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<b>let </b><font color="Maroon">AS</font> be   <a href="diraf.html#NM1">AffinSpace</a>;<br/><b>let </b><font color="Maroon">q</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>;<br/><b>let </b><font color="Maroon">c</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>;<br/><b>let </b><font color="Maroon">a'</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>;<br/><b>let </b><font color="Maroon">b'</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>;<br/><b>let </b><font color="Maroon">c'</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>;<br/><b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AS</font>;<br/><b>let </b><font color="Maroon">P</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AS</font>;<br/><b>let </b><font color="Maroon">C</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AS</font>;<br/>





















<b>assume </b><b>that </b><br/><a NAME="E1:63"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> )
 <b>and </b><br/><a NAME="E2:63"/><i><font color="Green">E35</font></i>: 
not <font color="Maroon">A</font>,<font color="Maroon">P</font>,<font color="Maroon">C</font> <a href="aff_4.html#R2" target="_self">is_coplanar</a> 
 <b>and </b><br/><a NAME="E3:63"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c</font> )
 <b>and </b><br/><a NAME="E4:63"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">a'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">b'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp; <font color="Maroon">c'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> )
 <b>and </b><br/><a NAME="E5:63"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">A</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">P</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">C</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  )
 <b>and </b><br/><a NAME="E6:63"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">P</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">C</font> )
 <b>and </b><br/><a NAME="E7:63"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a'</font>,<font color="Maroon">b'</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">c</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a'</font>,<font color="Maroon">c'</font> )
 ;<br/>

<b>consider </b><font color="Maroon">X</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AS</font><b> such that </b><br/><a NAME="E9:63"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">P</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> &amp; <font color="Maroon">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> )
 <b>and </b><br/><a NAME="E10:63"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">X</font> <a href="aff_4.html#NR1" target="_self">is_plane</a> 
 <b>by </b><i><a class="ref" href="aff_4.html#T12" target="_self">Th12</a>, <a class="ref" href="aff_4.html#T16" target="_self">Th16</a>, <a class="txt" href="aff_4.html#E5:63"><i><font color="Green">E43</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">Y</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AS</font><b> such that </b><br/><a NAME="E12:63"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> )
 <b>and </b><br/><a NAME="E13:63"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">Y</font> <a href="aff_4.html#NR1" target="_self">is_plane</a> 
 <b>by </b><i><a class="ref" href="aff_4.html#T12" target="_self">Th12</a>, <a class="ref" href="aff_4.html#T16" target="_self">Th16</a>, <a class="txt" href="aff_4.html#E5:63"><i><font color="Green">E43</font></i></a></i>;<br/>
<a NAME="E14:63"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">P</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="aff_4.html#T12" target="_self">Th12</a>, <a class="ref" href="aff_4.html#D2" target="_self">Def2</a>, <a class="ref" href="aff_4.html#T16" target="_self">Th16</a>, <a class="txt" href="aff_4.html#E10:63"><i><font color="Green">E49</font></i></a></i>;<br/>
<a NAME="E15:63"/><b>then </b><i><font color="Green">E61</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font> )
 
