color="Olive" title="b8">b'</font> holds <br/><font color="Olive" title="b4">a</font>,<font color="Olive" title="b9">c'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b6">c</font>,<font color="Olive" title="b7">a'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">M</font>, <font color="Maroon" title="c4">N</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font>;<br/><b>let </b><font color="Maroon" title="c5">o</font>, <font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c7">b</font>, <font color="Maroon" title="c8">c</font>, <font color="Maroon" title="c9">a'</font>, <font color="Maroon" title="c10">b'</font>, <font color="Maroon" title="c11">c'</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">AS</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:73_1"/><i><font color="Green" title="E58">A2</font></i>: 
( <font color="Maroon" title="c3">M</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <font color="Maroon" title="c4">N</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  )
 <b>and </b><br/><a NAME="E2:73_1"/><i><font color="Green" title="E59">A3</font></i>: 
<font color="Maroon" title="c3">M</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">N</font>
 <b>and </b><br/><a NAME="E3:73_1"/><i><font color="Green" title="E60">A4</font></i>: 
( <font color="Maroon" title="c5">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c5">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">N</font> )
 <b>and </b><br/><a NAME="E4:73_1"/><i><font color="Green" title="E61">A5</font></i>: 
( <font color="Maroon" title="c5">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">a</font> &amp; <font color="Maroon" title="c5">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">a'</font> &amp; <font color="Maroon" title="c5">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">b</font> &amp; <font color="Maroon" title="c5">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">b'</font> &amp; <font color="Maroon" title="c5">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">c</font> &amp; <font color="Maroon" title="c5">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">c'</font> )
 <b>and </b><br/><a NAME="E5:73_1"/><i><font color="Green" title="E62">A6</font></i>: 
( <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c8">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c9">a'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">N</font> &amp; <font color="Maroon" title="c10">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">N</font> &amp; <font color="Maroon" title="c11">c'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">N</font> )
 <b>and </b><br/><a NAME="E6:73_1"/><i><font color="Green" title="E63">A7</font></i>: 
( <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c10">b'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c9">a'</font> &amp; <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c11">c'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c10">b'</font> )
 ;<br/>

