<?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>, <font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>11</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>





















<b>assume </b><b>that </b><br/><a NAME="E1:63"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> &amp; <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>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> )
 <b>and </b><br/><a NAME="E2:63"/><i><font color="Green">E53</font></i>: 
not <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="aff_4.html#R2" target="_self">is_coplanar</a> 
 <b>and </b><br/><a NAME="E3:63"/><i><font color="Green">E54</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> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>and </b><br/><a NAME="E4:63"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>7</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>11</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> )
 <b>and </b><br/><a NAME="E5:63"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">c<sub>10</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">c<sub>11</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a>  )
 <b>and </b><br/><a NAME="E6:63"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>11</sub></font> )
 <b>and </b><br/><a NAME="E7:63"/><i><font color="Green">E58</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>6</sub></font>,<font color="Maroon">c<sub>7</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>6</sub></font>,<font color="Maroon">c<sub>8</sub></font> )
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>12</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="E9:63"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>12</sub></font> )
 <b>and </b><br/><a NAME="E10:63"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="aff_4.html#NR1" target="_self">is_plane</a> 
 <b>by </b><i><a class="txt" href="aff_4.html#E1:63"><i><font color="Green">E52</font></i></a>, <a class="txt" href="aff_4.html#E5:63"><i><font color="Green">E56</font></i></a>, <a class="ref" href="aff_4.html#T38" target="_self">Th38</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>13</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="E12:63"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>13</sub></font> )
 <b>and </b><br/><a NAME="E13:63"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="aff_4.html#NR1" target="_self">is_plane</a> 
 <b>by </b><i><a class="txt" href="aff_4.html#E1:63"><i><font color="Green">E52</font></i></a>, <a class="txt" href="aff_4.html#E5:63"><i><font color="Green">E56</font></i></a>, <a class="ref" href="aff_4.html#T38" target="_self">Th38</a></i>;<br/>
<a NAME="E14:63"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="aff_4.html#E1:63"><i><font color="Green">E52</font></i></a>, <a class="txt" href="aff_4.html#E2:63"><i><font color="Green">E53</font></i></a>, <a class="txt" href="aff_4.html#E5:63"><i><font color="Green">E56</font></i></a>, <a class="ref" href="aff_4.html#T47" target="_self">Th47</a></i>;<br/>
<a NAME="E15:63"/><b>then </b><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>4</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>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="aff_4.html#E1:63"><i><font color="Green">E52</font></i></a>, <a class="txt" href="aff_4.html#E3:63"><i><font color="Green">E54</font></i></a>, <a class="txt" href="aff_4.html#E4:63"><i><font color="Green">E55</font></i></a>, <a class="txt" href="aff_4.html#E5:63"><i><font color="Green">E56</font></i></a>, <a class="txt" href="aff_4.html#E6:63"><i><font color="Green">E57</font></i></a>, <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/>
<div><i><font color="Green">E65</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:63_1"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><a NAME="E2:63_1"/><b>then </b><i><font color="Green">E67</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E1:63"><i><font color="Green">E52</font></i></a>, <a class="txt" href="aff_4.html#E3:63"><i><font color="Green">E54</font></i></a>, <a class="txt" href="aff_4.html#E4:63"><i><font color="Green">E55</font></i></a>, <a class="txt" href="aff_4.html#E5:63"><i><font color="Green">E56</font></i></a>, <a class="txt" href="aff_4.html#E6:63"><i><font color="Green">E57</font></i></a>, <a class="txt" href="aff_4.html#E7:63"><i><font color="Green">E58</font></i></a>, <a class="ref" href="aff_4.html#T8" target="_self">Th8</a></i>;<br/><a