<?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> 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>8</sub></font>, <font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</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="E1:67"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is  not   <a href="diraf.html#NM2">AffinPlane</a>
 <b>and </b><br/><a NAME="E2:67"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="aff_1.html#R4">//</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="aff_1.html#R4">//</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>and </b><br/><a NAME="E3:67"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>5</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>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> )
 <b>and </b><br/><a NAME="E4:67"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <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>  )
 <b>and </b><br/><a NAME="E5:67"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>9</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>10</sub></font> )
 <b>and </b><br/><a NAME="E6:67"/><i><font color="Green">E60</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>3</sub></font>,<font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:67_1"/>
<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="aff_4.html#R2" target="_self">is_coplanar</a> 
 ;<br/><b>then consider </b><font color="Maroon">c<sub>11</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="E3:67_1"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>11</sub></font> )
 <b>and </b><br/><a NAME="E4:67_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="aff_4.html#NR1" target="_self">is_plane</a> 
 <b>by </b><i><a class="ref" href="aff_4.html#D5" target="_self">Def5</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>12</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="E6:67_1"/><i><font color="Green">E63</font></i>: 
not <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E1:67"><i><font color="Green">E55</font></i></a>, <a class="txt" href="aff_4.html#E4:67_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="aff_4.html#T48" target="_self">Th48</a></i>;<br/><b>set </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>12</sub></font> <a href="aff_4.html#K2" target="_self">*</a> <font color="Maroon">c<sub>8</sub></font>;<br/><a NAME="E7:67_1"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font> <a href="aff_4.html#K2" target="_self">*</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="aff_1.html#R4">//</a> <font color="Maroon">c<sub>12</sub></font> <a href="aff_4.html#K2" target="_self">*</a> <font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="txt" href="aff_4.html#E4:67"><i><font color="Green">E58</font></i></a>, <a class="ref" href="aff_4.html#D3" target="_self">Def3</a></i>;<br/><a NAME="E8:67_1"/><b>then </b><i><font color="Green">E65</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="aff_4.html#K2" target="_self">*</a> <font color="Maroon">c<sub>8</sub></font> <a href="aff_1.html#R4">//</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="aff_4.html#K2" target="_self">*</a> <font color="Maroon">c<sub>8</sub></font> <a href="aff_1.html#R4">//</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="txt" href="aff_4.html#E2:67"><i><font color="Green">E56</font></i></a>, <a class="ref" href="aff_1.html#T58">AFF_1:58</a></i>;<br/><a NAME="E9:67_1"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="aff_4.html#K2" target="_self">*</a> <font color="Maroon">c<sub>8</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a> 
 <b>by </b><i><a class="txt" href="aff_4.html#E4:67"><i><font color="Green">E58</font></i></a>, <a class="ref" href="aff_4.html#T27" target="_self">Th27</a></i>;<br/><a NAME="E10:67_1"/><i><font color="Green">E67</font></i>: 
not <font color="Maroon">c<sub>12</sub></font> <a href="aff_4.html#K2" target="_self">*</a> <font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E6:67_1"><i><font color="Green">E63</font></i></a>, <a class="txt" href="aff_4.html#E7:67_1"><i><font color="Green">E64</font></i></a></i>;<br/><a NAME="E11:67_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="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font> <a href="aff_4.html#K2" target="_self">*</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_1"><b>proof </b></a><div class="add">

<div><i><font color="Green">E68</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:67_1_1_1"/><i><font color="Green">E69</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>
 ;<br/><b>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="E3:67_1_1_1"/><i><font color="Green">E70</font></i>: 
<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>12</sub></font>,<font color="Maroon">c<sub>14</sub></font>
 <b>and </b><br/><a NAME="E4:67_1_1_1"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="ref" href="diraf.html#T47">DIRAF:47</a></i>;<br/><a NAME="E5:67_1_1_1"/>
<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="analoaf.html#R2">//</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#E3:67_1_1_1"><i><font color="Green">E70</font></i></a>, <a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/><a NAME="E6:67_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="aff_1.html#R2">//</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E3:67"><i><font color="Green">E57</font></i></a>, <a class="txt" href="aff_4.html#E4:67"><i><font color="Green">E58</font></i></a>, <a class="txt" href="aff_4.html#E1:67_1_1_1"><i><font color="Green">E69</font></i></a>, <a class="ref" href="aff_1.html#T41">AFF_1:41</a></i>;<br/><a NAME="E7:67_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="aff_1.html#R2">//</a> <font color="Maroon">c<sub>12</sub></font> <a href="aff_4.html#K2" target="_self">*</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E7:67_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="aff_4.html#T3" target="_self">Th3</a></i>;<br/><a NAME="E8:67_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font> <a href="aff_4.html#K2" target="_self">*</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E7:67_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="aff_4.html#T2" target="_self">Th2</a></i>;<br/><b>hence </b><a NAME="E9:67_1_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="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font> <a href="aff_4.html#K2" target="_self">*</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="aff_4.html#E4:67_1_1_1"><i><font color="Green">E71</font></i></a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:67_1_1_2"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><b>take </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>12</sub></font>;<br/><b>thus </b><a NAME="E2:67_1_1_2"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font> <a href="aff_4.html#K2" target="_self">*</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E4:67"><i><font color="Green">E58</font></i></a>, <a class="ref" href="aff_4.html#D3" target="_self">Def3</a></i>;<br/><b>thus </b><a NAME="E3:67_1_1_2"/>
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E1:67_1_1_2"><i><font color="Green">E69</font></i></a>, <a class="ref" href="aff_1.html#T11">AFF_1:11</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:67_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="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font> <a href="aff_4.html#K2" target="_self">*</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="aff_4.html#E1:67_1_1"><i><font color="Green">E68</font></i></a></i>;<br/>