<b>consider </b><font color="Maroon" title="c12">Y</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font><b> such that </b><br/><a NAME="E8:73_1"/><i><font color="Green" title="E64">A8</font></i>: 
( <font color="Maroon" title="c3">M</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">Y</font> &amp; <font color="Maroon" title="c4">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">Y</font> )
 <b>and </b><br/><a NAME="E9:73_1"/><i><font color="Green" title="E65">A9</font></i>: 
<font color="Maroon" title="c12">Y</font> <a href="aff_4.html#NR1" title="AFF_4:NR.1">is_plane</a> 
 <b>by </b><i><a class="txt" href="afproj.html#E1:73_1"><i><font color="Green" title="E58">A2</font></i></a>, <a class="txt" href="afproj.html#E3:73_1"><i><font color="Green" title="E60">A4</font></i></a>, <a class="ref" href="aff_4.html#T38" title="AFF_4:th.38">AFF_4:38</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">p</font> = <font color="Maroon" title="c5">o</font>, <font color="Maroon" title="c14">a1</font> = <font color="Maroon" title="c9">a'</font>, <font color="Maroon" title="c15">a2</font> = <font color="Maroon" title="c11">c'</font>, <font color="Maroon" title="c16">a3</font> = <font color="Maroon" title="c10">b'</font>, <font color="Maroon" title="c17">b1</font> = <font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c18">b2</font> = <font color="Maroon" title="c8">c</font>, <font color="Maroon" title="c19">b3</font> = <font color="Maroon" title="c7">b</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <span class="p1">(<span class="default"><a href="afproj.html#K13" title="AFPROJ:func.13">IncProjSp_of</a> <font color="Maroon" title="c1">AS</font></span>)</span> <b>by </b><i><a class="ref" href="afproj.html#T20" target="_self" title="AFPROJ:th.20">Th20</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c20">A3</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">M</font>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>, <font color="Maroon" title="c21">B3</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">N</font>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="incsp_1.html#U2" title="INCSP_1:sel.2">Lines</a> of <span class="p1">(<span class="default"><a href="afproj.html#K13" title="AFPROJ:func.13">IncProjSp_of</a> <font color="Maroon" title="c1">AS</font></span>)</span> <b>by </b><i><a class="txt" href="afproj.html#E1:73_1"><i><font color="Green" title="E58">A2</font></i></a>, <a class="ref" href="afproj.html#T23" target="_self" title="AFPROJ:th.23">Th23</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c22">C3'</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="afproj.html#K6" title="AFPROJ:func.6">PDir</a> <font color="Maroon" title="c12">Y</font></span>)</span>,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="incsp_1.html#U2" title="INCSP_1:sel.2">Lines</a> of <span class="p1">(<span class="default"><a href="afproj.html#K13" title="AFPROJ:func.13">IncProjSp_of</a> <font color="Maroon" title="c1">AS</font></span>)</span> <b>by </b><i><a class="txt" href="afproj.html#E9:73_1"><i><font color="Green" title="E65">A9</font></i></a>, <a class="ref" href="afproj.html#T23" target="_self" title="AFPROJ:th.23">Th23</a></i>;<br/>
<a NAME="E13:73_1"/><i><font color="Green" title="E66">A10</font></i>: 
( <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c10">b'</font> &amp; <font color="Maroon" title="c10">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c10">b'</font> &amp; <font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c9">a'</font> &amp; <font color="Maroon" title="c9">a'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c9">a'</font> &amp; <font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c11">c'</font> &amp; <font color="Maroon" title="c11">c'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c11">c'</font> &amp; <font color="Maroon" title="c8">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c10">b'</font> &amp; <font color="Maroon" title="c10">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c10">b'</font> &amp; <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c11">c'</font> &amp; <font color="Maroon" title="c11">c'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c11">c'</font> &amp; <font color="Maroon" title="c8">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c9">a'</font> &amp; <font color="Maroon" title="c9">a'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c9">a'</font> )
 
<b>by </b><i><a class="ref" href="aff_1.html#T26" title="AFF_1:th.26">AFF_1:26</a></i>;<br/>
<a NAME="E14:73_1"/><i><font color="Green" title="E67">A11</font></i>: 
( <font color="Maroon" title="c6">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">b'</font> &amp; <font color="Maroon" title="c7">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">a'</font> &amp; <font color="Maroon" title="c7">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">c'</font> &amp; <font color="Maroon" title="c8">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">b'</font> &amp; <font color="Maroon" title="c6">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">c'</font> &amp; <font color="Maroon" title="c8">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">a'</font> )
 
<b>by </b><i><a class="txt" href="afproj.html#E1:73_1"><i><font color="Green" title="E58">A2</font></i></a>, <a class="txt" href="afproj.html#E2:73_1"><i><font color="Green" title="E59">A3</font></i></a>, <a class="txt" href="afproj.html#E3:73_1"><i><font color="Green" title="E60">A4</font></i></a>, <a class="txt" href="afproj.html#E4:73_1"><i><font color="Green" title="E61">A5</font></i></a>, <a class="txt" href="afproj.html#E5:73_1"><i><font color="Green" title="E62">A6</font></i></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></i>;<br/>
<a NAME="E15:73_1"/><b>then </b><i><font color="Green" title="E68">A12</font></i>: 
(  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c10">b'</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp;  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c9">a'</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp;  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c11">c'</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp;  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c10">b'</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp;  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c11">c'</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp;  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c9">a'</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  )
 
<b>by </b><i><a class="ref" href="aff_1.html#D3" title="AFF_1:def.3">AFF_1:def 3</a></i>;<br/>
<a NAME="E16:73_1"/>
(  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c10">b'</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c9">a'</font> &amp;  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c11">c'</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c10">b'</font> )
 