<b>by </b><i><a class="ref" href="aff_4.html#T12" target="_self">Th12</a>, <a class="txt" href="aff_4.html#E17"><i><font color="Green">Lemma54</font></i></a>, <a class="ref" href="aff_4.html#T14" target="_self">Th14</a>, <a class="ref" href="aff_4.html#T16" target="_self">Th16</a>, <a class="txt" href="aff_4.html#E12:63"><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">E62</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">E63</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a'</font>
 ;<br/><a NAME="E2:63_1"/><b>then </b><i><font color="Green">E64</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b'</font>
 <b>by </b><i><a class="ref" href="aff_4.html#T12" target="_self">Th12</a>, <a class="txt" href="aff_4.html#E17"><i><font color="Green">Lemma54</font></i></a>, <a class="ref" href="aff_4.html#T14" target="_self">Th14</a>, <a class="ref" href="aff_4.html#T16" target="_self">Th16</a>, <a class="txt" href="aff_4.html#E12:63"><i><font color="Green">E57</font></i></a>, , </i>;<br/><a NAME="E3:63_1"/>
<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c'</font>
 <b>by </b><i><a class="ref" href="aff_4.html#T12" target="_self">Th12</a>, <a class="txt" href="aff_4.html#E17"><i><font color="Green">Lemma54</font></i></a>, <a class="ref" href="aff_4.html#T14" target="_self">Th14</a>, <a class="ref" href="aff_4.html#T16" target="_self">Th16</a>, <a class="txt" href="aff_4.html#E12:63"><i><font color="Green">E57</font></i></a>, , , </i>;<br/><b>hence </b><a NAME="E4:63_1"/>
<font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b'</font>,<font color="Maroon">c'</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E24"><i><font color="Green">Lemma67</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">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a'</font>
 ;<br/><a NAME="E2:63_2"/><b>then </b><i><font color="Green">E69</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c'</font>
 <b>by </b><i><a class="ref" href="aff_4.html#T12" target="_self">Th12</a>, <a class="txt" href="aff_4.html#E17"><i><font color="Green">Lemma54</font></i></a>, <a class="ref" href="aff_4.html#T14" target="_self">Th14</a>, <a class="ref" href="aff_4.html#T16" target="_self">Th16</a>, <a class="txt" href="aff_4.html#E12:63"><i><font color="Green">E57</font></i></a>, , </i>;<br/><div><i><font color="Green">E70</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">E71</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a'</font>
 ;<br/><a NAME="E2:63_2_1"/><b>then </b><i><font color="Green">E72</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b'</font>
 <b>by </b><i><a class="ref" href="aff_4.html#T12" target="_self">Th12</a>, <a class="txt" href="aff_4.html#E17"><i><font color="Green">Lemma54</font></i></a>, <a class="ref" href="aff_4.html#T14" target="_self">Th14</a>, <a class="ref" href="aff_4.html#T16" target="_self">Th16</a>, <a class="txt" href="aff_4.html#E12:63"><i><font color="Green">E57</font></i></a>, , </i>;<br/><a NAME="E3:63_2_1"/>
<font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c'</font>
 <b>by </b><i><a class="ref" href="aff_4.html#T12" target="_self">Th12</a>, <a class="txt" href="aff_4.html#E17"><i><font color="Green">Lemma54</font></i></a>, <a class="ref" href="aff_4.html#T14" target="_self">Th14</a>, <a class="ref" href="aff_4.html#T16" target="_self">Th16</a>, <a class="txt" href="aff_4.html#E12:63"><i><font color="Green">E57</font></i></a>, , <a class="txt" href="aff_4.html#E2:63_2"><i><font color="Green">E69</font></i></a>, </i>;<br/><b>hence </b><a NAME="E4:63_2_1"/>
<font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b'</font>,<font color="Maroon">c'</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E3:63_2"><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">E73</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a'</font>
 ;<br/><b>assume </b><a NAME="E2:63_2_2"/><i><font color="Green">E74</font></i>: 
not <font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b'</font>,<font color="Maroon">c'</font>
 ;<br/><a NAME="E3:63_2_2"/><i><font color="Green">E75</font></i>: 
( <font color="Maroon">a'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b'</font> &amp; <font color="Maroon">a'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c'</font> &amp; <font color="Maroon">c'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b'</font> )
 <b>by </b><i><a class="ref" href="aff_4.html#T12" target="_self">Th12</a>, <a class="ref" href="aff_4.html#T14" target="_self">Th14</a>, <a class="ref" href="aff_4.html#T16" target="_self">Th16</a>, <a class="txt" href="aff_4.html#E12:63"><i><font color="Green">E57</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#E24"><i><font color="Green">Lemma67</font></i></a>, <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/><b>set </b><font color="Maroon">BC</font> =  <a href="aff_1.html#K2">Line</a> <font color="Maroon">b</font>,<font color="Maroon">c</font>;<br/><b>set </b><font color="Maroon">BC'</font> =  <a href="aff_1.html#K2">Line</a> <font color="Maroon">b'</font>,<font color="Maroon">c'</font>;<br/><b>set </b><font color="Maroon">AB</font> =  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>;<br/><b>set </b><font color="Maroon">AB'</font> =  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a'</font>,<font color="Maroon">b'</font>;<br/><b>set </b><font color="Maroon">AC</font> =  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">c</font>;<br/><b>set </b><font color="Maroon">AC'</font> =  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a'</font>,<font color="Maroon">c'</font>;<br/><a NAME="E4:63_2_2"/><i><font color="Green">E76</font></i>: 
