<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">AS</font> be   <a href="diraf.html#NM1">AffinSpace</a>;<br/><b>let </b><font color="Maroon">X</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AS</font>;<br/><b>let </b><font color="Maroon">Y</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AS</font>;<br/>





<b>assume </b><a NAME="E1:72"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">X</font> <a href="aff_4.html#NR1" target="_self">is_plane</a>  &amp; <font color="Maroon">Y</font> <a href="aff_4.html#NR1" target="_self">is_plane</a>  )
 ;<br/>

<div><i><font color="Green">E35</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="72_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:72_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">X</font> <a href="aff_4.html#R1" target="_self">'||'</a> <font color="Maroon">Y</font>
 ;<br/><b>consider </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>, <font color="Maroon">b</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>, <font color="Maroon">c</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font><b> such that </b><br/><a NAME="E3:72_1"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 <b>and </b><br/><a NAME="E4:72_1"/><i><font color="Green">E43</font></i>: 
not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>
 <b>by </b><i>, <a class="ref" href="aff_4.html#T3" target="_self">Th3</a></i>;<br/><a NAME="E5:72_1"/><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">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">b</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_1.html#T16">AFF_1:16</a></i>;<br/><b>set </b><font color="Maroon">A</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">P</font> =  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">c</font>;<br/><a NAME="E6:72_1"/><i><font color="Green">E45</font></i>: 
( <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">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">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;  <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">c</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  )
 <b>by </b><i>, <a class="ref" href="aff_1.html#T26">AFF_1:26</a>, <a class="ref" href="aff_1.html#D3">AFF_1:def 3</a></i>;<br/><b>consider </b><font color="Maroon">a'</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>, <font color="Maroon">b'</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>, <font color="Maroon">c'</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font><b> such that </b><br/><a NAME="E8:72_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">a'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>and </b><br/><a NAME="E9:72_1"/>
( <font color="Maroon">b'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">c'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a'</font>,<font color="Maroon">b'</font>,<font color="Maroon">c'</font> )
 <b>by </b><i>, <a class="ref" href="aff_4.html#T3" target="_self">Th3</a></i>;<br/><b>take </b><font color="Maroon">A</font> =  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>;<br/><b>take </b><font color="Maroon">P</font> =  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">c</font>;<br/><b>take </b><font color="Maroon">M</font> = <font color="Maroon">a'</font> <a href="aff_4.html#K2" target="_self">*</a> <font color="Maroon">A</font>;<br/><b>take </b><font color="Maroon">N</font> = <font color="Maroon">a'</font> <a href="aff_4.html#K2" target="_self">*</a> <font color="Maroon">P</font>;<br/><a NAME="E10:72_1"/><i><font color="Green">E49</font></i>: 
not <font color="Maroon">A</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">P</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:72_1_1"/>
<font color="Maroon">A</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">P</font>
 ;<br/>

<a NAME="E2:72_1_1"/><b>then </b>
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="ref" href="aff_4.html#T18" target="_self">Th18</a>, <a class="ref" href="aff_1.html#T59">AFF_1:59</a></i>;<br/>
<b>hence </b><a NAME="E3:72_1_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="aff_4.html#T18" target="_self">Th18</a>, <a class="ref" href="aff_1.html#T33">AFF_1:33</a></i>;<br/>