<b>by </b><i><a class="txt" href="afproj.html#E6:73_1"><i><font color="Green" title="E63">A7</font></i></a>, <a class="txt" href="afproj.html#E14:73_1"><i><font color="Green" title="E67">A11</font></i></a>, <a class="ref" href="aff_1.html#T51" title="AFF_1:th.51">AFF_1:51</a></i>;<br/>
<a NAME="E17:73_1"/><b>then </b><i><font color="Green" title="E69">A13</font></i>: 
(  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c10">b'</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c9">a'</font> &amp;  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c11">c'</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c10">b'</font> )
 
<b>by </b><i><a class="txt" href="afproj.html#E15:73_1"><i><font color="Green" title="E68">A12</font></i></a>, <a class="ref" href="aff_4.html#T40" title="AFF_4:th.40">AFF_4:40</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c23">A1</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c11">c'</font></span>)</span>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>, <font color="Maroon" title="c24">A2</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c9">a'</font></span>)</span>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>, <font color="Maroon" title="c25">B1</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c10">b'</font></span>)</span>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>, <font color="Maroon" title="c26">B2</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c10">b'</font></span>)</span>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>, <font color="Maroon" title="c27">C1</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c9">a'</font></span>)</span>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>, <font color="Maroon" title="c28">C2</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c11">c'</font></span>)</span>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="incsp_1.html#U2" title="INCSP_1:sel.2">Lines</a> of <span class="p1">(<span class="default"><a href="afproj.html#K13" title="AFPROJ:func.13">IncProjSp_of</a> <font color="Maroon" title="c1">AS</font></span>)</span> <b>by </b><i><a class="txt" href="afproj.html#E15:73_1"><i><font color="Green" title="E68">A12</font></i></a>, <a class="ref" href="afproj.html#T23" target="_self" title="AFPROJ:th.23">Th23</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c29">c1</font> =  <a href="afproj.html#K5" title="AFPROJ:func.5">LDir</a> <span class="p1">(<span class="default"><a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c11">c'</font></span>)</span>, <font color="Maroon" title="c30">c2</font> =  <a href="afproj.html#K5" title="AFPROJ:func.5">LDir</a> <span class="p1">(<span class="default"><a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c10">b'</font></span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <span class="p1">(<span class="default"><a href="afproj.html#K13" title="AFPROJ:func.13">IncProjSp_of</a> <font color="Maroon" title="c1">AS</font></span>)</span> <b>by </b><i><a class="txt" href="afproj.html#E15:73_1"><i><font color="Green" title="E68">A12</font></i></a>, <a class="ref" href="afproj.html#T20" target="_self" title="AFPROJ:th.20">Th20</a></i>;<br/>
<a NAME="E20:73_1"/><i><font color="Green" title="E70">A14</font></i>: 
( <font color="Maroon" title="c13">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c20">A3</font> &amp; <font color="Maroon" title="c17">b1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c20">A3</font> &amp; <font color="Maroon" title="c18">b2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c20">A3</font> &amp; <font color="Maroon" title="c19">b3</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c20">A3</font> &amp; <font color="Maroon" title="c13">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c21">B3</font> &amp; <font color="Maroon" title="c14">a1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c21">B3</font> &amp; <font color="Maroon" title="c15">a2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c21">B3</font> &amp; <font color="Maroon" title="c16">a3</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c21">B3</font> )
 
<b>by </b><i><a class="txt" href="afproj.html#E1:73_1"><i><font color="Green" title="E58">A2</font></i></a>, <a class="txt" href="afproj.html#E3:73_1"><i><font color="Green" title="E60">A4</font></i></a>, <a class="txt" href="afproj.html#E5:73_1"><i><font color="Green" title="E62">A6</font></i></a>, <a class="ref" href="afproj.html#T26" target="_self" title="AFPROJ:th.26">Th26</a></i>;<br/>
<a NAME="E21:73_1"/><i><font color="Green" title="E71">A15</font></i>: 
( <font color="Maroon" title="c15">a2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c23">A1</font> &amp; <font color="Maroon" title="c19">b3</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c23">A1</font> &amp; <font color="Maroon" title="c29">c1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c23">A1</font> &amp; <font color="Maroon" title="c16">a3</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c26">B2</font> &amp; <font color="Maroon" title="c18">b2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c26">B2</font> &amp; <font color="Maroon" title="c29">c1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c26">B2</font> &amp; <font color="Maroon" title="c17">b1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c25">B1</font> &amp; <font color="Maroon" title="c16">a3</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c25">B1</font> &amp; <font color="Maroon" title="c30">c2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c25">B1</font> &amp; <font color="Maroon" title="c19">b3</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c24">A2</font> &amp; <font color="Maroon" title="c14">a1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c24">A2</font> &amp; <font color="Maroon" title="c30">c2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c24">A2</font> )
 
