<?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>let </b><font color="Maroon">c<sub>2</sub></font>, <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><a NAME="E1:71"/><i><font color="Green">E58</font></i>: 
( not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>2</sub></font>,<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#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>2</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>6</sub></font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:71_1"/><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>7</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="E3:71_1"/><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>7</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>and </b><br/><a NAME="E4:71_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>7</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" target="_self">Th37</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:71_1_1"/><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>7</sub></font>
 ;<br/><a NAME="E2:71_1_1"/><i><font color="Green">E63</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="aff_4.html#E1:71"><i><font color="Green">E58</font></i></a>, <a class="ref" href="aff_1.html#T16">AFF_1:16</a></i>;<br/><b>set </b><font color="Maroon">c<sub>8</sub></font> =  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>9</sub></font> =  <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="E3:71_1_1"/><i><font color="Green">E64</font></i>: 
( <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>3</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>2</sub></font>,<font color="Maroon">c<sub>3</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;  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</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>2</sub></font>,<font color="Maroon">c<sub>3</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> )
 <b>by </b><i><a class="txt" href="aff_4.html#E1:71"><i><font color="Green">E58</font></i></a>, <a class="txt" href="aff_4.html#E2:71_1_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="aff_1.html#T26">AFF_1:26</a>, <a class="ref" href="aff_1.html#T51">AFF_1:51</a>, <a class="ref" href="aff_1.html#D3">AFF_1:def 3</a></i>;<br/><a NAME="E4:71_1_1"/><i><font color="Green">E65</font></i>: 
 <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E3:71_1"><i><font color="Green">E60</font></i></a>, <a class="txt" href="aff_4.html#E4:71_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="aff_4.html#E2:71_1_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="aff_4.html#T19" target="_self">Th19</a></i>;<br/><a NAME="E5:71_1_1"/><i><font color="Green">E66</font></i>: 
not <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:71_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/>

<a NAME="E2:71_1_1_1"/><b>then </b>
 <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="tarski.html#R1">c=</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="aff_4.html#E4:71_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="aff_4.html#E3:71_1_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="aff_4.html#E4:71_1_1"><i><font color="Green">E65</font></i></a>, <a class="ref" href="aff_4.html#T23" target="_self">Th23</a></i>;<br/>
<b>hence </b><a NAME="E3:71_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_4.html#E1:71_1_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="aff_4.html#E3:71_1_1"><i><font color="Green">E64</font></i></a></i>;<br/>


</div><b>end;</b></div><b>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="E7:71_1_1"/><i><font color="Green">E67</font></i>: 
(  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>10</sub></font> &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="tarski.html#R1">c=</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>and </b><br/><a NAME="E8:71_1_1"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="aff_4.html#NR1" target="_self">is_plane</a> 
 <b>by </b><i><a class="txt" href="aff_4.html#E3:71_1_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="aff_4.html#T39" target="_self">Th39</a></i>;<br/><b>set </b><font color="Maroon">c<sub>11</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/><b>set </b><font color="Maroon">c<sub>12</sub></font> =  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font>;<br/><a NAME="E9:71_1_1"/><i><font color="Green">E69</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="txt" href="aff_4.html#E3:71_1_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="aff_4.html#E7:71_1_1"><i><font color="Green">E67</font></i></a></i>;<br/><a NAME="E10:71_1_1"/><i><font color="Green">E70</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;  <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; <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> &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> )
 <b>by </b><i><a class="txt" href="aff_4.html#E3:71_1"><i><font color="Green">E60</font></i></a>, <a class="txt" href="aff_4.html#E1:71_1_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="aff_4.html#E5:71_1_1"><i><font color="Green">E66</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/><a NAME="E11:71_1_1"/><i><font color="Green">E71</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="tarski.html#R1">c=</a> <font color="Maroon">c<sub>10</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>10</sub></font> )
 <b>by </b><i><a class="txt" href="aff_4.html#E3:71_1"><i><font color="Green">E60</font></i></a>, <a class="txt" href="aff_4.html#E1:71_1_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="aff_4.html#E3:71_1_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="aff_4.html#E5:71_1_1"><i><font color="Green">E66</font></i></a>, <a class="txt" href="aff_4.html#E7:71_1_1"><i><font color="Green">E67</font></i></a>, <a class="txt" href="aff_4.html#E8:71_1_1"><i><font color="Green">E68</font></i></a>, <a class="ref" href="aff_4.html#T19" target="_self">Th19</a></i>;<br/><a NAME="E12:71_1_1"/><i><font color="Green">E72</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="hidden.html#NR2" target="_self">&lt;&gt;</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>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:71_1_1_2"/>
 <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="hidden.html#R1">=</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>
 ;<br/>

