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<b>let </b><font color="Maroon" title="c1">AS</font> be   <a href="diraf.html#NM1" title="DIRAF:NM.1">AffinSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">q</font>, <font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">a'</font>, <font color="Maroon" title="c5">b</font>, <font color="Maroon" title="c6">b'</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font>;<br/><b>let </b><font color="Maroon" title="c7">M</font>, <font color="Maroon" title="c8">N</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font>;<br/>















<b>assume </b><b>that </b><br/><a NAME="E1:9"/><i><font color="Green" title="E9">A1</font></i>: 
( <font color="Maroon" title="c2">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">M</font> &amp; <font color="Maroon" title="c2">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">N</font> &amp; <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">M</font> &amp; <font color="Maroon" title="c4">a'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">M</font> &amp; <font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">N</font> &amp; <font color="Maroon" title="c6">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">N</font> )
 <b>and </b><br/><a NAME="E2:9"/><i><font color="Green" title="E10">A2</font></i>: 
( <font color="Maroon" title="c2">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">a</font> &amp; <font color="Maroon" title="c2">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c7">M</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">N</font> )
 <b>and </b><br/><a NAME="E3:9"/><i><font color="Green" title="E11">A3</font></i>: 
( <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">a'</font>,<font color="Maroon" title="c6">b'</font> or <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c3">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b'</font>,<font color="Maroon" title="c4">a'</font> )
 <b>and </b><br/><a NAME="E4:9"/><i><font color="Green" title="E12">A4</font></i>: 
( <font color="Maroon" title="c7">M</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <font color="Maroon" title="c8">N</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  )
 <b>and </b><br/><a NAME="E5:9"/><i><font color="Green" title="E13">A5</font></i>: 
<font color="Maroon" title="c2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a'</font>
 ;<br/>

<a NAME="E6:9"/><i><font color="Green" title="E14">A6</font></i>: 
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">a'</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">b'</font> )
 
<b>by </b><i><a class="txt" href="aff_4.html#E1:9"><i><font color="Green" title="E9">A1</font></i></a>, <a class="txt" href="aff_4.html#E4:9"><i><font color="Green" title="E12">A4</font></i></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></i>;<br/>
<a NAME="E7:9"/><i><font color="Green" title="E15">A7</font></i>: 
not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:9_1"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">b</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c9">A</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font><b> such that </b><br/><a NAME="E3:9_1"/><i><font color="Green" title="E16">A8</font></i>: 
( <font color="Maroon" title="c9">A</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <font color="Maroon" title="c2">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">A</font> &amp; <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">A</font> &amp; <font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">A</font> )
 <b>by </b><i><a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></i>;<br/>
<a NAME="E4:9_1"/>
( <font color="Maroon" title="c7">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">A</font> &amp; <font color="Maroon" title="c8">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">A</font> )
 
<b>by </b><i><a class="txt" href="aff_4.html#E1:9"><i><font color="Green" title="E9">A1</font></i></a>, <a class="txt" href="aff_4.html#E2:9"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="aff_4.html#E4:9"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="aff_4.html#E3:9_1"><i><font color="Green" title="E16">A8</font></i></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></i>;<br/>
<b>hence </b><a NAME="E5:9_1"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_4.html#E2:9"><i><font color="Green" title="E10">A2</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:9"/>
<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">a'</font>,<font color="Maroon" title="c6">b'</font>
 
<b>by </b><i><a class="txt" href="aff_4.html#E3:9"><i><font color="Green" title="E11">A3</font></i></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></i>;<br/>
<b>hence </b><a NAME="E9:9"/>
<font color="Maroon" title="c2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b'</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E5:9"><i><font color="Green" title="E13">A5</font></i></a>, <a class="txt" href="aff_4.html#E6:9"><i><font color="Green" title="E14">A6</font></i></a>, <a class="txt" href="aff_4.html#E7:9"><i><font color="Green" title="E15">A7</font></i></a>, <a class="ref" href="aff_1.html#T69" title="AFF_1:th.69">AFF_1:69</a></i>;<br/>


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