<?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="E1:75"/><i><font color="Green" title="E56">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#V9" title="INCPROJ:attr.9">Desarguesian</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="E2:75"/>
for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">P</font>, <font color="Olive" title="b3">C</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="b4">o</font>, <font color="Olive" title="b5">a</font>, <font color="Olive" title="b6">b</font>, <font color="Olive" title="b7">c</font>, <font color="Olive" title="b8">a'</font>, <font color="Olive" title="b9">b'</font>, <font color="Olive" title="b10">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="b4">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> &amp; <font color="Olive" title="b4">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">P</font> &amp; <font color="Olive" title="b4">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">C</font> &amp; <font color="Olive" title="b4">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b5">a</font> &amp; <font color="Olive" title="b4">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b6">b</font> &amp; <font color="Olive" title="b4">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b7">c</font> &amp; <font color="Olive" title="b5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> &amp; <font color="Olive" title="b8">a'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> &amp; <font color="Olive" title="b6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">P</font> &amp; <font color="Olive" title="b9">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">P</font> &amp; <font color="Olive" title="b7">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">C</font> &amp; <font color="Olive" title="b10">c'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">C</font> &amp; <font color="Olive" title="b1">A</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <font color="Olive" title="b2">P</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <font color="Olive" title="b3">C</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <font color="Olive" title="b1">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">P</font> &amp; <font color="Olive" title="b1">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">C</font> &amp; <font color="Olive" title="b5">a</font>,<font color="Olive" title="b6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b8">a'</font>,<font color="Olive" title="b9">b'</font> &amp; <font color="Olive" title="b5">a</font>,<font color="Olive" title="b7">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b8">a'</font>,<font color="Olive" title="b10">c'</font> holds <br/><font color="Olive" title="b6">b</font>,<font color="Olive" title="b7">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b9">b'</font>,<font color="Olive" title="b10">c'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c4">P</font>, <font color="Maroon" title="c5">C</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="c6">o</font>, <font color="Maroon" title="c7">a</font>, <font color="Maroon" title="c8">b</font>, <font color="Maroon" title="c9">c</font>, <font color="Maroon" title="c10">a'</font>, <font color="Maroon" title="c11">b'</font>, <font color="Maroon" title="c12">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="E1:75_1"/><i><font color="Green" title="E57">A2</font></i>: 
( <font color="Maroon" title="c6">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Maroon" title="c6">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P</font> &amp; <font color="Maroon" title="c6">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">C</font> )
 <b>and </b><br/><a NAME="E2:75_1"/><i><font color="Green" title="E58">A3</font></i>: 
( <font color="Maroon" title="c6">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">a</font> &amp; <font color="Maroon" title="c6">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">b</font> &amp; <font color="Maroon" title="c6">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">c</font> )
 <b>and </b><br/><a NAME="E3:75_1"/><i><font color="Green" title="E59">A4</font></i>: 
( <font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Maroon" title="c10">a'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Maroon" title="c8">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P</font> &amp; <font color="Maroon" title="c11">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P</font> &amp; <font color="Maroon" title="c9">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">C</font> &amp; <font color="Maroon" title="c12">c'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">C</font> )
 <b>and </b><br/><a NAME="E4:75_1"/><i><font color="Green" title="E60">A5</font></i>: 
( <font color="Maroon" title="c3">A</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <font color="Maroon" title="c4">P</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <font color="Maroon" title="c5">C</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  )