NAME="E3:63_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E1:63"><i><font color="Green">E52</font></i></a>, <a class="txt" href="aff_4.html#E3:63"><i><font color="Green">E54</font></i></a>, <a class="txt" href="aff_4.html#E4:63"><i><font color="Green">E55</font></i></a>, <a class="txt" href="aff_4.html#E5:63"><i><font color="Green">E56</font></i></a>, <a class="txt" href="aff_4.html#E6:63"><i><font color="Green">E57</font></i></a>, <a class="txt" href="aff_4.html#E7:63"><i><font color="Green">E58</font></i></a>, <a class="txt" href="aff_4.html#E1:63_1"><i><font color="Green">E66</font></i></a>, <a class="ref" href="aff_4.html#T8" target="_self">Th8</a></i>;<br/><b>hence </b><a NAME="E4:63_1"/>
<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E2:63_1"><i><font color="Green">E67</font></i></a>, <a class="ref" href="aff_1.html#T12">AFF_1:12</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:63_2"/><i><font color="Green">E66</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>6</sub></font>
 ;<br/><a NAME="E2:63_2"/><b>then </b><i><font color="Green">E67</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>8</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E1:63"><i><font color="Green">E52</font></i></a>, <a class="txt" href="aff_4.html#E3:63"><i><font color="Green">E54</font></i></a>, <a class="txt" href="aff_4.html#E4:63"><i><font color="Green">E55</font></i></a>, <a class="txt" href="aff_4.html#E5:63"><i><font color="Green">E56</font></i></a>, <a class="txt" href="aff_4.html#E6:63"><i><font color="Green">E57</font></i></a>, <a class="txt" href="aff_4.html#E7:63"><i><font color="Green">E58</font></i></a>, <a class="ref" href="aff_4.html#T8" target="_self">Th8</a></i>;<br/><div><i><font color="Green">E68</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="63_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:63_2_1"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><a NAME="E2:63_2_1"/><b>then </b><i><font color="Green">E70</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E1:63"><i><font color="Green">E52</font></i></a>, <a class="txt" href="aff_4.html#E3:63"><i><font color="Green">E54</font></i></a>, <a class="txt" href="aff_4.html#E4:63"><i><font color="Green">E55</font></i></a>, <a class="txt" href="aff_4.html#E5:63"><i><font color="Green">E56</font></i></a>, <a class="txt" href="aff_4.html#E6:63"><i><font color="Green">E57</font></i></a>, <a class="txt" href="aff_4.html#E7:63"><i><font color="Green">E58</font></i></a>, <a class="ref" href="aff_4.html#T9" target="_self">Th9</a></i>;<br/><a NAME="E3:63_2_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E1:63"><i><font color="Green">E52</font></i></a>, <a class="txt" href="aff_4.html#E3:63"><i><font color="Green">E54</font></i></a>, <a class="txt" href="aff_4.html#E4:63"><i><font color="Green">E55</font></i></a>, <a class="txt" href="aff_4.html#E5:63"><i><font color="Green">E56</font></i></a>, <a class="txt" href="aff_4.html#E6:63"><i><font color="Green">E57</font></i></a>, <a class="txt" href="aff_4.html#E7:63"><i><font color="Green">E58</font></i></a>, <a class="txt" href="aff_4.html#E1:63_2_1"><i><font color="Green">E69</font></i></a>, <a class="ref" href="aff_4.html#T9" target="_self">Th9</a></i>;<br/><b>hence </b><a NAME="E4:63_2_1"/>
<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E2:63_2_1"><i><font color="Green">E70</font></i></a>, <a class="ref" href="aff_1.html#T11">AFF_1:11</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:63_2_2"/><i><font color="Green">E69</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>6</sub></font>
 ;<br/><b>assume </b><a NAME="E2:63_2_2"/><i><font color="Green">E70</font></i>: 
not <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>
 ;<br/><a NAME="E3:63_2_2"/><i><font color="Green">E71</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="aff_4.html#E1:63"><i><font color="Green">E52</font></i></a>, <a class="txt" href="aff_4.html#E4:63"><i><font color="Green">E55</font></i></a>, <a class="txt" href="aff_4.html#E5:63"><i><font color="Green">E56</font></i></a>, <a class="txt" href="aff_4.html#E6:63"><i><font color="Green">E57</font></i></a>, <a class="txt" href="aff_4.html#E14:63"><i><font color="Green">E63</font></i></a>, <a class="txt" href="aff_4.html#E1:63_2"><i><font color="Green">E66</font></i></a>, <a class="txt" href="aff_4.html#E2:63_2"><i><font color="Green">E67</font></i></a>, <a class="ref" href="aff_1.html#T30">AFF_1:30</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>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>15</sub></font> =  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>16</sub></font> =  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>17</sub></font> =  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>18</sub></font> =  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>19</sub></font> =  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font>;<br/><a NAME="E4:63_2_2"/><i><font color="Green">E72</font></i>: 