</div><b>end;</b></div><a NAME="E11:72_1"/>
( <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> &amp; <font color="Maroon">P</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> )
 <b>by </b><i>, , , </i>;<br/><b>hence </b><a NAME="E12:72_1"/>
( not <font color="Maroon">A</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">P</font> &amp; <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> &amp; <font color="Maroon">P</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> &amp; <font color="Maroon">M</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">N</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> &amp; ( <font color="Maroon">A</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">M</font> or <font color="Maroon">M</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">A</font> ) &amp; ( <font color="Maroon">P</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">N</font> or <font color="Maroon">N</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">P</font> ) )
 <b>by </b><i>, <a class="ref" href="aff_4.html#T18" target="_self">Th18</a>, , <a class="txt" href="aff_4.html#E24"><i><font color="Green">Lemma67</font></i></a>, , <a class="txt" href="aff_4.html#E3:72_1"><i><font color="Green">E37</font></i></a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_2"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">A</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AS</font>, <font color="Maroon">P</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AS</font>, <font color="Maroon">M</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AS</font>, <font color="Maroon">N</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AS</font><b> such that </b><a NAME="E1:72_2"/><i><font color="Green">E57</font></i>: 
not <font color="Maroon">A</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">P</font>
 <b>and </b><br/><a NAME="E2:72_2"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> &amp; <font color="Maroon">P</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> &amp; <font color="Maroon">M</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">N</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> )
 <b>and </b><br/><a NAME="E3:72_2"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">A</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">M</font> or <font color="Maroon">M</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">A</font> )
 <b>and </b><br/><a NAME="E4:72_2"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">P</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">N</font> or <font color="Maroon">N</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">P</font> )
 ;<br/><a NAME="E5:72_2"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">A</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">M</font> &amp; <font color="Maroon">P</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">N</font> )
 <b>by </b><i><a class="txt" href="aff_4.html#E6:72_2"><i><font color="Green">E63</font></i></a>, <a class="txt" href="aff_4.html#E8:72_2"><i><font color="Green">E64</font></i></a></i>;<br/><a NAME="E6:72_2"/><i><font color="Green">E63</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">M</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">N</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  )
 <b>by </b><i><a class="txt" href="aff_4.html#E6:72_2"><i><font color="Green">E63</font></i></a>, <a class="txt" href="aff_4.html#E8:72_2"><i><font color="Green">E64</font></i></a>, <a class="ref" href="aff_1.html#T50">AFF_1:50</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="E8:72_2"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> )
 <b>by </b><i><a class="txt" href="aff_4.html#E2:72_2"><i><font color="Green">E59</font></i></a>, <a class="txt" href="aff_4.html#E4:72_2"><i><font color="Green">E61</font></i></a>, <a class="txt" href="aff_4.html#E5:72_2"><i><font color="Green">E62</font></i></a>, </i>;<br/><a NAME="E9:72_2"/>
not <font color="Maroon">M</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">N</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:72_2_1"/>
<font color="Maroon">M</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">N</font>
 ;<br/>

<a NAME="E2:72_2_1"/><b>then </b>
<font color="Maroon">P</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">M</font>
 