<b>by </b><i><a class="txt" href="afproj.html#E13:73_1"><i><font color="Green" title="E66">A10</font></i></a>, <a class="txt" href="afproj.html#E15:73_1"><i><font color="Green" title="E68">A12</font></i></a>, <a class="txt" href="afproj.html#E17:73_1"><i><font color="Green" title="E69">A13</font></i></a>, <a class="ref" href="afproj.html#T26" target="_self" title="AFPROJ:th.26">Th26</a>, <a class="ref" href="afproj.html#T28" target="_self" title="AFPROJ:th.28">Th28</a>, <a class="ref" href="afproj.html#T30" target="_self" title="AFPROJ:th.30">Th30</a></i>;<br/>
<a NAME="E22:73_1"/><i><font color="Green" title="E72">A16</font></i>: 
( <font color="Maroon" title="c17">b1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c28">C2</font> &amp; <font color="Maroon" title="c15">a2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c28">C2</font> &amp; <font color="Maroon" title="c18">b2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c27">C1</font> &amp; <font color="Maroon" title="c14">a1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c27">C1</font> )
 
<b>by </b><i><a class="txt" href="afproj.html#E13:73_1"><i><font color="Green" title="E66">A10</font></i></a>, <a class="txt" href="afproj.html#E15:73_1"><i><font color="Green" title="E68">A12</font></i></a>, <a class="ref" href="afproj.html#T26" target="_self" title="AFPROJ:th.26">Th26</a></i>;<br/>
<a NAME="E23:73_1"/>
(  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c11">c'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">Y</font> &amp;  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c10">b'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">Y</font> )
 
<b>by </b><i><a class="txt" href="afproj.html#E5:73_1"><i><font color="Green" title="E62">A6</font></i></a>, <a class="txt" href="afproj.html#E8:73_1"><i><font color="Green" title="E64">A8</font></i></a>, <a class="txt" href="afproj.html#E9:73_1"><i><font color="Green" title="E65">A9</font></i></a>, <a class="txt" href="afproj.html#E14:73_1"><i><font color="Green" title="E67">A11</font></i></a>, <a class="ref" href="aff_4.html#T19" title="AFF_4:th.19">AFF_4:19</a></i>;<br/>
<a NAME="E24:73_1"/><b>then </b><i><font color="Green" title="E73">A17</font></i>: 
( <font color="Maroon" title="c29">c1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c22">C3'</font> &amp; <font color="Maroon" title="c30">c2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c22">C3'</font> )
 