(  <a href="aff_1.html#K2">Line</a> <font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp;  <a href="aff_1.html#K2">Line</a> <font color="Maroon">b'</font>,<font color="Maroon">c'</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp;  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">c</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp;  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a'</font>,<font color="Maroon">c'</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp;  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp;  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a'</font>,<font color="Maroon">b'</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  )
 <b>by </b><i>, , <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">E77</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">c</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">c</font> &amp; <font color="Maroon">a'</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a'</font>,<font color="Maroon">c'</font> &amp; <font color="Maroon">c'</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a'</font>,<font color="Maroon">c'</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">E78</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">b</font>,<font color="Maroon">c</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">b</font>,<font color="Maroon">c</font> &amp; <font color="Maroon">b'</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">b'</font>,<font color="Maroon">c'</font> &amp; <font color="Maroon">c'</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">b'</font>,<font color="Maroon">c'</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">E79</font></i>: 
(  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="aff_1.html#R4">//</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a'</font>,<font color="Maroon">b'</font> &amp;  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">c</font> <a href="aff_1.html#R4">//</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a'</font>,<font color="Maroon">c'</font> )
 <b>by </b><i>, , , <a class="ref" href="aff_1.html#T51">AFF_1:51</a></i>;<br/><a NAME="E8:63_2_2"/><i><font color="Green">E80</font></i>: 
(  <a href="aff_1.html#K2">Line</a> <font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> &amp;  <a href="aff_1.html#K2">Line</a> <font color="Maroon">b'</font>,<font color="Maroon">c'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> )
 <b>by </b><i><a class="ref" href="aff_4.html#T14" target="_self">Th14</a>, <a class="txt" href="aff_4.html#E13:63"><i><font color="Green">E59</font></i></a>, <a class="txt" href="aff_4.html#E14:63"><i><font color="Green">E60</font></i></a>, , , </i>;<br/><a NAME="E9:63_2_2"/><i><font color="Green">E81</font></i>: 
(  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">c</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> &amp;  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a'</font>,<font color="Maroon">c'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> )
 <b>by </b><i><a class="ref" href="aff_4.html#T14" target="_self">Th14</a>, <a class="txt" href="aff_4.html#E15:63"><i><font color="Green">E61</font></i></a>, <a class="txt" href="aff_4.html#E16:63"><i><font color="Green">E62</font></i></a>, , , </i>;<br/><a NAME="E10:63_2_2"/>
not  <a href="aff_1.html#K2">Line</a> <font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="aff_1.html#R4">//</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">b'</font>,<font color="Maroon">c'</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E3:63_2"><i><font color="Green">E70</font></i></a>, <a class="txt" href="aff_4.html#E2:63_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">p</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font><b> such that </b><br/><a NAME="E12:63_2_2"/><i><font color="Green">E82</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">b</font>,<font color="Maroon">c</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">b'</font>,<font color="Maroon">c'</font> )
 <b>by </b><i><a class="txt" href="aff_4.html#E14:63"><i><font color="Green">E60</font></i></a>, , <a class="txt" href="aff_4.html#E4:63_2_2"><i><font color="Green">E76</font></i></a>, </i>;<br/><b>set </b><font color="Maroon">K</font> = <font color="Maroon">p</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">a</font>,<font color="Maroon">b</font></span>)</span>;<br/><a NAME="E13:63_2_2"/><i><font color="Green">E83</font></i>: 
( <font color="Maroon">p</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">a</font>,<font color="Maroon">b</font></span>)</span> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">p</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">a</font>,<font color="Maroon">b</font></span>)</span> &amp;  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">p</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">a</font>,<font color="Maroon">b</font></span>)</span> &amp;  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a'</font>,<font color="Maroon">b'</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">p</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">a</font>,<font color="Maroon">b</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">p</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">a</font>,<font color="Maroon">b</font></span>)</span> <a href="aff_1.html#NR2" target="_self">is_line</a> 
 <b>by </b><i>, </i>;<br/>