 <b>and </b><br/><a NAME="E5:75_1"/><i><font color="Green" title="E61">A6</font></i>: 
( <font color="Maroon" title="c3">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">P</font> &amp; <font color="Maroon" title="c3">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">C</font> )
 <b>and </b><br/><a NAME="E6:75_1"/><i><font color="Green" title="E62">A7</font></i>: 
( <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">a'</font>,<font color="Maroon" title="c11">b'</font> &amp; <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c9">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">a'</font>,<font color="Maroon" title="c12">c'</font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:75_1_1"/><i><font color="Green" title="E63">A8</font></i>: 
<font color="Maroon" title="c4">P</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">C</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:75_1_1_1"/><i><font color="Green" title="E64">A9</font></i>: 
( <font color="Maroon" title="c7">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">a'</font> &amp; <font color="Maroon" title="c6">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">a'</font> )
 ;<br/><a NAME="E2:75_1_1_1"/><b>then </b><i><font color="Green" title="E65">A10</font></i>: 
( <font color="Maroon" title="c6">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">b'</font> &amp; <font color="Maroon" title="c6">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">c'</font> &amp; <font color="Maroon" title="c8">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">b'</font> &amp; <font color="Maroon" title="c9">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">c'</font> )
 <b>by </b><i><a class="txt" href="afproj.html#E1:75_1"><i><font color="Green" title="E57">A2</font></i></a>, <a class="txt" href="afproj.html#E2:75_1"><i><font color="Green" title="E58">A3</font></i></a>, <a class="txt" href="afproj.html#E3:75_1"><i><font color="Green" title="E59">A4</font></i></a>, <a class="txt" href="afproj.html#E4:75_1"><i><font color="Green" title="E60">A5</font></i></a>, <a class="txt" href="afproj.html#E5:75_1"><i><font color="Green" title="E61">A6</font></i></a>, <a class="txt" href="afproj.html#E6:75_1"><i><font color="Green" title="E62">A7</font></i></a>, <a class="ref" href="aff_4.html#T8" title="AFF_4:th.8">AFF_4:8</a>, <a class="ref" href="aff_4.html#T9" title="AFF_4:th.9">AFF_4:9</a></i>;<br/><a NAME="E3:75_1_1_1"/><b>then </b><i><font color="Green" title="E66">A11</font></i>: 
( <font color="Maroon" title="c7">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">b</font> &amp; <font color="Maroon" title="c8">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">c</font> &amp; <font color="Maroon" title="c7">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">c</font> &amp; <font color="Maroon" title="c10">a'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">b'</font> &amp; <font color="Maroon" title="c11">b'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">c'</font> &amp; <font color="Maroon" title="c10">a'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">c'</font> )
 <b>by </b><i><a class="txt" href="afproj.html#E1:75_1"><i><font color="Green" title="E57">A2</font></i></a>, <a class="txt" href="afproj.html#E2:75_1"><i><font color="Green" title="E58">A3</font></i></a>, <a class="txt" href="afproj.html#E3:75_1"><i><font color="Green" title="E59">A4</font></i></a>, <a class="txt" href="afproj.html#E4:75_1"><i><font color="Green" title="E60">A5</font></i></a>, <a class="txt" href="afproj.html#E5:75_1"><i><font color="Green" title="E61">A6</font></i></a>, <a class="txt" href="afproj.html#E1:75_1_1"><i><font color="Green" title="E63">A8</font></i></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></i>;<br/><a NAME="E4:75_1_1_1"/><b>then </b><i><font color="Green" title="E67">A12</font></i>: 
(  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">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="c8">b</font>,<font color="Maroon" title="c9">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="c7">a</font>,<font color="Maroon" title="c9">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="c10">a'</font>,<font color="Maroon" title="c11">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="c11">b'</font>,<font color="Maroon" title="c12">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="c10">a'</font>,<font color="Maroon" title="c12">c'</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="E5:75_1_1_1"/><i><font color="Green" title="E68">A13</font></i>: 