<b>by </b><i><a class="txt" href="afproj.html#E9:73_1"><i><font color="Green" title="E65">A9</font></i></a>, <a class="txt" href="afproj.html#E15:73_1"><i><font color="Green" title="E68">A12</font></i></a>, <a class="ref" href="afproj.html#T31" target="_self" title="AFPROJ:th.31">Th31</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:73_1_1"/><i><font color="Green" title="E74">A18</font></i>: 
( <font color="Maroon" title="c17">b1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c18">b2</font> &amp; <font color="Maroon" title="c18">b2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c19">b3</font> &amp; <font color="Maroon" title="c19">b3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c17">b1</font> )
 ;<br/><a NAME="E2:73_1_1"/>
( <font color="Maroon" title="c14">a1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">a2</font> &amp; <font color="Maroon" title="c15">a2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c16">a3</font> &amp; <font color="Maroon" title="c14">a1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c16">a3</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:73_1_1_1"/><i><font color="Green" title="E75">A19</font></i>: 
(  not <font color="Maroon" title="c14">a1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">a2</font> or  not <font color="Maroon" title="c15">a2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c16">a3</font> or  not <font color="Maroon" title="c14">a1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c16">a3</font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:73_1_1_1_1"/>
<font color="Maroon" title="c9">a'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">c'</font>
 ;<br/><a NAME="E2:73_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c10">b'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c10">b'</font>
 <b>by </b><i><a class="txt" href="afproj.html#E6:73_1"><i><font color="Green" title="E63">A7</font></i></a>, <a class="txt" href="afproj.html#E14:73_1"><i><font color="Green" title="E67">A11</font></i></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></i>;<br/><b>hence </b><a NAME="E3:73_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="afproj.html#E1:73_1"><i><font color="Green" title="E58">A2</font></i></a>, <a class="txt" href="afproj.html#E2:73_1"><i><font color="Green" title="E59">A3</font></i></a>, <a class="txt" href="afproj.html#E3:73_1"><i><font color="Green" title="E60">A4</font></i></a>, <a class="txt" href="afproj.html#E4:73_1"><i><font color="Green" title="E61">A5</font></i></a>, <a class="txt" href="afproj.html#E5:73_1"><i><font color="Green" title="E62">A6</font></i></a>, <a class="txt" href="afproj.html#E1:73_1_1"><i><font color="Green" title="E74">A18</font></i></a>, <a class="ref" href="aff_4.html#T9" title="AFF_4:th.9">AFF_4:9</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:73_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="afproj.html#E1:73_1"><i><font color="Green" title="E58">A2</font></i></a>, <a class="txt" href="afproj.html#E2:73_1"><i><font color="Green" title="E59">A3</font></i></a>, <a class="txt" href="afproj.html#E3:73_1"><i><font color="Green" title="E60">A4</font></i></a>, <a class="txt" href="afproj.html#E4:73_1"><i><font color="Green" title="E61">A5</font></i></a>, <a class="txt" href="afproj.html#E5:73_1"><i><font color="Green" title="E62">A6</font></i></a>, <a class="txt" href="afproj.html#E6:73_1"><i><font color="Green" title="E63">A7</font></i></a>, <a class="txt" href="afproj.html#E1:73_1_1"><i><font color="Green" title="E74">A18</font></i></a>, <a class="txt" href="afproj.html#E1:73_1_1_1"><i><font color="Green" title="E75">A19</font></i></a>, <a class="ref" href="aff_4.html#T9" title="AFF_4:th.9">AFF_4:9</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:73_1_1"/><b>then </b><i><font color="Green" title="E75">A20</font></i>: 
<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">a1</font>,<font color="Maroon" title="c15">a2</font>,<font color="Maroon" title="c16">a3</font> <a href="incproj.html#R2" title="INCPROJ:pred.2">are_mutually_different</a> 
 <b>by </b><i><a class="txt" href="afproj.html#E4:73_1"><i><font color="Green" title="E61">A5</font></i></a>, <a class="ref" href="incproj.html#D6" title="INCPROJ:def.6">INCPROJ:def 6</a></i>;<br/><a NAME="E4:73_1_1"/><i><font color="Green" title="E76">A21</font></i>: 
<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c17">b1</font>,<font color="Maroon" title="c18">b2</font>,<font color="Maroon" title="c19">b3</font> <a href="incproj.html#R2" title="INCPROJ:pred.2">are_mutually_different</a> 
 <b>by </b><i><a class="txt" href="afproj.html#E4:73_1"><i><font color="Green" title="E61">A5</font></i></a>, <a class="txt" href="afproj.html#E1:73_1_1"><i><font color="Green" title="E74">A18</font></i></a>, <a class="ref" href="incproj.html#D6" title="INCPROJ:def.6">INCPROJ:def 6</a></i>;<br/><a NAME="E5:73_1_1"/><i><font color="Green" title="E77">A22</font></i>: 
( <font color="Maroon" title="c20">A3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c21">B3</font> &amp; <font color="Maroon" title="c13">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c20">A3</font> &amp; <font color="Maroon" title="c13">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c21">B3</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_1_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:73_1_1_2"/>
<font color="Maroon" title="c20">A3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c21">B3</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:73_1_1_2_1"/>
<font color="Maroon" title="c20">A3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">B3</font>
 ;<br/>