(  <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>  &amp;  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</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>6</sub></font>,<font color="Maroon">c<sub>8</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>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>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a>  )
 <b>by </b><i><a class="txt" href="aff_4.html#E15:63"><i><font color="Green">E64</font></i></a>, <a class="txt" href="aff_4.html#E3:63_2_2"><i><font color="Green">E71</font></i></a>, <a class="ref" href="aff_1.html#D3">AFF_1:def 3</a></i>;<br/><a NAME="E5:63_2_2"/><i><font color="Green">E73</font></i>: 
( <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>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>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>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="ref" href="aff_1.html#T26">AFF_1:26</a></i>;<br/><a NAME="E6:63_2_2"/><i><font color="Green">E74</font></i>: 
( <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> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="ref" href="aff_1.html#T26">AFF_1:26</a></i>;<br/><a NAME="E7:63_2_2"/><i><font color="Green">E75</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>6</sub></font>,<font color="Maroon">c<sub>7</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>6</sub></font>,<font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="txt" href="aff_4.html#E7:63"><i><font color="Green">E58</font></i></a>, <a class="txt" href="aff_4.html#E15:63"><i><font color="Green">E64</font></i></a>, <a class="txt" href="aff_4.html#E3:63_2_2"><i><font color="Green">E71</font></i></a>, <a class="ref" href="aff_1.html#T51">AFF_1:51</a></i>;<br/><a NAME="E8:63_2_2"/><i><font color="Green">E76</font></i>: 
(  <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="tarski.html#R1">c=</a> <font color="Maroon">c<sub>12</sub></font> &amp;  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>12</sub></font> )
 <b>by </b><i><a class="txt" href="aff_4.html#E4:63"><i><font color="Green">E55</font></i></a>, <a class="txt" href="aff_4.html#E9:63"><i><font color="Green">E59</font></i></a>, <a class="txt" href="aff_4.html#E10:63"><i><font color="Green">E60</font></i></a>, <a class="txt" href="aff_4.html#E15:63"><i><font color="Green">E64</font></i></a>, <a class="txt" href="aff_4.html#E3:63_2_2"><i><font color="Green">E71</font></i></a>, <a class="ref" href="aff_4.html#T19" target="_self">Th19</a></i>;<br/><a NAME="E9:63_2_2"/><i><font color="Green">E77</font></i>: 
(  <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>13</sub></font> &amp;  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>13</sub></font> )
 <b>by </b><i><a class="txt" href="aff_4.html#E4:63"><i><font color="Green">E55</font></i></a>, <a class="txt" href="aff_4.html#E12:63"><i><font color="Green">E61</font></i></a>, <a class="txt" href="aff_4.html#E13:63"><i><font color="Green">E62</font></i></a>, <a class="txt" href="aff_4.html#E15:63"><i><font color="Green">E64</font></i></a>, <a class="txt" href="aff_4.html#E3:63_2_2"><i><font color="Green">E71</font></i></a>, <a class="ref" href="aff_4.html#T19" target="_self">Th19</a></i>;<br/><a NAME="E10:63_2_2"/>
not  <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#R4">//</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E2:63_2_2"><i><font color="Green">E70</font></i></a>, <a class="txt" href="aff_4.html#E6:63_2_2"><i><font color="Green">E74</font></i></a>, <a class="ref" href="aff_1.html#T53">AFF_1:53</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>20</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="E12:63_2_2"/><i><font color="Green">E78</font></i>: 