<b>by </b><i><a class="txt" href="aff_4.html#E8:72_2"><i><font color="Green">E64</font></i></a>, <a class="ref" href="aff_1.html#T58">AFF_1:58</a></i>;<br/>
<b>hence </b><a NAME="E3:72_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_4.html#E4:72_2"><i><font color="Green">E61</font></i></a>, <a class="txt" href="aff_4.html#E6:72_2"><i><font color="Green">E63</font></i></a>, <a class="ref" href="aff_1.html#T58">AFF_1:58</a></i>;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon">q</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font><b> such that </b><br/><a NAME="E11:72_2"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> )
 <b>by </b><i><a class="txt" href="aff_4.html#E2:72_2"><i><font color="Green">E59</font></i></a>, <a class="txt" href="aff_4.html#E5:72_2"><i><font color="Green">E62</font></i></a>, <a class="txt" href="aff_4.html#E11:72_2"><i><font color="Green">E66</font></i></a>, </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_2_2"><b>now </b></a><div class="add"><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">Z</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AS</font>;<br/><b>assume </b><a NAME="E1:72_2_2"/><i><font color="Green">E69</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">Z</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> )
 ;<br/><a NAME="E2:72_2_2"/><b>then </b><i><font color="Green">E70</font></i>: 
( <font color="Maroon">Z</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">a</font> <a href="aff_4.html#K2" target="_self">*</a> <font color="Maroon">Z</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> <a href="aff_4.html#K2" target="_self">*</a> <font color="Maroon">Z</font> )
 <b>by </b><i/>;<br/><div><i><font color="Green">E71</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="72_2_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:72_2_2_1"/>
<font color="Maroon">Z</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">A</font>
 ;<br/><a NAME="E2:72_2_2_1"/><b>then </b>
<font color="Maroon">Z</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">M</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E6:72_2"><i><font color="Green">E63</font></i></a>, <a class="ref" href="aff_1.html#T58">AFF_1:58</a></i>;<br/><a NAME="E3:72_2_2_1"/><b>then </b>
<font color="Maroon">a</font> <a href="aff_4.html#K2" target="_self">*</a> <font color="Maroon">Z</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">M</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E2:72_2_2"><i><font color="Green">E70</font></i></a>, <a class="ref" href="aff_1.html#T58">AFF_1:58</a></i>;<br/><b>hence </b><a NAME="E4:72_2_2_1"/>
<font color="Maroon">a</font> <a href="aff_4.html#K2" target="_self">*</a> <font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E2:72_2"><i><font color="Green">E59</font></i></a>, <a class="txt" href="aff_4.html#E5:72_2"><i><font color="Green">E62</font></i></a>, <a class="txt" href="aff_4.html#E1:72_2_2"><i><font color="Green">E69</font></i></a>, <a class="txt" href="aff_4.html#E2:72_2_2"><i><font color="Green">E70</font></i></a>, </i>;<br/></div><b>end;</b></div><div><i><font color="Green">E72</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="72_2_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:72_2_2_2"/>
<font color="Maroon">Z</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">P</font>
 ;<br/><a NAME="E2:72_2_2_2"/><b>then </b>
<font color="Maroon">Z</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">N</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E8:72_2"><i><font color="Green">E64</font></i></a>, <a class="ref" href="aff_1.html#T58">AFF_1:58</a></i>;<br/><a NAME="E3:72_2_2_2"/><b>then </b>
<font color="Maroon">a</font> <a href="aff_4.html#K2" target="_self">*</a> <font color="Maroon">Z</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">N</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E2:72_2_2"><i><font color="Green">E70</font></i></a>, <a class="ref" href="aff_1.html#T58">AFF_1:58</a></i>;<br/><b>hence </b><a NAME="E4:72_2_2_2"/>
<font color="Maroon">a</font> <a href="aff_4.html#K2" target="_self">*</a> <font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E2:72_2"><i><font color="Green">E59</font></i></a>, <a class="txt" href="aff_4.html#E5:72_2"><i><font color="Green">E62</font></i></a>, <a class="txt" href="aff_4.html#E1:72_2_2"><i><font color="Green">E69</font></i></a>, <a class="txt" href="aff_4.html#E2:72_2_2"><i><font color="Green">E70</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_2_2_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:72_2_2_3"/><i><font color="Green">E73</font></i>: 
( not <font color="Maroon">Z</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">A</font> &amp; not <font color="Maroon">Z</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">P</font> )
 ;<br/><b>consider </b><font color="Maroon">b</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font><b> such that </b><br/><a NAME="E3:72_2_2_3"/><i><font color="Green">E74</font></i>: 
( <font color="Maroon">p</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#R2">in</a> <font color="Maroon">A</font> )
 <b>by </b><i><a class="txt" href="aff_4.html#E11:72_2"><i><font color="Green">E66</font></i></a>, <a class="ref" href="aff_1.html#T32">AFF_1:32</a></i>;<br/><b>set </b><font color="Maroon">Z1</font> = <font color="Maroon">b</font> <a href="aff_4.html#K2" target="_self">*</a> <font color="Maroon">Z</font>;<br/><a NAME="E4:72_2_2_3"/><i><font color="Green">E75</font></i>: 
( <font color="Maroon">b</font> <a href="aff_4.html#K2" target="_self">*</a> <font color="Maroon">Z</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b</font> <a href="aff_4.html#K2" target="_self">*</a> <font color="Maroon">Z</font> &amp; <font color="Maroon">Z</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">b</font> <a href="aff_4.html#K2" target="_self">*</a> <font color="Maroon">Z</font> )
 <b>by </b><i><a class="txt" href="aff_4.html#E1:72_2_2"><i><font color="Green">E69</font></i></a>, , </i>;<br/><a NAME="E5:72_2_2_3"/><i><font color="Green">E76</font></i>: 
not <font color="Maroon">b</font> <a href="aff_4.html#K2" target="_self">*</a> <font color="Maroon">Z</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">P</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_2_2_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:72_2_2_3_1"/>
<font color="Maroon">b</font> <a href="aff_4.html#K2" target="_self">*</a> <font color="Maroon">Z</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">P</font>
 ;<br/>

<a NAME="E2:72_2_2_3_1"/><b>then </b>
( <font color="Maroon">b</font> <a href="aff_4.html#K2" target="_self">*</a> <font color="Maroon">Z</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">P</font> &amp; <font color="Maroon">Z</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">b</font> <a href="aff_4.html#K2" target="_self">*</a> <font color="Maroon">Z</font> )
 
<b>by </b><i><a class="txt" href="aff_4.html#E1:72_2_2"><i><font color="Green">E69</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E3:72_2_2_3_1"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_4.html#E1:72_2_2_3"><i><font color="Green">E73</font></i></a>, <a class="ref" href="aff_1.html#T58">AFF_1:58</a></i>;<br/>