( <font color="Maroon" title="c7">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">a</font>,<font color="Maroon" title="c8">b</font> &amp; <font color="Maroon" title="c8">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">a</font>,<font color="Maroon" title="c8">b</font> &amp; <font color="Maroon" title="c8">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">b</font>,<font color="Maroon" title="c9">c</font> &amp; <font color="Maroon" title="c9">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">b</font>,<font color="Maroon" title="c9">c</font> &amp; <font color="Maroon" title="c7">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">a</font>,<font color="Maroon" title="c9">c</font> &amp; <font color="Maroon" title="c9">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">a</font>,<font color="Maroon" title="c9">c</font> &amp; <font color="Maroon" title="c10">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="c10">a'</font>,<font color="Maroon" title="c11">b'</font> &amp; <font color="Maroon" title="c11">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="c10">a'</font>,<font color="Maroon" title="c11">b'</font> &amp; <font color="Maroon" title="c11">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="c11">b'</font>,<font color="Maroon" title="c12">c'</font> &amp; <font color="Maroon" title="c12">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="c11">b'</font>,<font color="Maroon" title="c12">c'</font> &amp; <font color="Maroon" title="c10">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="c10">a'</font>,<font color="Maroon" title="c12">c'</font> &amp; <font color="Maroon" title="c12">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="c10">a'</font>,<font color="Maroon" title="c12">c'</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="E6:75_1_1_1"/><i><font color="Green" title="E69">A14</font></i>: 
(  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">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="c10">a'</font>,<font color="Maroon" title="c11">b'</font> &amp;  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c9">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="c10">a'</font>,<font color="Maroon" title="c12">c'</font> )
 <b>by </b><i><a class="txt" href="afproj.html#E6:75_1"><i><font color="Green" title="E62">A7</font></i></a>, <a class="txt" href="afproj.html#E3:75_1_1_1"><i><font color="Green" title="E66">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="E7:75_1_1_1"/><b>then </b><i><font color="Green" title="E70">A15</font></i>: 
(  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">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="c10">a'</font>,<font color="Maroon" title="c11">b'</font> &amp;  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c9">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="c10">a'</font>,<font color="Maroon" title="c12">c'</font> )
 <b>by </b><i><a class="txt" href="afproj.html#E4:75_1_1_1"><i><font color="Green" title="E67">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="c13">p</font> = <font color="Maroon" title="c6">o</font>, <font color="Maroon" title="c14">a1</font> = <font color="Maroon" title="c7">a</font>, <font color="Maroon" title="c15">b1</font> = <font color="Maroon" title="c10">a'</font>, <font color="Maroon" title="c16">a2</font> = <font color="Maroon" title="c8">b</font>, <font color="Maroon" title="c17">b2</font> = <font color="Maroon" title="c11">b'</font>, <font color="Maroon" title="c18">a3</font> = <font color="Maroon" title="c9">c</font>, <font color="Maroon" title="c19">b3</font> = <font color="Maroon" title="c12">c'</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">C1</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">A</font>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>, <font color="Maroon" title="c21">C2</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">P</font>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>, <font color="Maroon" title="c22">C3'</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">C</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#E4:75_1"><i><font color="Green" title="E60">A5</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="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="c8">b</font>,<font color="Maroon" title="c9">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">a</font>,<font color="Maroon" title="c9">c</font></span>)</span>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>, <font color="Maroon" title="c25">A3</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">a</font>,<font color="Maroon" title="c8">b</font></span>)</span>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>, <font color="Maroon" title="c26">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="c11">b'</font>,<font color="Maroon" title="c12">c'</font></span>)</span>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>, <font color="Maroon" title="c27">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="c10">a'</font>,<font color="Maroon" title="c12">c'</font></span>)</span>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>, <font color="Maroon" title="c28">B3</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="c10">a'</font>,<font