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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="diraf.html#NM1">AffinSpace</a>;<br/>

<b>assume </b><a NAME="E1:76"/><i><font color="Green">E60</font></i>: 
 <a href="afproj.html#K13" target="_self">IncProjSp_of</a> <font color="Maroon">c<sub>1</sub></font> is <a href="incproj.html#V9">Fanoian</a>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>2</sub></font> =  <a href="afproj.html#K13" target="_self">IncProjSp_of</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<a NAME="E2:76"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="analoaf.html#R2">//</a> <font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font> &amp; <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>3</sub></font> <a href="analoaf.html#R2">//</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>4</sub></font> &amp; <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>4</sub></font> <a href="analoaf.html#R2">//</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="analoaf.html#R2">//</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>3</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:76_1"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> )
 <b>and </b><br/><a NAME="E2:76_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<b>assume </b><a NAME="E3:76_1"/><i><font color="Green">E63</font></i>: 
not <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>4</sub></font> as    <a href="subset_1.html#M1">Element</a> of the <a href="incsp_1.html#U1">Points</a> of <span class="p1">(<span class="default"><a href="afproj.html#K13" target="_self">IncProjSp_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="ref" href="afproj.html#T20" target="_self">Th20</a></i>;<br/>
<a NAME="E5:76_1"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="txt" href="afproj.html#E3:76_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="aff_1.html#T11">AFF_1:11</a>, <a class="ref" href="aff_1.html#T12">AFF_1:12</a></i>;<br/>
<a NAME="E6:76_1"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="ref" href="aff_1.html#T26">AFF_1:26</a></i>;<br/>
<a NAME="E7:76_1"/><i><font color="Green">E66</font></i>: 
(  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> &amp;  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_1"><b>proof </b></a><div class="add">

<div><i><font color="Green">E67</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:76_1_1_1"/>
 <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>
 ;<br/><a NAME="E2:76_1_1_1"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="afproj.html#E6:76_1"><i><font color="Green">E65</font></i></a>, <a class="ref" href="aff_1.html#D2">AFF_1:def 2</a></i>;<br/><b>hence </b><a NAME="E3:76_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="afproj.html#E3:76_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:76_1_1_2"/>
 <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>
 ;<br/><a NAME="E2:76_1_1_2"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="afproj.html#E6:76_1"><i><font color="Green">E65</font></i></a>, <a class="ref" href="aff_1.html#D2">AFF_1:def 2</a></i>;<br/><a NAME="E3:76_1_1_2"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/><b>hence </b><a NAME="E4:76_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="afproj.html#E3:76_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:76_1_1"/>
(  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> &amp;  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="afproj.html#E1:76_1_1"><i><font color="Green">E67</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:76_1"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_2"><b>proof </b></a><div class="add">

<div><i><font color="Green">E68</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:76_1_2_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><a NAME="E2:76_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="afproj.html#E1:76_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/><a NAME="E3:76_1_2_1"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/><a NAME="E4:76_1_2_1"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/><b>hence </b><a NAME="E5:76_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="afproj.html#E3:76_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:76_1_2_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><a NAME="E2:76_1_2_2"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="afproj.html#E1:76_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/><a NAME="E3:76_1_2_2"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/><a NAME="E4:76_1_2_2"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/><b>hence </b><a NAME="E5:76_1_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="afproj.html#E3:76_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:76_1_2"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="afproj.html#E1:76_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="afproj.html#E3:76_1"><i><font color="Green">E63</font></i></a>, <a class="txt" href="afproj.html#E1:76_1_2"><i><font color="Green">E68</font></i></a>, <a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:76_1"/><b>then </b><i><font color="Green">E68</font></i>: 
(  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp;  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp;  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp;  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp;  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp;  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a>  )
 