<a NAME="E2:71_1_1_2"/><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="hidden.html#R1">=</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="aff_4.html#E2:71_1_1"><i><font color="Green">E63</font></i></a>, <a class="txt" href="aff_4.html#E3:71_1_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="aff_4.html#E10:71_1_1"><i><font color="Green">E70</font></i></a>, <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/>
<b>hence </b><a NAME="E3:71_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_4.html#E1:71_1_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="aff_4.html#E4:71_1_1"><i><font color="Green">E65</font></i></a>, <a class="txt" href="aff_4.html#E10:71_1_1"><i><font color="Green">E70</font></i></a></i>;<br/>


</div><b>end;</b></div><div><i><font color="Green">E73</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1_1_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:71_1_1_3"/><i><font color="Green">E74</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#R4">//</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>
 ;<br/><b>set </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="aff_4.html#K2" target="_self">*</a> <span class="p1">(<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>;<br/><a NAME="E2:71_1_1_3"/><i><font color="Green">E75</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="aff_4.html#K2" target="_self">*</a> <span class="p1">(<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> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="aff_4.html#K2" target="_self">*</a> <span class="p1">(<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> &amp;  <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#R4">//</a> <font color="Maroon">c<sub>4</sub></font> <a href="aff_4.html#K2" target="_self">*</a> <span class="p1">(<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> )
 <b>by </b><i><a class="txt" href="aff_4.html#E10:71_1_1"><i><font color="Green">E70</font></i></a>, <a class="ref" href="aff_4.html#D3" target="_self">Def3</a>, <a class="ref" href="aff_4.html#T27" target="_self">Th27</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>14</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="E4:71_1_1_3"/><i><font color="Green">E76</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="aff_4.html#K2" target="_self">*</a> <span class="p1">(<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> &amp; <font color="Maroon">c<sub>2</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>14</sub></font> )
 <b>by </b><i><a class="txt" href="aff_4.html#E10:71_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="aff_4.html#E9"><i><font color="Green">Lemma7</font></i></a></i>;<br/><a NAME="E5:71_1_1_3"/>
 <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="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font> <a href="aff_4.html#K2" target="_self">*</a> <span class="p1">(<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>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1_1_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:71_1_1_3_1"/>
 <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="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="aff_4.html#K2" target="_self">*</a> <span class="p1">(<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>
 ;<br/>

<a NAME="E2:71_1_1_3_1"/><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="hidden.html#R1">=</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="aff_4.html#E2:71_1_1"><i><font color="Green">E63</font></i></a>, <a class="txt" href="aff_4.html#E10:71_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="aff_4.html#E2:71_1_1_3"><i><font color="Green">E75</font></i></a>, <a class="ref" href="aff_1.html#T71">AFF_1:71</a></i>;<br/>
<a NAME="E3:71_1_1_3_1"/><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>7</sub></font>
 
<b>by </b><i><a class="txt" href="aff_4.html#E3:71_1"><i><font color="Green">E60</font></i></a>, <a class="txt" href="aff_4.html#E4:71_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="aff_4.html#E2:71_1_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="aff_4.html#T19" target="_self">Th19</a></i>;<br/>
<b>hence </b><a NAME="E4:71_1_1_3_1"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_4.html#E1:71_1_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="aff_4.html#E10:71_1_1"><i><font color="Green">E70</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E6:71_1_1_3"/><b>then </b>
<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>6</sub></font>,<font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E1:71"><i><font color="Green">E58</font></i></a>, <a class="txt" href="aff_4.html#E1:71_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="aff_4.html#E10:71_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="aff_4.html#E12:71_1_1"><i><font color="Green">E72</font></i></a>, <a class="txt" href="aff_4.html#E1:71_1_1_3"><i><font color="Green">E74</font></i></a>, <a class="txt" href="aff_4.html#E2:71_1_1_3"><i><font color="Green">E75</font></i></a>, <a class="txt" href="aff_4.html#E4:71_1_1_3"><i><font color="Green">E76</font></i></a>, <a class="ref" href="aff_4.html#T51" target="_self">Th51</a></i>;<br/><b>hence </b><a NAME="E7:71_1_1_3"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">c<sub>2</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="Olive">b<sub>1</sub></font> &amp; <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>6</sub></font>,<font color="Olive">b<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="aff_4.html#E4:71_1_1_3"><i><font color="Green">E76</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1_1_4"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">c<sub>13</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><a NAME="E1:71_1_1_4"/><i><font color="Green">E74</font></i>: 
( <font color="Maroon">c<sub>13</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>13</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> )
 ;<br/><a NAME="E2:71_1_1_4"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>2</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1_1_4_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:71_1_1_4_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E2:71_1_1_4_1"/><b>then </b>
 <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</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>
 