color="Maroon" title="c11">b'</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#E4:75_1_1_1"><i><font color="Green" title="E67">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">s</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">a</font>,<font color="Maroon" title="c8">b</font></span>)</span>, <font color="Maroon" title="c30">r</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">a</font>,<font color="Maroon" title="c9">c</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#E4:75_1_1_1"><i><font color="Green" title="E67">A12</font></i></a>, <a class="ref" href="afproj.html#T20" target="_self" title="AFPROJ:th.20">Th20</a></i>;<br/><a NAME="E12:75_1_1_1"/><i><font color="Green" title="E71">A16</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="c21">C2</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="c22">C3'</font> &amp; <font color="Maroon" title="c16">a2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c21">C2</font> &amp; <font color="Maroon" title="c17">b2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c21">C2</font> &amp; <font color="Maroon" title="c18">a3</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="c19">b3</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#E1:75_1"><i><font color="Green" title="E57">A2</font></i></a>, <a class="txt" href="afproj.html#E3:75_1"><i><font color="Green" title="E59">A4</font></i></a>, <a class="txt" href="afproj.html#E4:75_1"><i><font color="Green" title="E60">A5</font></i></a>, <a class="ref" href="afproj.html#T26" target="_self" title="AFPROJ:th.26">Th26</a></i>;<br/><a NAME="E13:75_1_1_1"/><i><font color="Green" title="E72">A17</font></i>: 
( <font color="Maroon" title="c16">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="c18">a3</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="c17">b2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c26">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="c26">B1</font> )
 <b>by </b><i><a class="txt" href="afproj.html#E4:75_1_1_1"><i><font color="Green" title="E67">A12</font></i></a>, <a class="txt" href="afproj.html#E5:75_1_1_1"><i><font color="Green" title="E68">A13</font></i></a>, <a class="ref" href="afproj.html#T26" target="_self" title="AFPROJ:th.26">Th26</a></i>;<br/><div><i><font color="Green" title="E73">A18</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:75_1_1_1_1"/>
<font color="Maroon" title="c21">C2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">C3'</font>
 ;<br/><b>then </b><font color="Maroon" title="c4">P</font> = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">C</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="c5">C</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="E3:75_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="afproj.html#E1:75_1_1"><i><font color="Green" title="E63">A8</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E15:75_1_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="c23">A1</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="c26">B1</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1_1_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:75_1_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="c23">A1</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="c26">B1</font> )
 ;<br/>

<a NAME="E2:75_1_1_1_2"/><b>then </b>
( <font color="Maroon" title="c18">a3</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c21">C2</font> or <font color="Maroon" title="c19">b3</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c21">C2</font> )
 
<b>by </b><i><a class="txt" href="afproj.html#E2:75_1"><i><font color="Green" title="E58">A3</font></i></a>, <a class="txt" href="afproj.html#E2:75_1_1_1"><i><font color="Green" title="E65">A10</font></i></a>, <a class="txt" href="afproj.html#E12:75_1_1_1"><i><font color="Green" title="E71">A16</font></i></a>, <a class="txt" href="afproj.html#E13:75_1_1_1"><i><font color="Green" title="E72">A17</font></i></a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></i>;<br/>
<b>hence </b><a NAME="E3:75_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="afproj.html#E2:75_1"><i><font color="Green" title="E58">A3</font></i></a>, <a class="txt" href="afproj.html#E2:75_1_1_1"><i><font color="Green" title="E65">A10</font></i></a>, <a class="txt" href="afproj.html#E12:75_1_1_1"><i><font color="Green" title="E71">A16</font></i></a>, <a class="txt" href="afproj.html#E14:75_1_1_1"><i><font color="Green" title="E73">A18</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">t</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="E17:75_1_1_1"/><i><font color="Green" title="E74">A19</font></i>: 
( <font color="Maroon" title="c31">t</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="c31">t</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c26">B1</font> )
