<?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="E1:69"/><i><font color="Green">E58</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#V6">2-dimensional</a>
 ;<br/>

<b>assume </b><a NAME="E2:69"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is  not   <a href="diraf.html#NM2">AffinPlane</a>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>consider </b><font color="Maroon">c<sub>4</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="E5:69"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>and </b><br/><a NAME="E6:69"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="aff_4.html#NR1" target="_self">is_plane</a> 
 <b>by </b><i><a class="ref" href="aff_4.html#T37">AFF_4:37</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E8:69"/><i><font color="Green">E62</font></i>: 
not <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="afproj.html#E2:69"><i><font color="Green">E59</font></i></a>, <a class="txt" href="afproj.html#E6:69"><i><font color="Green">E61</font></i></a>, <a class="ref" href="aff_4.html#T48">AFF_4:48</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> =  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>;<br/>
<a NAME="E9:69"/><i><font color="Green">E63</font></i>: 
(  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>2</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>2</sub></font>,<font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="txt" href="afproj.html#E5:69"><i><font color="Green">E60</font></i></a>, <a class="txt" href="afproj.html#E8:69"><i><font color="Green">E62</font></i></a>, <a class="ref" href="aff_1.html#T26">AFF_1:26</a>, <a class="ref" href="aff_1.html#D3">AFF_1:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>7</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>4</sub></font></span>)</span>,2</span><a href="tarski.html#K4">]</a></span>, <font color="Maroon">c<sub>8</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>2</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span>,1</span><a href="tarski.html#K4">]</a></span> as   <a href="incsp_1.html#NM2">LINE</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#E6:69"><i><font color="Green">E61</font></i></a>, <a class="ref" href="afproj.html#T23" target="_self">Th23</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="incsp_1.html#NM1">POINT</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> such that </b><br/><a NAME="E12:69"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="txt" href="afproj.html#E1:69"><i><font color="Green">E58</font></i></a>, <a class="ref" href="incproj.html#D14">INCPROJ:def 14</a></i>;<br/>
<a NAME="E13:69"/>
<font color="Maroon">c<sub>9</sub></font> is  not   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="afproj.html#E6:69"><i><font color="Green">E61</font></i></a>, <a class="txt" href="afproj.html#E12:69"><i><font color="Green">E64</font></i></a>, <a class="ref" href="afproj.html#T27" target="_self">Th27</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>10</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="E15:69"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="afproj.html#K5" target="_self">LDir</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a>  )
 <b>by </b><i><a class="ref" href="afproj.html#T20" target="_self">Th20</a></i>;<br/>
<a NAME="E16:69"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="aff_4.html#R1">'||'</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="aff_4.html#R1">'||'</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="txt" href="afproj.html#E6:69"><i><font color="Green">E61</font></i></a>, <a class="txt" href="afproj.html#E9:69"><i><font color="Green">E63</font></i></a>, <a class="txt" href="afproj.html#E12:69"><i><font color="Green">E64</font></i></a>, <a class="txt" href="afproj.html#E15:69"><i><font color="Green">E65</font></i></a>, <a class="ref" href="afproj.html#T28" target="_self">Th28</a>, <a class="ref" href="afproj.html#T29" target="_self">Th29</a></i>;<br/>
<a NAME="E17:69"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="aff_1.html#R4">//</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="afproj.html#E9:69"><i><font color="Green">E63</font></i></a>, <a class="txt" href="afproj.html#E15:69"><i><font color="Green">E65</font></i></a>, <a class="ref" href="aff_4.html#T40">AFF_4:40</a></i>;<br/>
<a NAME="E18:69"/><b>then </b>
 <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="aff_4.html#R1">'||'</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="afproj.html#E16:69"><i><font color="Green">E66</font></i></a>, <a class="ref" href="aff_4.html#T56">AFF_4:56</a></i>;<br/>
<a NAME="E19:69"/><b>then </b>
 <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="afproj.html#E5:69"><i><font color="Green">E60</font></i></a>, <a class="txt" href="afproj.html#E6:69"><i><font color="Green">E61</font></i></a>, <a class="txt" href="afproj.html#E9:69"><i><font color="Green">E63</font></i></a>, <a class="ref" href="aff_4.html#T43">AFF_4:43</a></i>;<br/>
<b>hence </b><a NAME="E20:69"/>
contradiction
 <b>by </b><i><a class="txt" href="afproj.html#E8:69"><i><font color="Green">E62</font></i></a>, <a class="txt" href="afproj.html#E9:69"><i><font color="Green">E63</font></i></a></i>;<br/>


</div>