</div><b>end;</b></div><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="E13:67_1"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font> <a href="aff_4.html#K2" target="_self">*</a> <font color="Maroon">c<sub>8</sub></font>
 <b>and </b><br/><a NAME="E14:67_1"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>14</sub></font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:67_1_2"/><i><font color="Green">E70</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>
 ;<br/><a NAME="E2:67_1_2"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="aff_4.html#K2" target="_self">*</a> <font color="Maroon">c<sub>8</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#E3:67_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="aff_4.html#E6:67_1"><i><font color="Green">E63</font></i></a>, <a class="txt" href="aff_4.html#E7:67_1"><i><font color="Green">E64</font></i></a></i>;<br/><a NAME="E3:67_1_2"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="aff_4.html#K2" target="_self">*</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E3:67_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="aff_4.html#E6:67_1"><i><font color="Green">E63</font></i></a>, <a class="txt" href="aff_4.html#E7:67_1"><i><font color="Green">E64</font></i></a></i>;<br/><a NAME="E4:67_1_2"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="aff_4.html#K2" target="_self">*</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E3:67_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="aff_4.html#E6:67_1"><i><font color="Green">E63</font></i></a>, <a class="txt" href="aff_4.html#E7:67_1"><i><font color="Green">E64</font></i></a></i>;<br/><a NAME="E5:67_1_2"/><i><font color="Green">E74</font></i>: 
not <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="aff_4.html#K2" target="_self">*</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="aff_4.html#R2" target="_self">is_coplanar</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:67_1_2_1"/>
<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="aff_4.html#K2" target="_self">*</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="aff_4.html#R2" target="_self">is_coplanar</a> 
 ;<br/>

<a NAME="E2:67_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="aff_4.html#K2" target="_self">*</a> <font color="Maroon">c<sub>8</sub></font> <a href="aff_4.html#R2" target="_self">is_coplanar</a> 
 
<b>by </b><i><a class="ref" href="aff_4.html#T44" target="_self">Th44</a></i>;<br/>
<b>hence </b><a NAME="E3:67_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_4.html#E4:67"><i><font color="Green">E58</font></i></a>, <a class="txt" href="aff_4.html#E5:67"><i><font color="Green">E59</font></i></a>, <a class="txt" href="aff_4.html#E3:67_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="aff_4.html#E4:67_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="aff_4.html#E10:67_1"><i><font color="Green">E67</font></i></a>, <a class="ref" href="aff_4.html#T46" target="_self">Th46</a></i>;<br/>


</div><b>end;</b></div><a NAME="E6:67_1_2"/><i><font color="Green">E75</font></i>: 
not <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="aff_4.html#K2" target="_self">*</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="aff_4.html#R2" target="_self">is_coplanar</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_2_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:67_1_2_2"/>
<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="aff_4.html#K2" target="_self">*</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="aff_4.html#R2" target="_self">is_coplanar</a> 
 ;<br/>

<a NAME="E2:67_1_2_2"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="aff_4.html#K2" target="_self">*</a> <font color="Maroon">c<sub>8</sub></font> <a href="aff_4.html#R2" target="_self">is_coplanar</a> 
 
<b>by </b><i><a class="ref" href="aff_4.html#T44" target="_self">Th44</a></i>;<br/>
<b>hence </b><a NAME="E3:67_1_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_4.html#E4:67"><i><font color="Green">E58</font></i></a>, <a class="txt" href="aff_4.html#E5:67"><i><font color="Green">E59</font></i></a>, <a class="txt" href="aff_4.html#E3:67_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="aff_4.html#E4:67_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="aff_4.html#E10:67_1"><i><font color="Green">E67</font></i></a>, <a class="ref" href="aff_4.html#T46" target="_self">Th46</a></i>;<br/>


</div><b>end;</b></div><a NAME="E7:67_1_2"/><i><font color="Green">E76</font></i>: 
not <font color="Maroon">c<sub>12</sub></font> <a href="aff_4.html#K2" target="_self">*</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="aff_4.html#R2" target="_self">is_coplanar</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_2_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:67_1_2_3"/>
<font color="Maroon">c<sub>12</sub></font> <a href="aff_4.html#K2" target="_self">*</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="aff_4.html#R2" target="_self">is_coplanar</a> 
 ;<br/>

<a NAME="E2:67_1_2_3"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="aff_4.html#K2" target="_self">*</a> <font color="Maroon">c<sub>8</sub></font> <a href="aff_4.html#R2" target="_self">is_coplanar</a> 
 