</div><b>end;</b></div><a NAME="E6:72_2_2_3"/>
<font color="Maroon">b</font> <a href="aff_4.html#K2" target="_self">*</a> <font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E2:72_2"><i><font color="Green">E59</font></i></a>, <a class="txt" href="aff_4.html#E5:72_2"><i><font color="Green">E62</font></i></a>, <a class="txt" href="aff_4.html#E1:72_2_2"><i><font color="Green">E69</font></i></a>, <a class="txt" href="aff_4.html#E3:72_2_2_3"><i><font color="Green">E74</font></i></a>, <a class="txt" href="aff_4.html#E1"><i><font color="Green">Lemma32</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">c</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font><b> such that </b><br/><a NAME="E8:72_2_2_3"/><i><font color="Green">E77</font></i>: 
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b</font> <a href="aff_4.html#K2" target="_self">*</a> <font color="Maroon">Z</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> )
 <b>by </b><i><a class="txt" href="aff_4.html#E2:72_2"><i><font color="Green">E59</font></i></a>, <a class="txt" href="aff_4.html#E5:72_2"><i><font color="Green">E62</font></i></a>, <a class="txt" href="aff_4.html#E11:72_2"><i><font color="Green">E66</font></i></a>, <a class="txt" href="aff_4.html#E4:72_2_2_3"><i><font color="Green">E75</font></i></a>, <a class="txt" href="aff_4.html#E5:72_2_2_3"><i><font color="Green">E76</font></i></a>, </i>;<br/><a NAME="E9:72_2_2_3"/><i><font color="Green">E78</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="txt" href="aff_4.html#E11:72_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="txt" href="aff_4.html#E1:72_2_2_3"><i><font color="Green">E73</font></i></a>, <a class="txt" href="aff_4.html#E3:72_2_2_3"><i><font color="Green">E74</font></i></a>, <a class="txt" href="aff_4.html#E4:72_2_2_3"><i><font color="Green">E75</font></i></a>, <a class="txt" href="aff_4.html#E8:72_2_2_3"><i><font color="Green">E77</font></i></a>, <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/><b>consider </b><font color="Maroon">b'</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font><b> such that </b><br/><a NAME="E11:72_2_2_3"/><i><font color="Green">E79</font></i>: 
( <font color="Maroon">q</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#R2">in</a> <font color="Maroon">M</font> )
 <b>by </b><i><a class="txt" href="aff_4.html#E11:72_2"><i><font color="Green">E66</font></i></a>, <a class="ref" href="aff_1.html#T32">AFF_1:32</a></i>;<br/><a NAME="E12:72_2_2_3"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">p</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">q</font>,<font color="Maroon">b'</font>
 <b>by </b><i><a class="ref" href="aff_4.html#T18" target="_self">Th18</a>, <a class="txt" href="aff_4.html#E24"><i><font color="Green">Lemma67</font></i></a>, <a class="txt" href="aff_4.html#E25"><i><font color="Green">Lemma68</font></i></a>, <a class="txt" href="aff_4.html#E3:72_2_2_3"><i><font color="Green">E74</font></i></a>, <a class="txt" href="aff_4.html#E11:72_2_2_3"><i><font color="Green">E79</font></i></a>, <a class="ref" href="aff_1.html#T53">AFF_1:53</a></i>;<br/><a NAME="E13:72_2_2_3"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E4:72_2"><i><font color="Green">E61</font></i></a>, <a class="txt" href="aff_4.html#E11:72_2"><i><font color="Green">E66</font></i></a>, <a class="ref" href="aff_1.html#T55">AFF_1:55</a></i>;<br/><a NAME="E14:72_2_2_3"/><i><font color="Green">E82</font></i>: 
not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">p</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_2_2_3_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:72_2_2_3_2"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">p</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>
 ;<br/>