<b>then </b><font color="Maroon" title="c3">M</font> = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">N</font>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
<b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
<br/>.= 
<font color="Maroon" title="c4">N</font>
<b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
;<br/>
<b>hence </b><a NAME="E3:73_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="afproj.html#E2:73_1"><i><font color="Green" title="E59">A3</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:73_1_1_2"/>
( <font color="Maroon" title="c13">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c20">A3</font> &amp; <font color="Maroon" title="c13">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c21">B3</font> )
 <b>by </b><i><a class="txt" href="afproj.html#E1:73_1"><i><font color="Green" title="E58">A2</font></i></a>, <a class="txt" href="afproj.html#E3:73_1"><i><font color="Green" title="E60">A4</font></i></a>, <a class="ref" href="afproj.html#T26" target="_self" title="AFPROJ:th.26">Th26</a></i>;<br/>


</div><b>end;</b></div><a NAME="E6:73_1_1"/>
( not <font color="Maroon" title="c13">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c27">C1</font> &amp; not <font color="Maroon" title="c13">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c28">C2</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_1_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:73_1_1_3"/>
( <font color="Maroon" title="c13">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c27">C1</font> or <font color="Maroon" title="c13">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c28">C2</font> )
 ;<br/>

<a NAME="E2:73_1_1_3"/><b>then </b>
( <font color="Maroon" title="c14">a1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c20">A3</font> or <font color="Maroon" title="c15">a2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c20">A3</font> )
 