<b>by </b><i><a class="ref" href="aff_4.html#T44" target="_self">Th44</a></i>;<br/>
<b>hence </b><a NAME="E3:67_1_2_3"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_4.html#E4:67"><i><font color="Green">E58</font></i></a>, <a class="txt" href="aff_4.html#E3:67_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="aff_4.html#E4:67_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="aff_4.html#E10:67_1"><i><font color="Green">E67</font></i></a>, <a class="txt" href="aff_4.html#E1:67_1_2"><i><font color="Green">E70</font></i></a>, <a class="ref" href="aff_4.html#T46" target="_self">Th46</a></i>;<br/>


</div><b>end;</b></div><a NAME="E8:67_1_2"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E2:67"><i><font color="Green">E56</font></i></a>, <a class="txt" href="aff_4.html#E3:67"><i><font color="Green">E57</font></i></a>, <a class="txt" href="aff_4.html#E4:67"><i><font color="Green">E58</font></i></a>, <a class="txt" href="aff_4.html#E5:67"><i><font color="Green">E59</font></i></a>, <a class="txt" href="aff_4.html#E6:67"><i><font color="Green">E60</font></i></a>, <a class="txt" href="aff_4.html#E7:67_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="aff_4.html#E9:67_1"><i><font color="Green">E66</font></i></a>, <a class="txt" href="aff_4.html#E13:67_1"><i><font color="Green">E68</font></i></a>, <a class="txt" href="aff_4.html#E14:67_1"><i><font color="Green">E69</font></i></a>, <a class="txt" href="aff_4.html#E2:67_1_2"><i><font color="Green">E71</font></i></a>, <a class="txt" href="aff_4.html#E5:67_1_2"><i><font color="Green">E74</font></i></a>, <a class="txt" href="aff_4.html#E66"><i><font color="Green">Lemma53</font></i></a></i>;<br/><a NAME="E9:67_1_2"/>
<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E2:67"><i><font color="Green">E56</font></i></a>, <a class="txt" href="aff_4.html#E3:67"><i><font color="Green">E57</font></i></a>, <a class="txt" href="aff_4.html#E4:67"><i><font color="Green">E58</font></i></a>, <a class="txt" href="aff_4.html#E5:67"><i><font color="Green">E59</font></i></a>, <a class="txt" href="aff_4.html#E6:67"><i><font color="Green">E60</font></i></a>, <a class="txt" href="aff_4.html#E7:67_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="aff_4.html#E9:67_1"><i><font color="Green">E66</font></i></a>, <a class="txt" href="aff_4.html#E13:67_1"><i><font color="Green">E68</font></i></a>, <a class="txt" href="aff_4.html#E14:67_1"><i><font color="Green">E69</font></i></a>, <a class="txt" href="aff_4.html#E2:67_1_2"><i><font color="Green">E71</font></i></a>, <a class="txt" href="aff_4.html#E6:67_1_2"><i><font color="Green">E75</font></i></a>, <a class="txt" href="aff_4.html#E66"><i><font color="Green">Lemma53</font></i></a></i>;<br/><b>hence </b><a NAME="E10:67_1_2"/>
<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E3:67"><i><font color="Green">E57</font></i></a>, <a class="txt" href="aff_4.html#E4:67"><i><font color="Green">E58</font></i></a>, <a class="txt" href="aff_4.html#E7:67_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="aff_4.html#E8:67_1"><i><font color="Green">E65</font></i></a>, <a class="txt" href="aff_4.html#E9:67_1"><i><font color="Green">E66</font></i></a>, <a class="txt" href="aff_4.html#E13:67_1"><i><font color="Green">E68</font></i></a>, <a class="txt" href="aff_4.html#E3:67_1_2"><i><font color="Green">E72</font></i></a>, <a class="txt" href="aff_4.html#E4:67_1_2"><i><font color="Green">E73</font></i></a>, <a class="txt" href="aff_4.html#E7:67_1_2"><i><font color="Green">E76</font></i></a>, <a class="txt" href="aff_4.html#E8:67_1_2"><i><font color="Green">E77</font></i></a>, <a class="txt" href="aff_4.html#E66"><i><font color="Green">Lemma53</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E16:67_1"/>
<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E3:67"><i><font color="Green">E57</font></i></a>, <a class="txt" href="aff_4.html#E4:67"><i><font color="Green">E58</font></i></a>, <a class="ref" href="aff_1.html#T65">AFF_1:65</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:67"/>
<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E2:67"><i><font color="Green">E56</font></i></a>, <a class="txt" href="aff_4.html#E3:67"><i><font color="Green">E57</font></i></a>, <a class="txt" href="aff_4.html#E4:67"><i><font color="Green">E58</font></i></a>, <a class="txt" href="aff_4.html#E5:67"><i><font color="Green">E59</font></i></a>, <a class="txt" href="aff_4.html#E6:67"><i><font color="Green">E60</font></i></a>, <a class="txt" href="aff_4.html#E66"><i><font color="Green">Lemma53</font></i></a></i>;<br/>


</div>