( <font color="Maroon">c<sub>20</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>20</sub></font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="txt" href="aff_4.html#E10:63"><i><font color="Green">E60</font></i></a>, <a class="txt" href="aff_4.html#E4:63_2_2"><i><font color="Green">E72</font></i></a>, <a class="txt" href="aff_4.html#E8:63_2_2"><i><font color="Green">E76</font></i></a>, <a class="ref" href="aff_4.html#T22" target="_self">Th22</a></i>;<br/><b>set </b><font color="Maroon">c<sub>21</sub></font> = <font color="Maroon">c<sub>20</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>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>;<br/><a NAME="E13:63_2_2"/><i><font color="Green">E79</font></i>: 
( <font color="Maroon">c<sub>20</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>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>20</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>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> &amp;  <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> <font color="Maroon">c<sub>20</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>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> &amp;  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="aff_1.html#R4">//</a> <font color="Maroon">c<sub>20</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>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_2_2_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:63_2_2_1"/>
<font color="Maroon">c<sub>20</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>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="aff_1.html#NR2" target="_self">is_line</a> 
 <b>by </b><i><a class="txt" href="aff_4.html#E4:63_2_2"><i><font color="Green">E72</font></i></a>, <a class="ref" href="aff_4.html#T27" target="_self">Th27</a></i>;<br/>

<b>thus </b><a NAME="E2:63_2_2_1"/>
( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>20</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>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> &amp;  <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> <font color="Maroon">c<sub>20</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>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="aff_4.html#E4:63_2_2"><i><font color="Green">E72</font></i></a>, <a class="ref" href="aff_4.html#D3" target="_self">Def3</a></i>;<br/>

<b>hence </b><a NAME="E3:63_2_2_1"/>
 <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="aff_1.html#R4">//</a> <font color="Maroon">c<sub>20</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>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="aff_4.html#E7:63_2_2"><i><font color="Green">E75</font></i></a>, <a class="ref" href="aff_1.html#T58">AFF_1:58</a></i>;<br/>


</div><b>end;</b></div><a NAME="E14:63_2_2"/><i><font color="Green">E80</font></i>: 
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>20</sub></font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>20</sub></font> )
 <b>by </b><i><a class="txt" href="aff_4.html#E4:63_2_2"><i><font color="Green">E72</font></i></a>, <a class="txt" href="aff_4.html#E6:63_2_2"><i><font color="Green">E74</font></i></a>, <a class="txt" href="aff_4.html#E12:63_2_2"><i><font color="Green">E78</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>22</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="E16:63_2_2"/><i><font color="Green">E81</font></i>: 
(  <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>22</sub></font> &amp; <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>20</sub></font>,<font color="Maroon">c<sub>22</sub></font> )
 <b>by </b><i><a class="txt" href="aff_4.html#E15:63"><i><font color="Green">E64</font></i></a>, <a class="ref" href="aff_4.html#T1" target="_self">Th1</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>23</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="E18:63_2_2"/><i><font color="Green">E82</font></i>: 
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>23</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>23</sub></font> )
 <b>by </b><i><a class="txt" href="aff_4.html#E3:63_2_2"><i><font color="Green">E71</font></i></a>, <a class="txt" href="aff_4.html#E14:63_2_2"><i><font color="Green">E80</font></i></a>, <a class="ref" href="aff_4.html#T1" target="_self">Th1</a></i>;<br/><a NAME="E19:63_2_2"/><i><font color="Green">E83</font></i>: 
( <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>20</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>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>20</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>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_2_2_2"><b>proof </b></a><div class="add">