<b>by </b><i><a class="txt" href="aff_4.html#E2:71_1_1"><i><font color="Green">E63</font></i></a>, <a class="txt" href="aff_4.html#E3:71_1_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="aff_4.html#E10:71_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="aff_4.html#E1:71_1_1_4"><i><font color="Green">E74</font></i></a>, <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/>
<b>hence </b><a NAME="E3:71_1_1_4_1"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_4.html#E4:71_1_1"><i><font color="Green">E65</font></i></a>, <a class="txt" href="aff_4.html#E5:71_1_1"><i><font color="Green">E66</font></i></a>, <a class="txt" href="aff_4.html#E10:71_1_1"><i><font color="Green">E70</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:71_1_1_4"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1_1_4_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:71_1_1_4_2"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E2:71_1_1_4_2"/><b>then </b>
 <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.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="aff_4.html#E2:71_1_1"><i><font color="Green">E63</font></i></a>, <a class="txt" href="aff_4.html#E3:71_1_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="aff_4.html#E10:71_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="aff_4.html#E1:71_1_1_4"><i><font color="Green">E74</font></i></a>, <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/>
<b>hence </b><a NAME="E3:71_1_1_4_2"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_4.html#E1:71_1_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="aff_4.html#E4:71_1_1"><i><font color="Green">E65</font></i></a>, <a class="txt" href="aff_4.html#E10:71_1_1"><i><font color="Green">E70</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E4:71_1_1_4"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E1:71"><i><font color="Green">E58</font></i></a>, <a class="txt" href="aff_4.html#E3:71_1"><i><font color="Green">E60</font></i></a>, <a class="txt" href="aff_4.html#E4:71_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="aff_4.html#E1:71_1_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="aff_4.html#E8:71_1_1"><i><font color="Green">E68</font></i></a>, <a class="txt" href="aff_4.html#E9:71_1_1"><i><font color="Green">E69</font></i></a>, <a class="txt" href="aff_4.html#E11:71_1_1"><i><font color="Green">E71</font></i></a>, <a class="txt" href="aff_4.html#E1:71_1_1_4"><i><font color="Green">E74</font></i></a>, <a class="ref" href="aff_4.html#T25" target="_self">Th25</a></i>;<br/><b>set </b><font color="Maroon">c<sub>14</sub></font> =  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>4</sub></font>;<br/><a NAME="E5:71_1_1_4"/><i><font color="Green">E78</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>13</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>13</sub></font>,<font color="Maroon">c<sub>4</sub></font> &amp;  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a>  )
 <b>by </b><i><a class="txt" href="aff_4.html#E4:71_1_1_4"><i><font color="Green">E77</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/><a NAME="E6:71_1_1_4"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E10:71_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="aff_4.html#E1:71_1_1_4"><i><font color="Green">E74</font></i></a>, <a class="ref" href="aff_1.html#T33">AFF_1:33</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>15</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="E8:71_1_1_4"/><i><font color="Green">E79</font></i>: 
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>15</sub></font>
 <b>and </b><br/><a NAME="E9:71_1_1_4"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">c<sub>2</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>15</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E2:71_1_1_4"><i><font color="Green">E75</font></i></a>, <a class="ref" href="aff_4.html#T1" target="_self">Th1</a></i>;<br/><a NAME="E10:71_1_1_4"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E4:71_1_1_4"><i><font color="Green">E77</font></i></a>, <a class="txt" href="aff_4.html#E5:71_1_1_4"><i><font color="Green">E78</font></i></a>, <a class="txt" href="aff_4.html#E8:71_1_1_4"><i><font color="Green">E79</font></i></a>, <a class="ref" href="aff_1.html#T39">AFF_1:39</a></i>;<br/><a NAME="E11:71_1_1_4"/>
 <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="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1_1_4_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:71_1_1_4_3"/>
 <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="hidden.html#R1">=</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E2:71_1_1_4_3"/><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="hidden.html#R1">=</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="aff_4.html#E2:71_1_1"><i><font color="Green">E63</font></i></a>, <a class="txt" href="aff_4.html#E10:71_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="aff_4.html#E5:71_1_1_4"><i><font color="Green">E78</font></i></a>, <a class="ref" href="aff_1.html#T71">AFF_1:71</a></i>;<br/>
<a NAME="E3:71_1_1_4_3"/><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>7</sub></font>
 