 <b>by </b><i><a class="txt" href="afproj.html#E12:75_1_1_1"><i><font color="Green" title="E71">A16</font></i></a>, <a class="txt" href="afproj.html#E13:75_1_1_1"><i><font color="Green" title="E72">A17</font></i></a>, <a class="txt" href="afproj.html#E14:75_1_1_1"><i><font color="Green" title="E73">A18</font></i></a>, <a class="ref" href="incproj.html#D13" title="INCPROJ:def.13">INCPROJ:def 13</a></i>;<br/><a NAME="E18:75_1_1_1"/>
( <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">C1</font> &amp; <font color="Maroon" title="c15">b1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c20">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="c20">C1</font> )
 <b>by </b><i><a class="txt" href="afproj.html#E1:75_1"><i><font color="Green" title="E57">A2</font></i></a>, <a class="txt" href="afproj.html#E3:75_1"><i><font color="Green" title="E59">A4</font></i></a>, <a class="txt" href="afproj.html#E4:75_1"><i><font color="Green" title="E60">A5</font></i></a>, <a class="ref" href="afproj.html#T26" target="_self" title="AFPROJ:th.26">Th26</a></i>;<br/><a NAME="E19:75_1_1_1"/><b>then </b><i><font color="Green" title="E75">A20</font></i>: 
( <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">b1</font>,<font color="Maroon" title="c14">a1</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c20">C1</font> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c16">a2</font>,<font color="Maroon" title="c17">b2</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c21">C2</font> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c18">a3</font>,<font color="Maroon" title="c19">b3</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c22">C3'</font> )
 <b>by </b><i><a class="txt" href="afproj.html#E12:75_1_1_1"><i><font color="Green" title="E71">A16</font></i></a>, <a class="ref" href="incsp_1.html#T12" title="INCSP_1:th.12">INCSP_1:12</a></i>;<br/><a NAME="E20:75_1_1_1"/><i><font color="Green" title="E76">A21</font></i>: 
( <font color="Maroon" title="c16">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="c18">a3</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="c18">a3</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">r</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="c16">a2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c25">A3</font> &amp; <font color="Maroon" title="c29">s</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c25">A3</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="c25">A3</font> )
 <b>by </b><i><a class="txt" href="afproj.html#E4:75_1_1_1"><i><font color="Green" title="E67">A12</font></i></a>, <a class="txt" href="afproj.html#E5:75_1_1_1"><i><font color="Green" title="E68">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#T30" target="_self" title="AFPROJ:th.30">Th30</a></i>;<br/><a NAME="E21:75_1_1_1"/>
( <font color="Maroon" title="c17">b2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c26">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="c26">B1</font> &amp; <font color="Maroon" title="c15">b1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c27">B2</font> &amp; <font color="Maroon" title="c30">r</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c27">B2</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="c27">B2</font> &amp; <font color="Maroon" title="c15">b1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c28">B3</font> &amp; <font color="Maroon" title="c29">s</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c28">B3</font> &amp; <font color="Maroon" title="c17">b2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c28">B3</font> )
 <b>by </b><i><a class="txt" href="afproj.html#E4:75_1_1_1"><i><font color="Green" title="E67">A12</font></i></a>, <a class="txt" href="afproj.html#E5:75_1_1_1"><i><font color="Green" title="E68">A13</font></i></a>, <a class="txt" href="afproj.html#E7:75_1_1_1"><i><font color="Green" title="E70">A15</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></i>;<br/><a NAME="E22:75_1_1_1"/><b>then </b><i><font color="Green" title="E77">A22</font></i>: 
( <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c18">a3</font>,<font color="Maroon" title="c16">a2</font>,<font color="Maroon" title="c31">t</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c23">A1</font> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c18">a3</font>,<font color="Maroon" title="c30">r</font>,<font color="Maroon" title="c14">a1</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c24">A2</font> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c16">a2</font>,<font color="Maroon" title="c29">s</font>,<font color="Maroon" title="c14">a1</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c25">A3</font> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c31">t</font>,<font color="Maroon" title="c17">b2</font>,<font color="Maroon" title="c19">b3</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c26">B1</font> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c15">b1</font>,<font color="Maroon" title="c30">r</font>,<font color="Maroon" title="c19">b3</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c27">B2</font> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c15">b1</font>,<font color="Maroon" title="c29">s</font>,<font color="Maroon" title="c17">b2</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c28">B3</font> )