<b>by </b><i><a class="txt" href="afproj.html#E5:76_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="aff_1.html#D3">AFF_1:def 3</a></i>;<br/>
<a NAME="E10:76_1"/><i><font color="Green">E69</font></i>: 
(  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="aff_1.html#R4">//</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> &amp;  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="aff_1.html#R4">//</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> &amp;  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="aff_1.html#R4">//</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="txt" href="afproj.html#E1:76_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="afproj.html#E2:76_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="afproj.html#E5:76_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="afproj.html#E8:76_1"><i><font color="Green">E67</font></i></a>, <a class="ref" href="aff_1.html#T51">AFF_1:51</a></i>;<br/>
<a NAME="E11:76_1"/><b>then </b><i><font color="Green">E70</font></i>: 
(  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="aff_4.html#R1">'||'</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> &amp;  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="aff_4.html#R1">'||'</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> &amp;  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="aff_4.html#R1">'||'</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="txt" href="afproj.html#E9:76_1"><i><font color="Green">E68</font></i></a>, <a class="ref" href="aff_4.html#T40">AFF_4:40</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>,1</span><a href="tarski.html#K4">]</a></span>, <font color="Maroon">c<sub>12</sub></font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>,1</span><a href="tarski.html#K4">]</a></span>, <font color="Maroon">c<sub>13</sub></font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>,1</span><a href="tarski.html#K4">]</a></span>, <font color="Maroon">c<sub>14</sub></font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span>,1</span><a href="tarski.html#K4">]</a></span>, <font color="Maroon">c<sub>15</sub></font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>,1</span><a href="tarski.html#K4">]</a></span>, <font color="Maroon">c<sub>16</sub></font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span>,1</span><a href="tarski.html#K4">]</a></span> as    <a href="subset_1.html#M1">Element</a> of the <a href="incsp_1.html#U2">Lines</a> of <span class="p1">(<span class="default"><a href="afproj.html#K13" target="_self">IncProjSp_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="txt" href="afproj.html#E9:76_1"><i><font color="Green">E68</font></i></a>, <a class="ref" href="afproj.html#T23" target="_self">Th23</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>17</sub></font> =  <a href="afproj.html#K5" target="_self">LDir</a> <span class="p1">(<span class="default"><a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>, <font color="Maroon">c<sub>18</sub></font> =  <a href="afproj.html#K5" target="_self">LDir</a> <span class="p1">(<span class="default"><a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span>, <font color="Maroon">c<sub>19</sub></font> =  <a href="afproj.html#K5" target="_self">LDir</a> <span class="p1">(<span class="default"><a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> as    <a href="subset_1.html#M1">Element</a> of the <a href="incsp_1.html#U1">Points</a> of <span class="p1">(<span class="default"><a href="afproj.html#K13" target="_self">IncProjSp_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="txt" href="afproj.html#E9:76_1"><i><font color="Green">E68</font></i></a>, <a class="ref" href="afproj.html#T20" target="_self">Th20</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>20</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E15:76_1"/><i><font color="Green">E71</font></i>: 
(  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>20</sub></font> &amp;  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>20</sub></font> )
 <b>and </b><br/><a NAME="E16:76_1"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="aff_4.html#NR1" target="_self">is_plane</a> 
 <b>by </b><i><a class="txt" href="afproj.html#E6:76_1"><i><font color="Green">E65</font></i></a>, <a class="txt" href="afproj.html#E9:76_1"><i><font color="Green">E68</font></i></a>, <a class="ref" href="aff_4.html#T38">AFF_4:38</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>21</sub></font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="afproj.html#K6" target="_self">PDir</a> <font color="Maroon">c<sub>20</sub></font></span>)</span>,2</span><a href="tarski.html#K4">]</a></span> as    <a href="subset_1.html#M1">Element</a> of the <a href="incsp_1.html#U2">Lines</a> of <span class="p1">(<span class="default"><a href="afproj.html#K13" target="_self">IncProjSp_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="txt" href="afproj.html#E16:76_1"><i><font color="Green">E72</font></i></a>, <a class="ref" href="afproj.html#T23" target="_self">Th23</a></i>;<br/>
<a NAME="E18:76_1"/><i><font color="Green">E73</font></i>: 
(  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>20</sub></font> &amp;  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>20</sub></font> &amp;  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>20</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_3"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:76_1_3"/>
(  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>20</sub></font> &amp;  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>20</sub></font> )
 <b>by </b><i><a class="txt" href="afproj.html#E15:76_1"><i><font color="Green">E71</font></i></a></i>;<br/>

<a NAME="E2:76_1_3"/>
 <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="aff_4.html#K2">*</a> <span class="p1">(<span class="default"><a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="afproj.html#E6:76_1"><i><font color="Green">E65</font></i></a>, <a class="txt" href="afproj.html#E9:76_1"><i><font color="Green">E68</font></i></a>, <a class="txt" href="afproj.html#E10:76_1"><i><font color="Green">E69</font></i></a>, <a class="ref" href="aff_4.html#D3">AFF_4:def 3</a></i>;<br/>
<a NAME="E3:76_1_3"/><b>then </b>
 <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>20</sub></font>
 