<b>by </b><i><a class="txt" href="aff_4.html#E3:71_1"><i><font color="Green">E60</font></i></a>, <a class="txt" href="aff_4.html#E4:71_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="aff_4.html#E2:71_1_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="aff_4.html#T19" target="_self">Th19</a></i>;<br/>
<b>hence </b><a NAME="E4:71_1_1_4_3"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_4.html#E1:71_1_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="aff_4.html#E10:71_1_1"><i><font color="Green">E70</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E12:71_1_1_4"/><b>then </b>
<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>6</sub></font>,<font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E1:71"><i><font color="Green">E58</font></i></a>, <a class="txt" href="aff_4.html#E1:71_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="aff_4.html#E10:71_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="aff_4.html#E12:71_1_1"><i><font color="Green">E72</font></i></a>, <a class="txt" href="aff_4.html#E1:71_1_1_4"><i><font color="Green">E74</font></i></a>, <a class="txt" href="aff_4.html#E2:71_1_1_4"><i><font color="Green">E75</font></i></a>, <a class="txt" href="aff_4.html#E3:71_1_1_4"><i><font color="Green">E76</font></i></a>, <a class="txt" href="aff_4.html#E4:71_1_1_4"><i><font color="Green">E77</font></i></a>, <a class="txt" href="aff_4.html#E5:71_1_1_4"><i><font color="Green">E78</font></i></a>, <a class="txt" href="aff_4.html#E9:71_1_1_4"><i><font color="Green">E80</font></i></a>, <a class="txt" href="aff_4.html#E10:71_1_1_4"><i><font color="Green">E81</font></i></a>, <a class="ref" href="aff_4.html#T49" target="_self">Th49</a></i>;<br/><b>hence </b><a NAME="E13:71_1_1_4"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">c<sub>2</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="Olive">b<sub>1</sub></font> &amp; <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>6</sub></font>,<font color="Olive">b<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="aff_4.html#E9:71_1_1_4"><i><font color="Green">E80</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E15:71_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">c<sub>2</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="Olive">b<sub>1</sub></font> &amp; <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>6</sub></font>,<font color="Olive">b<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="aff_4.html#E8:71_1_1"><i><font color="Green">E68</font></i></a>, <a class="txt" href="aff_4.html#E10:71_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="aff_4.html#E11:71_1_1"><i><font color="Green">E71</font></i></a>, <a class="txt" href="aff_4.html#E13:71_1_1"><i><font color="Green">E73</font></i></a>, <a class="ref" href="aff_4.html#T22" target="_self">Th22</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:71_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">c<sub>2</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="Olive">b<sub>1</sub></font> &amp; <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>6</sub></font>,<font color="Olive">b<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="aff_4.html#E1:71"><i><font color="Green">E58</font></i></a>, <a class="txt" href="aff_4.html#E3:71_1"><i><font color="Green">E60</font></i></a>, <a class="txt" href="aff_4.html#E4:71_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="aff_4.html#E70"><i><font color="Green">Lemma56</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:71"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">c<sub>2</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="Olive">b<sub>1</sub></font> &amp; <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>6</sub></font>,<font color="Olive">b<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="aff_4.html#E1:71"><i><font color="Green">E58</font></i></a>, <a class="ref" href="aff_4.html#T53" target="_self">Th53</a></i>;<br/>


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