 <b>by </b><i><a class="txt" href="afproj.html#E17:75_1_1_1"><i><font color="Green" title="E74">A19</font></i></a>, <a class="txt" href="afproj.html#E20:75_1_1_1"><i><font color="Green" title="E76">A21</font></i></a>, <a class="ref" href="incsp_1.html#T12" title="INCSP_1:th.12">INCSP_1:12</a></i>;<br/><div><i><font color="Green" title="E78">A23</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1_1_1_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:75_1_1_1_3"/>
<font color="Maroon" title="c20">C1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">C2</font>
 ;<br/><b>then </b><font color="Maroon" title="c3">A</font> = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">P</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">P</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="E3:75_1_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="afproj.html#E5:75_1"><i><font color="Green" title="E61">A6</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1_1_1_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:75_1_1_1_4"/>
<font color="Maroon" title="c20">C1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">C3'</font>
 ;<br/><b>then </b><font color="Maroon" title="c3">A</font> = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">C</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="c5">C</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="E3:75_1_1_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="afproj.html#E5:75_1"><i><font color="Green" title="E61">A6</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E25:75_1_1_1"/><b>then </b>
<font color="Maroon" title="c20">C1</font>,<font color="Maroon" title="c21">C2</font>,<font color="Maroon" title="c22">C3'</font> <a href="zfmisc_1.html#R1" title="ZFMISC_1:pred.1">are_mutually_different</a> 
 <b>by </b><i><a class="txt" href="afproj.html#E14:75_1_1_1"><i><font color="Green" title="E73">A18</font></i></a>, <a class="txt" href="afproj.html#E23:75_1_1_1"><i><font color="Green" title="E78">A23</font></i></a>, <a class="ref" href="zfmisc_1.html#D5" title="ZFMISC_1:def.5">ZFMISC_1:def 5</a></i>;<br/><b>then consider </b><font color="Maroon" title="c32">O</font> being    <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> such that </b><br/><a NAME="E27:75_1_1_1"/><i><font color="Green" title="E79">A24</font></i>: 
<span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c30">r</font>,<font color="Maroon" title="c29">s</font>,<font color="Maroon" title="c31">t</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c32">O</font>
 <b>by </b><i><a class="txt" href="afproj.html#E1:75"><i><font color="Green" title="E56">A1</font></i></a>, <a class="txt" href="afproj.html#E2:75_1"><i><font color="Green" title="E58">A3</font></i></a>, <a class="txt" href="afproj.html#E1:75_1_1_1"><i><font color="Green" title="E64">A9</font></i></a>, <a class="txt" href="afproj.html#E2:75_1_1_1"><i><font color="Green" title="E65">A10</font></i></a>, <a class="txt" href="afproj.html#E19:75_1_1_1"><i><font color="Green" title="E75">A20</font></i></a>, <a class="txt" href="afproj.html#E22:75_1_1_1"><i><font color="Green" title="E77">A22</font></i></a>, <a class="ref" href="incproj.html#D19" title="INCPROJ:def.19">INCPROJ:def 19</a></i>;<br/><a NAME="E28:75_1_1_1"/><i><font color="Green" title="E80">A25</font></i>: 
( <font color="Maroon" title="c30">r</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c32">O</font> &amp; <font color="Maroon" title="c29">s</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c32">O</font> &amp; <font color="Maroon" title="c31">t</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c32">O</font> )
 <b>by </b><i><a class="txt" href="afproj.html#E27:75_1_1_1"><i><font color="Green" title="E79">A24</font></i></a>, <a class="ref" href="incsp_1.html#T12" title="INCSP_1:th.12">INCSP_1:12</a></i>;<br/><div><i><font color="Green" title="E81">A26</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1_1_1_5"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:75_1_1_1_5"/>
<font color="Maroon" title="c30">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">s</font>
 ;<br/><a NAME="E2:75_1_1_1_5"/><b>then </b><i><font color="Green" title="E82">A27</font></i>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">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">a</font>,<font color="Maroon" title="c9">c</font>
 <b>by </b><i><a class="txt" href="afproj.html#E4:75_1_1_1"><i><font color="Green" title="E67">A12</font></i></a>, <a class="ref" href="afproj.html#T11" target="_self" title="AFPROJ:th.11">Th11</a></i>;<br/><a NAME="E3:75_1_1_1_5"/><b>then </b>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">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="c10">a'</font>,<font color="Maroon" title="c12">c'</font>