<b>by </b><i><a class="txt" href="afproj.html#E6:76_1"><i><font color="Green">E65</font></i></a>, <a class="txt" href="afproj.html#E9:76_1"><i><font color="Green">E68</font></i></a>, <a class="txt" href="afproj.html#E15:76_1"><i><font color="Green">E71</font></i></a>, <a class="txt" href="afproj.html#E16:76_1"><i><font color="Green">E72</font></i></a>, <a class="ref" href="aff_4.html#T28">AFF_4:28</a></i>;<br/>
<b>hence </b><a NAME="E4:76_1_3"/>
 <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="txt" href="afproj.html#E6:76_1"><i><font color="Green">E65</font></i></a>, <a class="txt" href="afproj.html#E8:76_1"><i><font color="Green">E67</font></i></a>, <a class="txt" href="afproj.html#E15:76_1"><i><font color="Green">E71</font></i></a>, <a class="txt" href="afproj.html#E16:76_1"><i><font color="Green">E72</font></i></a>, <a class="ref" href="aff_4.html#T19">AFF_4:19</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E19:76_1"/><i><font color="Green">E74</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>13</sub></font> )
 
<b>by </b><i><a class="txt" href="afproj.html#E6:76_1"><i><font color="Green">E65</font></i></a>, <a class="txt" href="afproj.html#E9:76_1"><i><font color="Green">E68</font></i></a>, <a class="txt" href="afproj.html#E11:76_1"><i><font color="Green">E70</font></i></a>, <a class="ref" href="afproj.html#T26" target="_self">Th26</a>, <a class="ref" href="afproj.html#T28" target="_self">Th28</a>, <a class="ref" href="afproj.html#T30" target="_self">Th30</a></i>;<br/>
<a NAME="E20:76_1"/><i><font color="Green">E75</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>16</sub></font> )
 
<b>by </b><i><a class="txt" href="afproj.html#E6:76_1"><i><font color="Green">E65</font></i></a>, <a class="txt" href="afproj.html#E9:76_1"><i><font color="Green">E68</font></i></a>, <a class="txt" href="afproj.html#E11:76_1"><i><font color="Green">E70</font></i></a>, <a class="ref" href="afproj.html#T26" target="_self">Th26</a>, <a class="ref" href="afproj.html#T28" target="_self">Th28</a>, <a class="ref" href="afproj.html#T30" target="_self">Th30</a></i>;<br/>
<a NAME="E21:76_1"/><i><font color="Green">E76</font></i>: 
( <font color="Maroon">c<sub>17</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>21</sub></font> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>21</sub></font> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>21</sub></font> )
 
<b>by </b><i><a class="txt" href="afproj.html#E9:76_1"><i><font color="Green">E68</font></i></a>, <a class="txt" href="afproj.html#E16:76_1"><i><font color="Green">E72</font></i></a>, <a class="txt" href="afproj.html#E18:76_1"><i><font color="Green">E73</font></i></a>, <a class="ref" href="afproj.html#T31" target="_self">Th31</a></i>;<br/>
<a NAME="E22:76_1"/><i><font color="Green">E77</font></i>: 
( not <font color="Maroon">c<sub>8</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>11</sub></font> &amp; not <font color="Maroon">c<sub>9</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>11</sub></font> &amp; not <font color="Maroon">c<sub>7</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>12</sub></font> &amp; not <font color="Maroon">c<sub>10</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>12</sub></font> &amp; not <font color="Maroon">c<sub>7</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>13</sub></font> &amp; not <font color="Maroon">c<sub>9</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>13</sub></font> &amp; not <font color="Maroon">c<sub>8</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>14</sub></font> &amp; not <font color="Maroon">c<sub>10</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>14</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_4"><b>proof </b></a><div class="add">