<a NAME="E2:72_2_2_3_2"/><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="txt" href="aff_4.html#E11:72_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="txt" href="aff_4.html#E3:72_2_2_3"><i><font color="Green">E74</font></i></a>, <a class="ref" href="aff_1.html#T39">AFF_1:39</a></i>;<br/>
<b>hence </b><a NAME="E3:72_2_2_3_2"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_4.html#E11:72_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="txt" href="aff_4.html#E8:72_2_2_3"><i><font color="Green">E77</font></i></a>, <a class="txt" href="aff_4.html#E9:72_2_2_3"><i><font color="Green">E78</font></i></a>, <a class="txt" href="aff_4.html#E13:72_2_2_3"><i><font color="Green">E81</font></i></a>, <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon">c'</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font><b> such that </b><br/><a NAME="E16:72_2_2_3"/><i><font color="Green">E83</font></i>: 
( <font color="Maroon">p</font>,<font color="Maroon">c</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">q</font>,<font color="Maroon">c'</font> &amp; <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#E11:72_2_2_3"><i><font color="Green">E79</font></i></a>, <a class="txt" href="aff_4.html#E12:72_2_2_3"><i><font color="Green">E80</font></i></a>, <a class="txt" href="aff_4.html#E1:72_2"><i><font color="Green">E57</font></i></a></i>;<br/><a NAME="E17:72_2_2_3"/><i><font color="Green">E84</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> )
 <b>by </b><i><a class="txt" href="aff_4.html#E14:72_2_2_3"><i><font color="Green">E82</font></i></a>, <a class="ref" href="aff_1.html#T16">AFF_1:16</a></i>;<br/><a NAME="E18:72_2_2_3"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">c'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E8:72_2"><i><font color="Green">E64</font></i></a>, <a class="txt" href="aff_4.html#E24"><i><font color="Green">Lemma67</font></i></a>, <a class="txt" href="aff_4.html#E25"><i><font color="Green">Lemma68</font></i></a>, <a class="txt" href="aff_4.html#E8:72_2_2_3"><i><font color="Green">E77</font></i></a>, <a class="txt" href="aff_4.html#E9:72_2_2_3"><i><font color="Green">E78</font></i></a>, <a class="txt" href="aff_4.html#E16:72_2_2_3"><i><font color="Green">E83</font></i></a>, </i>;<br/><b>set </b><font color="Maroon">Z2</font> =  <a href="aff_1.html#K2">Line</a> <font color="Maroon">b'</font>,<font color="Maroon">c'</font>;<br/><a NAME="E19:72_2_2_3"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">b'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_2_2_3_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:72_2_2_3_3"/>
<font color="Maroon">b'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c'</font>
 ;<br/>

<a NAME="E2:72_2_2_3_3"/><b>then </b>
( <font color="Maroon">p</font>,<font color="Maroon">c</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">q</font>,<font color="Maroon">b'</font> &amp; <font color="Maroon">q</font>,<font color="Maroon">b'</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">M</font> )
 
<b>by </b><i><a class="txt" href="aff_4.html#E11:72_2"><i><font color="Green">E66</font></i></a>, <a class="txt" href="aff_4.html#E25"><i><font color="Green">Lemma68</font></i></a>, <a class="txt" href="aff_4.html#E11:72_2_2_3"><i><font color="Green">E79</font></i></a>, <a class="txt" href="aff_4.html#E16:72_2_2_3"><i><font color="Green">E83</font></i></a>, <a class="ref" href="aff_1.html#T66">AFF_1:66</a></i>;<br/>
<a NAME="E3:72_2_2_3_3"/><b>then </b>
<font color="Maroon">p</font>,<font color="Maroon">c</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">M</font>
 
<b>by </b><i><a class="txt" href="aff_4.html#E11:72_2_2_3"><i><font color="Green">E79</font></i></a>, </i>;<br/>
<a NAME="E4:72_2_2_3_3"/><b>then </b>
<font color="Maroon">p</font>,<font color="Maroon">c</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="txt" href="aff_4.html#E6:72_2"><i><font color="Green">E63</font></i></a>, </i>;<br/>
<a NAME="E5:72_2_2_3_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="txt" href="aff_4.html#E24"><i><font color="Green">Lemma67</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E6:72_2_2_3_3"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_4.html#E11:72_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="txt" href="aff_4.html#E8:72_2_2_3"><i><font color="Green">E77</font></i></a>, <a class="txt" href="aff_4.html#E9:72_2_2_3"><i><font color="Green">E78</font></i></a>, <a class="txt" href="aff_4.html#E13:72_2_2_3"><i><font color="Green">E81</font></i></a>, <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/>