<a NAME="E1:63_2_2_2"/>
<font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>22</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="aff_4.html#E16:63_2_2"><i><font color="Green">E81</font></i></a>, <a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/>
<a NAME="E2:63_2_2_2"/><b>then </b>
<font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>22</sub></font> <a href="aff_1.html#R2">//</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="txt" href="aff_4.html#E15:63"><i><font color="Green">E64</font></i></a>, <a class="ref" href="aff_1.html#D4">AFF_1:def 4</a></i>;<br/>
<a NAME="E3:63_2_2_2"/><b>then </b>
<font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>22</sub></font> <a href="aff_1.html#R2">//</a> <font color="Maroon">c<sub>20</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>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="aff_4.html#E13:63_2_2"><i><font color="Green">E79</font></i></a>, <a class="ref" href="aff_4.html#T3" target="_self">Th3</a></i>;<br/>
<b>hence </b><a NAME="E4:63_2_2_2"/>
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>20</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>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="aff_4.html#E13:63_2_2"><i><font color="Green">E79</font></i></a>, <a class="ref" href="aff_4.html#T2" target="_self">Th2</a></i>;<br/>

<a NAME="E5:63_2_2_2"/>
<font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>23</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="aff_4.html#E18:63_2_2"><i><font color="Green">E82</font></i></a>, <a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/>
<a NAME="E6:63_2_2_2"/><b>then </b>
<font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>23</sub></font> <a href="aff_1.html#R2">//</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="aff_4.html#E3:63_2_2"><i><font color="Green">E71</font></i></a>, <a class="ref" href="aff_1.html#D4">AFF_1:def 4</a></i>;<br/>
<a NAME="E7:63_2_2_2"/><b>then </b>
<font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>23</sub></font> <a href="aff_1.html#R2">//</a> <font color="Maroon">c<sub>20</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>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="aff_4.html#E13:63_2_2"><i><font color="Green">E79</font></i></a>, <a class="ref" href="aff_4.html#T3" target="_self">Th3</a></i>;<br/>
<b>hence </b><a NAME="E8:63_2_2_2"/>
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>20</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>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="aff_4.html#E13:63_2_2"><i><font color="Green">E79</font></i></a>, <a class="ref" href="aff_4.html#T2" target="_self">Th2</a></i>;<br/>


</div><b>end;</b></div><a NAME="E20:63_2_2"/><i><font color="Green">E84</font></i>: 
( <font color="Maroon">c<sub>22</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>23</sub></font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="txt" href="aff_4.html#E15:63"><i><font color="Green">E64</font></i></a>, <a class="txt" href="aff_4.html#E3:63_2_2"><i><font color="Green">E71</font></i></a>, <a class="txt" href="aff_4.html#E4:63_2_2"><i><font color="Green">E72</font></i></a>, <a class="txt" href="aff_4.html#E5:63_2_2"><i><font color="Green">E73</font></i></a>, <a class="txt" href="aff_4.html#E16:63_2_2"><i><font color="Green">E81</font></i></a>, <a class="txt" href="aff_4.html#E18:63_2_2"><i><font color="Green">E82</font></i></a>, <a class="ref" href="aff_1.html#T39">AFF_1:39</a></i>;<br/><div><i><font color="Green">E85</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="63_2_2_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:63_2_2_3"/>
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font>
 ;<br/><a NAME="E2:63_2_2_3"/><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>