 <b>by </b><i><a class="txt" href="afproj.html#E6:75_1_1_1"><i><font color="Green" title="E69">A14</font></i></a>, <a class="ref" href="aff_1.html#T58" title="AFF_1:th.58">AFF_1:58</a></i>;<br/><a NAME="E4:75_1_1_1_5"/><b>then </b>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c10">a'</font>,<font color="Maroon" title="c11">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="c10">a'</font>,<font color="Maroon" title="c12">c'</font>
 <b>by </b><i><a class="txt" href="afproj.html#E6:75_1_1_1"><i><font color="Green" title="E69">A14</font></i></a>, <a class="ref" href="aff_1.html#T58" title="AFF_1:th.58">AFF_1:58</a></i>;<br/><a NAME="E5:75_1_1_1_5"/><b>then </b>
( <font color="Maroon" title="c9">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">a</font>,<font color="Maroon" title="c8">b</font> &amp; <font color="Maroon" title="c12">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="c10">a'</font>,<font color="Maroon" title="c11">b'</font> )
 <b>by </b><i><a class="txt" href="afproj.html#E5:75_1_1_1"><i><font color="Green" title="E68">A13</font></i></a>, <a class="txt" href="afproj.html#E2:75_1_1_1_5"><i><font color="Green" title="E82">A27</font></i></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></i>;<br/><b>hence </b><a NAME="E6:75_1_1_1_5"/>
<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">b'</font>,<font color="Maroon" title="c12">c'</font>
 <b>by </b><i><a class="txt" href="afproj.html#E5:75_1_1_1"><i><font color="Green" title="E68">A13</font></i></a>, <a class="txt" href="afproj.html#E6:75_1_1_1"><i><font color="Green" title="E69">A14</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><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1_1_1_6"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:75_1_1_1_6"/><i><font color="Green" title="E82">A28</font></i>: 
<font color="Maroon" title="c30">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c29">s</font>
 ;<br/><a NAME="E2:75_1_1_1_6"/>
 ex <font color="Olive" title="b1">X</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font> st <br/>( <font color="Maroon" title="c32">O</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="Olive" title="b1">X</font></span>)</span>,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b1">X</font> <a href="aff_4.html#NR1" title="AFF_4:NR.1">is_plane</a>  )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1_1_1_6_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon" title="c33">x</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">a</font>,<font color="Maroon" title="c8">b</font></span>)</span>, <font color="Maroon" title="c34">y</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">a</font>,<font color="Maroon" title="c9">c</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#K14" title="AFPROJ:func.14">ProjHorizon</a> <font color="Maroon" title="c1">AS</font></span>)</span> <b>by </b><i><a class="txt" href="afproj.html#E4:75_1_1_1"><i><font color="Green" title="E67">A12</font></i></a>, <a class="ref" href="afproj.html#T14" target="_self" title="AFPROJ:th.14">Th14</a></i>;<br/>
<a NAME="E2:75_1_1_1_6_1"/>
( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c33">x</font>,<font color="Maroon" title="c32">O</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="incsp_1.html#U3" title="INCSP_1:sel.3">Inc</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> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c34">y</font>,<font color="Maroon" title="c32">O</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="incsp_1.html#U3" title="INCSP_1:sel.3">Inc</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#E28:75_1_1_1"><i><font color="Green" title="E80">A25</font></i></a>, <a class="ref" href="incsp_1.html#D1" title="INCSP_1:def.1">INCSP_1:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c35">Z'</font> being    <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#K14" title="AFPROJ:func.14">ProjHorizon</a> <font color="Maroon" title="c1">AS</font></span>)</span><b> such that </b><br/><a NAME="E4:75_1_1_1_6_1"/><i><font color="Green" title="E83">A29</font></i>: 
<font color="Maroon" title="c32">O</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c35">Z'</font>,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="afproj.html#E1:75_1_1_1_6"><i><font color="Green" title="E82">A28</font></i></a>, <a class="ref" href="afproj.html#T41" target="_self" title="AFPROJ:th.41">Th41</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c36">X</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="E6:75_1_1_1_6_1"/><i><font color="Green" title="E84">A30</font></i>: 
( <font color="Maroon" title="c35">Z'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afproj.html#K6" title="AFPROJ:func.6">PDir</a> <font color="Maroon" title="c36">X</font> &amp; <font color="Maroon" title="c36">X</font> <a href="aff_4.html#NR1" title="AFF_4:NR.1">is_plane</a>  )
