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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">o</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">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">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>;<br/>













<b>assume </b><a NAME="E1:68"/><i><font color="Green">E27</font></i>: 
( not  <a href="aff_1.html#R1" target="_self">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">a</font>,<font color="Maroon">b</font> &amp;  <a href="aff_1.html#R1" target="_self">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">a</font>,<font color="Maroon">a'</font> &amp;  <a href="aff_1.html#R1" target="_self">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">b</font>,<font color="Maroon">b'</font> &amp;  <a href="aff_1.html#R1" target="_self">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">b</font>,<font color="Maroon">x</font> &amp; <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">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a'</font>,<font color="Maroon">x</font> )
 ;<br/>

<a NAME="E2:68"/><b>then </b><i><font color="Green">E39</font></i>: 
( <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font> &amp; <font color="Maroon">o</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">b</font> )
 
<b>by </b><i/>;<br/>
<b>assume </b><a NAME="E3:68"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">b'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font>
 ;<br/>

<a NAME="E4:68"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">a'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:68_1"/>
<font color="Maroon">a'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b'</font>
 ;<br/>

<a NAME="E2:68_1"/><b>then </b>
( <font color="Maroon">a'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font> &amp; <font color="Maroon">b'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font> )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E3:68_1"/>
contradiction
 <b>by </b><i>, , <a class="txt" href="aff_1.html#E5:68"><i><font color="Green">E52</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:68"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">a'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">o</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:68_2"/>
<font color="Maroon">a'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font>
 ;<br/>

<a NAME="E2:68_2"/><b>then </b>
( <font color="Maroon">b'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font> )
 
<b>by </b><i>, <a class="txt" href="aff_1.html#E5:68"><i><font color="Green">E52</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:68_2"/>
contradiction
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E6:68"/>
<font color="Maroon">a'</font>,<font color="Maroon">b'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a'</font>,<font color="Maroon">x</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E7:68"/><b>then </b><i><font color="Green">E68</font></i>: 
 <a href="aff_1.html#R1" target="_self">LIN</a> <font color="Maroon">a'</font>,<font color="Maroon">b'</font>,<font color="Maroon">x</font>
 
<b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">A</font> =  <a href="aff_1.html#K2" target="_self">Line</a> <font color="Maroon">o</font>,<font color="Maroon">a</font>;<br/>
<b>set </b><font color="Maroon">C</font> =  <a href="aff_1.html#K2" target="_self">Line</a> <font color="Maroon">o</font>,<font color="Maroon">b</font>;<br/>
<b>set </b><font color="Maroon">P</font> =  <a href="aff_1.html#K2" target="_self">Line</a> <font color="Maroon">a'</font>,<font color="Maroon">b'</font>;<br/>
<a NAME="E8:68"/><i><font color="Green">E69</font></i>: 
(  <a href="aff_1.html#K2" target="_self">Line</a> <font color="Maroon">o</font>,<font color="Maroon">a</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp;  <a href="aff_1.html#K2" target="_self">Line</a> <font color="Maroon">o</font>,<font color="Maroon">b</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp;  <a href="aff_1.html#K2" target="_self">Line</a> <font color="Maroon">a'</font>,<font color="Maroon">b'</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">o</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2" target="_self">Line</a> <font color="Maroon">o</font>,<font color="Maroon">a</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2" target="_self">Line</a> <font color="Maroon">o</font>,<font color="Maroon">a</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2" target="_self">Line</a> <font color="Maroon">o</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" target="_self">Line</a> <font color="Maroon">o</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" target="_self">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" target="_self">Line</a> <font color="Maroon">a'</font>,<font color="Maroon">b'</font> )
 
<b>by </b><i>, , , </i>;<br/>
<a NAME="E9:68"/><b>then </b><i><font color="Green">E70</font></i>: 
( <font color="Maroon">a'</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2" target="_self">Line</a> <font color="Maroon">o</font>,<font color="Maroon">a</font> &amp; <font color="Maroon">b'</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2" target="_self">Line</a> <font color="Maroon">o</font>,<font color="Maroon">b</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2" target="_self">Line</a> <font color="Maroon">o</font>,<font color="Maroon">b</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2" target="_self">Line</a> <font color="Maroon">a'</font>,<font color="Maroon">b'</font> )
 
<b>by </b><i>, , , <a class="ref" href="aff_1.html#T25" target="_self">Th25</a>, <a class="ref" href="aff_1.html#T14" target="_self">Th14</a></i>;<br/>
<a NAME="E10:68"/><b>then </b>
<font color="Maroon">a'</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2" target="_self">Line</a> <font color="Maroon">o</font>,<font color="Maroon">b</font>
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T26" target="_self">Th26</a>, <a class="ref" href="aff_1.html#T10" target="_self">Th10</a></i>;<br/>
<a NAME="E11:68"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2" target="_self">Line</a> <font color="Maroon">o</font>,<font color="Maroon">a</font>
 
<b>by </b><i><a class="txt" href="aff_1.html#E30"><i><font color="Green">Lemma57</font></i></a>, <a class="ref" href="aff_1.html#T26" target="_self">Th26</a>, <a class="ref" href="aff_1.html#T27" target="_self">Th27</a>, <a class="ref" href="aff_1.html#T10" target="_self">Th10</a></i>;<br/>
<b>hence </b><a NAME="E12:68"/>
contradiction
 <b>by </b><i>, <a class="ref" href="aff_1.html#T26" target="_self">Th26</a>, <a class="txt" href="aff_1.html#E13"><i><font color="Green">Lemma29</font></i></a></i>;<br/>


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