<b>thus </b><a NAME="E2:63_2_2_1"/>
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">p</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">a</font>,<font color="Maroon">b</font></span>)</span> &amp;  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">p</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">a</font>,<font color="Maroon">b</font></span>)</span> )
 <b>by </b><i>, </i>;<br/>

<b>hence </b><a NAME="E3:63_2_2_1"/>
 <a href="aff_1.html#K2">Line</a> <font color="Maroon">a'</font>,<font color="Maroon">b'</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">p</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">a</font>,<font color="Maroon">b</font></span>)</span>
 <b>by </b><i><a class="txt" href="aff_4.html#E3:63_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">E84</font></i>: 
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c</font>,<font color="Maroon">b</font>,<font color="Maroon">p</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c'</font>,<font color="Maroon">b'</font>,<font color="Maroon">p</font> )
 <b>by </b><i>, <a class="txt" href="aff_4.html#E2:63_2_2"><i><font color="Green">E74</font></i></a>, <a class="txt" href="aff_4.html#E6:63_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">x</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font><b> such that </b><br/><a NAME="E16:63_2_2"/><i><font color="Green">E85</font></i>: 
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c</font>,<font color="Maroon">a</font>,<font color="Maroon">x</font> &amp; <font color="Maroon">b</font>,<font color="Maroon">a</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">p</font>,<font color="Maroon">x</font> )
 <b>by </b><i>, </i>;<br/><b>consider </b><font color="Maroon">y</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font><b> such that </b><br/><a NAME="E18:63_2_2"/><i><font color="Green">E86</font></i>: 
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c'</font>,<font color="Maroon">a'</font>,<font color="Maroon">y</font> &amp; <font color="Maroon">b'</font>,<font color="Maroon">a'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">p</font>,<font color="Maroon">y</font> )
 <b>by </b><i>, <a class="txt" href="aff_4.html#E8:63_2_2"><i><font color="Green">E80</font></i></a>, </i>;<br/><a NAME="E19:63_2_2"/><i><font color="Green">E87</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">p</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">a</font>,<font color="Maroon">b</font></span>)</span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">p</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">a</font>,<font color="Maroon">b</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">p</font>,<font color="Maroon">x</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
 
<b>by </b><i><a class="txt" href="aff_4.html#E9:63_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">p</font>,<font color="Maroon">x</font> <a href="aff_1.html#R2">//</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
 
<b>by </b><i>, <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">p</font>,<font color="Maroon">x</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">p</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">a</font>,<font color="Maroon">b</font></span>)</span>
 
<b>by </b><i><a class="txt" href="aff_4.html#E7:63_2_2"><i><font color="Green">E79</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E4:63_2_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">p</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">a</font>,<font color="Maroon">b</font></span>)</span>
 <b>by </b><i><a class="txt" href="aff_4.html#E7:63_2_2"><i><font color="Green">E79</font></i></a>, </i>;<br/>

<a NAME="E5:63_2_2_2"/>
<font color="Maroon">p</font>,<font color="Maroon">y</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a'</font>,<font color="Maroon">b'</font>
 
<b>by </b><i><a class="txt" href="aff_4.html#E12:63_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">p</font>,<font color="Maroon">y</font> <a href="aff_1.html#R2">//</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a'</font>,<font color="Maroon">b'</font>
 
<b>by </b><i>, <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">p</font>,<font color="Maroon">y</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">p</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">a</font>,<font color="Maroon">b</font></span>)</span>
 
<b>by </b><i><a class="txt" href="aff_4.html#E7:63_2_2"><i><font color="Green">E79</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E8:63_2_2_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">p</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">a</font>,<font color="Maroon">b</font></span>)</span>
 <b>by </b><i><a class="txt" href="aff_4.html#E7:63_2_2"><i><font color="Green">E79</font></i></a>, </i>;<br/>


