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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">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">b'</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>;<br/><b>let </b><font color="Maroon">M</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AS</font>;<br/><b>let </b><font color="Maroon">N</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="E1:10"/><i><font color="Green">E34</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> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">a'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> &amp; <font color="Maroon">b'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> )
 <b>and </b><br/><a NAME="E2:10"/><i><font color="Green">E35</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">M</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">N</font> )
 <b>and </b><br/><a NAME="E3:10"/><i><font color="Green">E36</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> or <font color="Maroon">b</font>,<font color="Maroon">a</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b'</font>,<font color="Maroon">a'</font> )
 <b>and </b><br/><a NAME="E4:10"/><i><font color="Green">E37</font></i>: 
( <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>and </b><br/><a NAME="E5:10"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a'</font>
 ;<br/>

<a NAME="E6:10"/><i><font color="Green">E44</font></i>: 
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">q</font>,<font color="Maroon">a</font>,<font color="Maroon">a'</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">q</font>,<font color="Maroon">b</font>,<font color="Maroon">b'</font> )
 
<b>by </b><i>, , <a class="ref" href="aff_1.html#T33">AFF_1:33</a></i>;<br/>
<a NAME="E7:10"/><i><font color="Green">E45</font></i>: 
not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">q</font>,<font color="Maroon">a</font>,<font color="Maroon">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:10_1"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">q</font>,<font color="Maroon">a</font>,<font color="Maroon">b</font>
 ;<br/>

<b>then consider </b><font color="Maroon">A</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AS</font><b> such that </b><br/><a NAME="E3:10_1"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">A</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">q</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">A</font> )
 <b>by </b><i><a class="ref" href="aff_1.html#T33">AFF_1:33</a></i>;<br/>
<a NAME="E4:10_1"/>
( <font color="Maroon">M</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> &amp; <font color="Maroon">N</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> )
 
<b>by </b><i>, , , , <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/>
<b>hence </b><a NAME="E5:10_1"/>
contradiction
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E8:10"/><i><font color="Green">E49</font></i>: 
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">q</font>,<font color="Maroon">b</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">b</font> )
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T11">AFF_1:11</a>, <a class="ref" href="aff_1.html#T16">AFF_1:16</a></i>;<br/>
<a NAME="E9:10"/>
<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>
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/>
<b>hence </b><a NAME="E10:10"/>
<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_1.html#T70">AFF_1:70</a></i>;<br/>


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