<div><i><font color="Green">E78</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:76_1_4_1"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>11</sub></font> or <font color="Maroon">c<sub>9</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>11</sub></font> )
 ;<br/><a NAME="E2:76_1_4_1"/><b>then </b>
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> or <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="ref" href="afproj.html#T26" target="_self">Th26</a></i>;<br/><b>hence </b><a NAME="E3:76_1_4_1"/>
contradiction
 <b>by </b><i><a class="txt" href="afproj.html#E6:76_1"><i><font color="Green">E65</font></i></a>, <a class="txt" href="afproj.html#E7:76_1"><i><font color="Green">E66</font></i></a>, <a class="txt" href="afproj.html#E10:76_1"><i><font color="Green">E69</font></i></a>, <a class="ref" href="aff_1.html#T59">AFF_1:59</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E79</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_4_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:76_1_4_2"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>14</sub></font> or <font color="Maroon">c<sub>10</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>14</sub></font> )
 ;<br/><a NAME="E2:76_1_4_2"/><b>then </b>
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="ref" href="afproj.html#T26" target="_self">Th26</a></i>;<br/><b>hence </b><a NAME="E3:76_1_4_2"/>
contradiction
 <b>by </b><i><a class="txt" href="afproj.html#E6:76_1"><i><font color="Green">E65</font></i></a>, <a class="txt" href="afproj.html#E7:76_1"><i><font color="Green">E66</font></i></a>, <a class="txt" href="afproj.html#E10:76_1"><i><font color="Green">E69</font></i></a>, <a class="ref" href="aff_1.html#T59">AFF_1:59</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E80</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_4_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:76_1_4_3"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>12</sub></font> or <font color="Maroon">c<sub>10</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>12</sub></font> )
 ;<br/><a NAME="E2:76_1_4_3"/><b>then </b>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="ref" href="afproj.html#T26" target="_self">Th26</a></i>;<br/><b>hence </b><a NAME="E3:76_1_4_3"/>
contradiction
 <b>by </b><i><a class="txt" href="afproj.html#E6:76_1"><i><font color="Green">E65</font></i></a>, <a class="txt" href="afproj.html#E7:76_1"><i><font color="Green">E66</font></i></a>, <a class="txt" href="afproj.html#E10:76_1"><i><font color="Green">E69</font></i></a>, <a class="ref" href="aff_1.html#T59">AFF_1:59</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_4_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:76_1_4_4"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>13</sub></font> or <font color="Maroon">c<sub>9</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>13</sub></font> )
 ;<br/><a NAME="E2:76_1_4_4"/><b>then </b>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> or <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="ref" href="afproj.html#T26" target="_self">Th26</a></i>;<br/><b>hence </b><a NAME="E3:76_1_4_4"/>
contradiction
 <b>by </b><i><a class="txt" href="afproj.html#E6:76_1"><i><font color="Green">E65</font></i></a>, <a class="txt" href="afproj.html#E7:76_1"><i><font color="Green">E66</font></i></a>, <a class="txt" href="afproj.html#E10:76_1"><i><font color="Green">E69</font></i></a>, <a class="ref" href="aff_1.html#T59">AFF_1:59</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:76_1_4"/>
( not <font color="Maroon">c<sub>8</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>11</sub></font> &amp; not <font color="Maroon">c<sub>9</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>11</sub></font> &amp; not <font color="Maroon">c<sub>7</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>12</sub></font> &amp; not <font color="Maroon">c<sub>10</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>12</sub></font> &amp; not <font color="Maroon">c<sub>7</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>13</sub></font> &amp; not <font color="Maroon">c<sub>9</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>13</sub></font> &amp; not <font color="Maroon">c<sub>8</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>14</sub></font> &amp; not <font color="Maroon">c<sub>10</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>14</sub></font> )
 <b>by </b><i><a class="txt" href="afproj.html#E1:76_1_4"><i><font color="Green">E78</font></i></a>, <a class="txt" href="afproj.html#E2:76_1_4"><i><font color="Green">E79</font></i></a>, <a class="txt" href="afproj.html#E3:76_1_4"><i><font color="Green">E80</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E23:76_1"/><i><font color="Green">E78</font></i>: 
( <font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="incproj.html#R4">on</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="incproj.html#R4">on</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="incproj.html#R4">on</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="incproj.html#R4">on</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="incproj.html#R4">on</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="incproj.html#R4">on</a> <font color="Maroon">c<sub>16</sub></font> )
 
<b>by </b><i><a class="txt" href="afproj.html#E19:76_1"><i><font color="Green">E74</font></i></a>, <a class="txt" href="afproj.html#E20:76_1"><i><font color="Green">E75</font></i></a>, <a class="ref" href="incproj.html#D8">INCPROJ:def 8</a></i>;<br/>
<a NAME="E24:76_1"/>
<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>18</sub></font> <a href="incproj.html#R3">on</a> <font color="Maroon">c<sub>21</sub></font>
 
<b>by </b><i><a class="txt" href="afproj.html#E21:76_1"><i><font color="Green">E76</font></i></a>, <a class="ref" href="incproj.html#D7">INCPROJ:def 7</a></i>;<br/>
<b>hence </b><a NAME="E25:76_1"/>
contradiction
 <b>by </b><i><a class="txt" href="afproj.html#E1:76"><i><font color="Green">E60</font></i></a>, <a class="txt" href="afproj.html#E21:76_1"><i><font color="Green">E76</font></i></a>, <a class="txt" href="afproj.html#E22:76_1"><i><font color="Green">E77</font></i></a>, <a class="txt" href="afproj.html#E23:76_1"><i><font color="Green">E78</font></i></a>, <a class="ref" href="incproj.html#D18">INCPROJ:def 18</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:76"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="translac.html#V1">Fanoian</a>
 <b>by </b><i><a class="ref" href="papdesaf.html#D5">PAPDESAF:def 5</a></i>;<br/>


</div>