<b>by </b><i><a class="txt" href="afproj.html#E4:73_1"><i><font color="Green" title="E61">A5</font></i></a>, <a class="txt" href="afproj.html#E20:73_1"><i><font color="Green" title="E70">A14</font></i></a>, <a class="txt" href="afproj.html#E22:73_1"><i><font color="Green" title="E72">A16</font></i></a>, <a class="txt" href="afproj.html#E56"><i><font color="Green" title="E41">Lm2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:73_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="afproj.html#E4:73_1"><i><font color="Green" title="E61">A5</font></i></a>, <a class="txt" href="afproj.html#E20:73_1"><i><font color="Green" title="E70">A14</font></i></a>, <a class="txt" href="afproj.html#E5:73_1_1"><i><font color="Green" title="E77">A22</font></i></a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></i>;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon" title="c31">c3</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <span class="p1">(<span class="default"><a href="afproj.html#K13" title="AFPROJ:func.13">IncProjSp_of</a> <font color="Maroon" title="c1">AS</font></span>)</span><b> such that </b><br/><a NAME="E8:73_1_1"/><i><font color="Green" title="E78">A23</font></i>: 
( <font color="Maroon" title="c31">c3</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c27">C1</font> &amp; <font color="Maroon" title="c31">c3</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c28">C2</font> )
 <b>by </b><i><a class="txt" href="afproj.html#E20:73_1"><i><font color="Green" title="E70">A14</font></i></a>, <a class="txt" href="afproj.html#E22:73_1"><i><font color="Green" title="E72">A16</font></i></a>, <a class="txt" href="afproj.html#E5:73_1_1"><i><font color="Green" title="E77">A22</font></i></a>, <a class="ref" href="incproj.html#D13" title="INCPROJ:def.13">INCPROJ:def 13</a></i>;<br/><a NAME="E9:73_1_1"/><i><font color="Green" title="E79">A24</font></i>: 
( <font color="Maroon" title="c15">a2</font>,<font color="Maroon" title="c19">b3</font>,<font color="Maroon" title="c29">c1</font> <a href="incproj.html#R4" title="INCPROJ:pred.4">on</a> <font color="Maroon" title="c23">A1</font> &amp; <font color="Maroon" title="c16">a3</font>,<font color="Maroon" title="c17">b1</font>,<font color="Maroon" title="c30">c2</font> <a href="incproj.html#R4" title="INCPROJ:pred.4">on</a> <font color="Maroon" title="c25">B1</font> &amp; <font color="Maroon" title="c14">a1</font>,<font color="Maroon" title="c18">b2</font>,<font color="Maroon" title="c31">c3</font> <a href="incproj.html#R4" title="INCPROJ:pred.4">on</a> <font color="Maroon" title="c27">C1</font> &amp; <font color="Maroon" title="c14">a1</font>,<font color="Maroon" title="c19">b3</font>,<font color="Maroon" title="c30">c2</font> <a href="incproj.html#R4" title="INCPROJ:pred.4">on</a> <font color="Maroon" title="c24">A2</font> &amp; <font color="Maroon" title="c16">a3</font>,<font color="Maroon" title="c18">b2</font>,<font color="Maroon" title="c29">c1</font> <a href="incproj.html#R4" title="INCPROJ:pred.4">on</a> <font color="Maroon" title="c26">B2</font> &amp; <font color="Maroon" title="c15">a2</font>,<font color="Maroon" title="c17">b1</font>,<font color="Maroon" title="c31">c3</font> <a href="incproj.html#R4" title="INCPROJ:pred.4">on</a> <font color="Maroon" title="c28">C2</font> &amp; <font color="Maroon" title="c17">b1</font>,<font color="Maroon" title="c18">b2</font>,<font color="Maroon" title="c19">b3</font> <a href="incproj.html#R4" title="INCPROJ:pred.4">on</a> <font color="Maroon" title="c20">A3</font> &amp; <font color="Maroon" title="c14">a1</font>,<font color="Maroon" title="c15">a2</font>,<font color="Maroon" title="c16">a3</font> <a href="incproj.html#R4" title="INCPROJ:pred.4">on</a> <font color="Maroon" title="c21">B3</font> )
 <b>by </b><i><a class="txt" href="afproj.html#E20:73_1"><i><font color="Green" title="E70">A14</font></i></a>, <a class="txt" href="afproj.html#E21:73_1"><i><font color="Green" title="E71">A15</font></i></a>, <a class="txt" href="afproj.html#E22:73_1"><i><font color="Green" title="E72">A16</font></i></a>, <a class="txt" href="afproj.html#E8:73_1_1"><i><font color="Green" title="E78">A23</font></i></a>, <a class="ref" href="incproj.html#D8" title="INCPROJ:def.8">INCPROJ:def 8</a></i>;<br/><a NAME="E10:73_1_1"/>
<font color="Maroon" title="c29">c1</font>,<font color="Maroon" title="c30">c2</font> <a href="incproj.html#R3" title="INCPROJ:pred.3">on</a> <font color="Maroon" title="c22">C3'</font>
 <b>by </b><i><a class="txt" href="afproj.html#E24:73_1"><i><font color="Green" title="E73">A17</font></i></a>, <a class="ref" href="incproj.html#D7" title="INCPROJ:def.7">INCPROJ:def 7</a></i>;<br/><a NAME="E11:73_1_1"/><b>then </b>