 <b>by </b><i><a class="txt" href="aff_4.html#E5:63_2_2"><i><font color="Green">E73</font></i></a>, <a class="txt" href="aff_4.html#E7:63_2_2"><i><font color="Green">E75</font></i></a>, <a class="txt" href="aff_4.html#E20:63_2_2"><i><font color="Green">E84</font></i></a>, <a class="ref" href="aff_1.html#T59">AFF_1:59</a></i>;<br/><a NAME="E3:63_2_2_3"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E4:63"><i><font color="Green">E55</font></i></a>, <a class="txt" href="aff_4.html#E5:63"><i><font color="Green">E56</font></i></a>, <a class="txt" href="aff_4.html#E1:63_2_2"><i><font color="Green">E69</font></i></a>, <a class="txt" href="aff_4.html#E4:63_2_2"><i><font color="Green">E72</font></i></a>, <a class="txt" href="aff_4.html#E5:63_2_2"><i><font color="Green">E73</font></i></a>, <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/><b>hence </b><a NAME="E4:63_2_2_3"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_4.html#E1:63"><i><font color="Green">E52</font></i></a>, <a class="txt" href="aff_4.html#E3:63"><i><font color="Green">E54</font></i></a>, <a class="txt" href="aff_4.html#E4:63"><i><font color="Green">E55</font></i></a>, <a class="txt" href="aff_4.html#E5:63"><i><font color="Green">E56</font></i></a>, <a class="txt" href="aff_4.html#E6:63"><i><font color="Green">E57</font></i></a>, <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_2_2_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:63_2_2_4"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>23</sub></font>
 ;<br/><a NAME="E2:63_2_2_4"/><b>then </b>
<font color="Maroon">c<sub>20</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>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>23</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E13:63_2_2"><i><font color="Green">E79</font></i></a>, <a class="txt" href="aff_4.html#E19:63_2_2"><i><font color="Green">E83</font></i></a>, <a class="ref" href="aff_1.html#T71">AFF_1:71</a></i>;<br/><a NAME="E3:63_2_2_4"/><b>then </b>
<font color="Maroon">c<sub>20</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>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E13:63"><i><font color="Green">E62</font></i></a>, <a class="txt" href="aff_4.html#E9:63_2_2"><i><font color="Green">E77</font></i></a>, <a class="txt" href="aff_4.html#E20:63_2_2"><i><font color="Green">E84</font></i></a>, <a class="txt" href="aff_4.html#E1:63_2_2_4"><i><font color="Green">E86</font></i></a>, <a class="ref" href="aff_4.html#T19" target="_self">Th19</a></i>;<br/><a NAME="E4:63_2_2_4"/><b>then </b><i><font color="Green">E87</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>13</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E4:63"><i><font color="Green">E55</font></i></a>, <a class="txt" href="aff_4.html#E12:63"><i><font color="Green">E61</font></i></a>, <a class="txt" href="aff_4.html#E13:63"><i><font color="Green">E62</font></i></a>, <a class="txt" href="aff_4.html#E5:63_2_2"><i><font color="Green">E73</font></i></a>, <a class="txt" href="aff_4.html#E13:63_2_2"><i><font color="Green">E79</font></i></a>, <a class="ref" href="aff_4.html#T23" target="_self">Th23</a></i>;<br/><a NAME="E5:63_2_2_4"/>
<font color="Maroon">c<sub>10</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#E1:63"><i><font color="Green">E52</font></i></a>, <a class="txt" href="aff_4.html#E3:63"><i><font color="Green">E54</font></i></a>, <a class="txt" href="aff_4.html#E4:63"><i><font color="Green">E55</font></i></a>, <a class="txt" href="aff_4.html#E5:63"><i><font color="Green">E56</font></i></a>, <a class="ref" href="aff_1.html#T71">AFF_1:71</a></i>;<br/><a NAME="E6:63_2_2_4"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E1:63"><i><font color="Green">E52</font></i></a>, <a class="txt" href="aff_4.html#E3:63"><i><font color="Green">E54</font></i></a>, <a class="txt" href="aff_4.html#E12:63"><i><font color="Green">E61</font></i></a>, <a class="txt" href="aff_4.html#E13:63"><i><font color="Green">E62</font></i></a>, <a class="txt" href="aff_4.html#E5:63_2_2"><i><font color="Green">E73</font></i></a>, <a class="txt" href="aff_4.html#E4:63_2_2_4"><i><font color="Green">E87</font></i></a>, <a class="ref" href="aff_4.html#T19" target="_self">Th19</a></i>;<br/><b>hence </b><a NAME="E7:63_2_2_4"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_4.html#E2:63"><i><font color="Green">E53</font></i></a>, <a class="txt" href="aff_4.html#E12:63"><i><font color="Green">E61</font></i></a>, <a class="txt" href="aff_4.html#E13:63"><i><font color="Green">E62</font></i></a>, <a class="ref" href="aff_4.html#D5" target="_self">Def5</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E23:63_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_4.html#E21:63_2_2"><i><font color="Green">E85</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:63_2"/>
<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E3:63_2"><i><font color="Green">E68</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E18:63"/>
<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E16:63"><i><font color="Green">E65</font></i></a></i>;<br/>


</div>