</div><b>end;</b></div><a NAME="E20:63_2_2"/><i><font color="Green">E88</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">c</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a'</font>,<font color="Maroon">c'</font> )
 <b>by </b><i>, , , <a class="txt" href="aff_4.html#E1:63_2_2"><i><font color="Green">E73</font></i></a>, <a class="txt" href="aff_4.html#E9:63_2_2"><i><font color="Green">E81</font></i></a>, <a class="txt" href="aff_4.html#E12:63_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">E89</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">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 ;<br/><a NAME="E2:63_2_2_3"/><b>then </b>
<font color="Maroon">a'</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">c</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E1:63_2_2"><i><font color="Green">E73</font></i></a>, <a class="txt" href="aff_4.html#E3:63_2_2"><i><font color="Green">E75</font></i></a>, <a class="txt" href="aff_4.html#E14:63_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</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="aff_4.html#T14" target="_self">Th14</a>, <a class="ref" href="aff_4.html#T16" target="_self">Th16</a>, <a class="txt" href="aff_4.html#E2:63_2"><i><font color="Green">E69</font></i></a>, , <a class="txt" href="aff_4.html#E1:63_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="ref" href="aff_4.html#T12" target="_self">Th12</a>, <a class="txt" href="aff_4.html#E17"><i><font color="Green">Lemma54</font></i></a>, <a class="ref" href="aff_4.html#T14" target="_self">Th14</a>, <a class="ref" href="aff_4.html#T16" target="_self">Th16</a>, <a class="txt" href="aff_4.html#E12:63"><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">E90</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font>
 ;<br/><a NAME="E2:63_2_2_4"/><b>then </b>
<font color="Maroon">p</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">a</font>,<font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E7:63_2_2"><i><font color="Green">E79</font></i></a>, <a class="txt" href="aff_4.html#E13:63_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">p</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">a</font>,<font color="Maroon">b</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E16:63"><i><font color="Green">E62</font></i></a>, <a class="txt" href="aff_4.html#E5:63_2_2"><i><font color="Green">E77</font></i></a>, <a class="txt" href="aff_4.html#E14:63_2_2"><i><font color="Green">E84</font></i></a>, <a class="txt" href="aff_4.html#E18:63_2_2"><i><font color="Green">E86</font></i></a>, </i>;<br/><a NAME="E4:63_2_2_4"/><b>then </b><i><font color="Green">E91</font></i>: 
 <a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="aff_4.html#T14" target="_self">Th14</a>, <a class="txt" href="aff_4.html#E15:63"><i><font color="Green">E61</font></i></a>, <a class="txt" href="aff_4.html#E16:63"><i><font color="Green">E62</font></i></a>, <a class="txt" href="aff_4.html#E1:63_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">E79</font></i></a>, </i>;<br/><a NAME="E5:63_2_2_4"/>
<font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">q</font>,<font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="aff_4.html#T12" target="_self">Th12</a>, <a class="txt" href="aff_4.html#E17"><i><font color="Green">Lemma54</font></i></a>, <a class="ref" href="aff_4.html#T14" target="_self">Th14</a>, <a class="ref" href="aff_4.html#T16" target="_self">Th16</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">P</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="aff_4.html#T12" target="_self">Th12</a>, <a class="txt" href="aff_4.html#E17"><i><font color="Green">Lemma54</font></i></a>, <a class="txt" href="aff_4.html#E15:63"><i><font color="Green">E61</font></i></a>, <a class="txt" href="aff_4.html#E16:63"><i><font color="Green">E62</font></i></a>, <a class="txt" href="aff_4.html#E1:63_2_2"><i><font color="Green">E73</font></i></a>, <a class="txt" href="aff_4.html#E19:63_2_2"><i><font color="Green">E87</font></i></a>, </i>;<br/><b>hence </b><a NAME="E7:63_2_2_4"/>
contradiction
 <b>by </b><i><a class="ref" href="aff_4.html#D2" target="_self">Def2</a>, <a class="txt" href="aff_4.html#E15:63"><i><font color="Green">E61</font></i></a>, <a class="txt" href="aff_4.html#E16:63"><i><font color="Green">E62</font></i></a>, <a class="ref" href="aff_4.html#T8" target="_self">Th8</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#E16:63_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">b</font>,<font color="Maroon">c</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b'</font>,<font color="Maroon">c'</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E25"><i><font color="Green">Lemma68</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E18:63"/>
<font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b'</font>,<font color="Maroon">c'</font>
 <b>by </b><i><a class="ref" href="aff_4.html#T18" target="_self">Th18</a></i>;<br/>


</div>