</div><b>end;</b></div><a NAME="E20:72_2_2_3"/><b>then </b>
(  <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; <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>, <a class="ref" href="aff_1.html#D3">AFF_1:def 3</a></i>;<br/><a NAME="E21:72_2_2_3"/><b>then </b>
<font color="Maroon">b</font> <a href="aff_4.html#K2" target="_self">*</a> <font color="Maroon">Z</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#E4:72_2_2_3"><i><font color="Green">E75</font></i></a>, <a class="txt" href="aff_4.html#E8:72_2_2_3"><i><font color="Green">E77</font></i></a>, <a class="txt" href="aff_4.html#E16:72_2_2_3"><i><font color="Green">E83</font></i></a>, <a class="txt" href="aff_4.html#E17:72_2_2_3"><i><font color="Green">E84</font></i></a>, <a class="txt" href="aff_4.html#E19:72_2_2_3"><i><font color="Green">E86</font></i></a>, <a class="ref" href="aff_1.html#T52">AFF_1:52</a></i>;<br/><a NAME="E22:72_2_2_3"/><b>then </b><i><font color="Green">E87</font></i>: 
<font color="Maroon">Z</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#E4:72_2_2_3"><i><font color="Green">E75</font></i></a>, <a class="ref" href="aff_1.html#T58">AFF_1:58</a></i>;<br/><a NAME="E23:72_2_2_3"/><i><font color="Green">E88</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">Y</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E2:72_2"><i><font color="Green">E59</font></i></a>, <a class="txt" href="aff_4.html#E5:72_2"><i><font color="Green">E62</font></i></a>, <a class="txt" href="aff_4.html#E11:72_2_2_3"><i><font color="Green">E79</font></i></a>, <a class="txt" href="aff_4.html#E18:72_2_2_3"><i><font color="Green">E85</font></i></a>, <a class="txt" href="aff_4.html#E19:72_2_2_3"><i><font color="Green">E86</font></i></a>, </i>;<br/><a NAME="E24:72_2_2_3"/>
<font color="Maroon">a</font> <a href="aff_4.html#K2" target="_self">*</a> <font color="Maroon">Z</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#E2:72_2_2"><i><font color="Green">E70</font></i></a>, <a class="txt" href="aff_4.html#E22:72_2_2_3"><i><font color="Green">E87</font></i></a>, <a class="ref" href="aff_1.html#T58">AFF_1:58</a></i>;<br/><b>hence </b><a NAME="E25:72_2_2_3"/>
<font color="Maroon">a</font> <a href="aff_4.html#K2" target="_self">*</a> <font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E2:72_2"><i><font color="Green">E59</font></i></a>, <a class="txt" href="aff_4.html#E1:72_2_2"><i><font color="Green">E69</font></i></a>, <a class="txt" href="aff_4.html#E2:72_2_2"><i><font color="Green">E70</font></i></a>, <a class="txt" href="aff_4.html#E23:72_2_2_3"><i><font color="Green">E88</font></i></a>, </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:72_2_2"/>
<font color="Maroon">a</font> <a href="aff_4.html#K2" target="_self">*</a> <font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E3:72_2_2"><i><font color="Green">E71</font></i></a>, <a class="txt" href="aff_4.html#E4:72_2_2"><i><font color="Green">E72</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E13:72_2"/>
<font color="Maroon">X</font> <a href="aff_4.html#R1" target="_self">'||'</a> <font color="Maroon">Y</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:72"/>
( <font color="Maroon">X</font> <a href="aff_4.html#R1" target="_self">'||'</a> <font color="Maroon">Y</font> iff  ex <font color="Olive">A</font>, <font color="Olive">P</font>, <font color="Olive">M</font>, <font color="Olive">N</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AS</font> st <br/>( not <font color="Olive">A</font> <a href="aff_1.html#R4">//</a> <font color="Olive">P</font> &amp; <font color="Olive">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> &amp; <font color="Olive">P</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> &amp; <font color="Olive">M</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> &amp; <font color="Olive">N</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> &amp; ( <font color="Olive">A</font> <a href="aff_1.html#R4">//</a> <font color="Olive">M</font> or <font color="Olive">M</font> <a href="aff_1.html#R4">//</a> <font color="Olive">A</font> ) &amp; ( <font color="Olive">P</font> <a href="aff_1.html#R4">//</a> <font color="Olive">N</font> or <font color="Olive">N</font> <a href="aff_1.html#R4">//</a> <font color="Olive">P</font> ) ) )
 <b>by </b><i/>;<br/>


</div>