<font color="Maroon" title="c31">c3</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c22">C3'</font>
 <b>by </b><i><a class="txt" href="afproj.html#E1:73"><i><font color="Green" title="E57">A1</font></i></a>, <a class="txt" href="afproj.html#E3:73_1_1"><i><font color="Green" title="E75">A20</font></i></a>, <a class="txt" href="afproj.html#E4:73_1_1"><i><font color="Green" title="E76">A21</font></i></a>, <a class="txt" href="afproj.html#E5:73_1_1"><i><font color="Green" title="E77">A22</font></i></a>, <a class="txt" href="afproj.html#E9:73_1_1"><i><font color="Green" title="E79">A24</font></i></a>, <a class="ref" href="incproj.html#D20" title="INCPROJ:def.20">INCPROJ:def 20</a></i>;<br/><a NAME="E12:73_1_1"/><b>then </b>
<font color="Maroon" title="c31">c3</font> is  not   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font>
 <b>by </b><i><a class="txt" href="afproj.html#E9:73_1"><i><font color="Green" title="E65">A9</font></i></a>, <a class="ref" href="afproj.html#T27" target="_self" title="AFPROJ:th.27">Th27</a></i>;<br/><b>then consider </b><font color="Maroon" title="c32">Y</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font><b> such that </b><br/><a NAME="E14:73_1_1"/><i><font color="Green" title="E80">A25</font></i>: 
( <font color="Maroon" title="c31">c3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afproj.html#K5" title="AFPROJ:func.5">LDir</a> <font color="Maroon" title="c32">Y</font> &amp; <font color="Maroon" title="c32">Y</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  )
 <b>by </b><i><a class="ref" href="afproj.html#T20" target="_self" title="AFPROJ:th.20">Th20</a></i>;<br/><a NAME="E15:73_1_1"/>
( <font color="Maroon" title="c32">Y</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c11">c'</font> &amp; <font color="Maroon" title="c32">Y</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c9">a'</font> )
 <b>by </b><i><a class="txt" href="afproj.html#E15:73_1"><i><font color="Green" title="E68">A12</font></i></a>, <a class="txt" href="afproj.html#E8:73_1_1"><i><font color="Green" title="E78">A23</font></i></a>, <a class="txt" href="afproj.html#E14:73_1_1"><i><font color="Green" title="E80">A25</font></i></a>, <a class="ref" href="afproj.html#T28" target="_self" title="AFPROJ:th.28">Th28</a></i>;<br/><a NAME="E16:73_1_1"/><b>then </b>
( <font color="Maroon" title="c32">Y</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c11">c'</font> &amp; <font color="Maroon" title="c32">Y</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c9">a'</font> )
 <b>by </b><i><a class="txt" href="afproj.html#E15:73_1"><i><font color="Green" title="E68">A12</font></i></a>, <a class="txt" href="afproj.html#E14:73_1_1"><i><font color="Green" title="E80">A25</font></i></a>, <a class="ref" href="aff_4.html#T40" title="AFF_4:th.40">AFF_4:40</a></i>;<br/><a NAME="E17:73_1_1"/><b>then </b>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c11">c'</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c9">a'</font>
 <b>by </b><i><a class="ref" href="aff_1.html#T58" title="AFF_1:th.58">AFF_1:58</a></i>;<br/><b>hence </b><a NAME="E18:73_1_1"/>
<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c11">c'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c9">a'</font>
 <b>by </b><i><a class="txt" href="afproj.html#E13:73_1"><i><font color="Green" title="E66">A10</font></i></a>, <a class="ref" href="aff_1.html#T53" title="AFF_1:th.53">AFF_1:53</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E26:73_1"/>
<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c11">c'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c9">a'</font>
 <b>by </b><i><a class="txt" href="afproj.html#E1:73_1"><i><font color="Green" title="E58">A2</font></i></a>, <a class="txt" href="afproj.html#E2:73_1"><i><font color="Green" title="E59">A3</font></i></a>, <a class="txt" href="afproj.html#E3:73_1"><i><font color="Green" title="E60">A4</font></i></a>, <a class="txt" href="afproj.html#E4:73_1"><i><font color="Green" title="E61">A5</font></i></a>, <a class="txt" href="afproj.html#E5:73_1"><i><font color="Green" title="E62">A6</font></i></a>, <a class="txt" href="afproj.html#E6:73_1"><i><font color="Green" title="E63">A7</font></i></a>, <a class="ref" href="afproj.html#T48" target="_self" title="AFPROJ:th.48">Th48</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:73"/>
<font color="Maroon" title="c1">AS</font> is <a href="aff_2.html#V2" title="AFF_2:attr.2">Pappian</a>
 <b>by </b><i><a class="ref" href="aff_2.html#D2" title="AFF_2:def.2">AFF_2:def 2</a></i>;<br/>


</div>