 <b>by </b><i><a class="ref" href="afproj.html#T15" target="_self" title="AFPROJ:th.15">Th15</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c36">X</font>
;<br/>

<b>thus </b><a NAME="E7:75_1_1_1_6_1"/>
( <font color="Maroon" title="c32">O</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="c36">X</font></span>)</span>,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c36">X</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#E4:75_1_1_1_6_1"><i><font color="Green" title="E83">A29</font></i></a>, <a class="txt" href="afproj.html#E6:75_1_1_1_6_1"><i><font color="Green" title="E84">A30</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:75_1_1_1_6"/><b>then </b>
<font color="Maroon" title="c31">t</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#E28:75_1_1_1"><i><font color="Green" title="E80">A25</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="c33">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="E5:75_1_1_1_6"/><i><font color="Green" title="E83">A31</font></i>: 
( <font color="Maroon" title="c31">t</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="c33">Y</font> &amp; <font color="Maroon" title="c33">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="E6:75_1_1_1_6"/>
( <font color="Maroon" title="c33">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">b</font>,<font color="Maroon" title="c9">c</font> &amp; <font color="Maroon" title="c33">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="c11">b'</font>,<font color="Maroon" title="c12">c'</font> )
 <b>by </b><i><a class="txt" href="afproj.html#E4:75_1_1_1"><i><font color="Green" title="E67">A12</font></i></a>, <a class="txt" href="afproj.html#E17:75_1_1_1"><i><font color="Green" title="E74">A19</font></i></a>, <a class="txt" href="afproj.html#E5:75_1_1_1_6"><i><font color="Green" title="E83">A31</font></i></a>, <a class="ref" href="afproj.html#T28" target="_self" title="AFPROJ:th.28">Th28</a></i>;<br/><a NAME="E7:75_1_1_1_6"/><b>then </b>
( <font color="Maroon" title="c33">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">b</font>,<font color="Maroon" title="c9">c</font> &amp; <font color="Maroon" title="c33">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="c11">b'</font>,<font color="Maroon" title="c12">c'</font> )
 <b>by </b><i><a class="txt" href="afproj.html#E4:75_1_1_1"><i><font color="Green" title="E67">A12</font></i></a>, <a class="txt" href="afproj.html#E5:75_1_1_1_6"><i><font color="Green" title="E83">A31</font></i></a>, <a class="ref" href="aff_4.html#T40" title="AFF_4:th.40">AFF_4:40</a></i>;<br/><a NAME="E8:75_1_1_1_6"/><b>then </b>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">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="c11">b'</font>,<font color="Maroon" title="c12">c'</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="E9:75_1_1_1_6"/>
<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">b'</font>,<font color="Maroon" title="c12">c'</font>
 <b>by </b><i><a class="txt" href="afproj.html#E5:75_1_1_1"><i><font color="Green" title="E68">A13</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="E31:75_1_1_1"/>
<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">b'</font>,<font color="Maroon" title="c12">c'</font>
 <b>by </b><i><a class="txt" href="afproj.html#E29:75_1_1_1"><i><font color="Green" title="E81">A26</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:75_1_1"/>
<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">b'</font>,<font color="Maroon" title="c12">c'</font>
 <b>by </b><i><a class="txt" href="afproj.html#E1:75_1"><i><font color="Green" title="E57">A2</font></i></a>, <a class="txt" href="afproj.html#E2:75_1"><i><font color="Green" title="E58">A3</font></i></a>, <a class="txt" href="afproj.html#E3:75_1"><i><font color="Green" title="E59">A4</font></i></a>, <a class="txt" href="afproj.html#E4:75_1"><i><font color="Green" title="E60">A5</font></i></a>, <a class="txt" href="afproj.html#E5:75_1"><i><font color="Green" title="E61">A6</font></i></a>, <a class="txt" href="afproj.html#E6:75_1"><i><font color="Green" title="E62">A7</font></i></a>, <a class="ref" href="afproj.html#T50" target="_self" title="AFPROJ:th.50">Th50</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:75_1"/>
<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">b'</font>,<font color="Maroon" title="c12">c'</font>
 <b>by </b><i><a class="txt" href="afproj.html#E3:75_1"><i><font color="Green" title="E59">A4</font></i></a>, <a class="txt" href="afproj.html#E4:75_1"><i><font color="Green" title="E60">A5</font></i></a>, <a class="ref" href="aff_1.html#T65" title="AFF_1:th.65">AFF_1:65</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:75"/>
<font color="Maroon" title="c1">AS</font> is <a href="aff_2.html#V4" title="AFF_2:attr.4">Desarguesian</a>
 <b>by </b><i><a class="ref" href="aff_2.html#D4" title="AFF_2:def.4">AFF_2:def 4</a></i>;<br/>


</div>
