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<b>let </b><font color="Maroon">SAS</font> be   <a href="semi_af1.html#NM1">Semi_Affine_Space</a>;<br/><b>let </b><font color="Maroon">o</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">SAS</font>;<br/><b>let </b><font color="Maroon">p</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">SAS</font>;<br/>





<b>assume </b><a NAME="E1:109"/><i><font color="Green">E29</font></i>: 
 <a href="semi_af1.html#R5" target="_self">qtrap</a> <font color="Maroon">o</font>,<font color="Maroon">p</font>
 ;<br/>

<b>assume </b><a NAME="E2:109"/><i><font color="Green">E30</font></i>: 
not <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p</font>
 ;<br/>

<a NAME="E3:109"/>
 ex <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">SAS</font> st <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">x</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">SAS</font>, <font color="Maroon">y</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">SAS</font>, <font color="Maroon">z</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">SAS</font><b> such that </b><br/><a NAME="E2:109_1"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">z</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="semi_af1.html#T67" target="_self">Th67</a></i>;<br/>
<a NAME="E3:109_1"/>
( <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> or <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font> or <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">z</font> )
 
<b>by </b><i><a class="txt" href="semi_af1.html#E9:109"><i><font color="Green">E89</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:109_1"/>
 ex <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">SAS</font> st <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b</font>
 ;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">b</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">SAS</font><b> such that </b><br/><a NAME="E5:109"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font>
 ;<br/>
<b>consider </b><font color="Maroon">c</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">SAS</font><b> such that </b><br/><a NAME="E7:109"/><i><font color="Green">E84</font></i>: 
not <font color="Maroon">o</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o</font>,<font color="Maroon">c</font>
 <b>by </b><i><a class="txt" href="semi_af1.html#E9:109"><i><font color="Green">E89</font></i></a>, </i>;<br/>
<b>consider </b><font color="Maroon">d</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">SAS</font><b> such that </b><br/><a NAME="E9:109"/><i><font color="Green">E89</font></i>: 
( <font color="Maroon">o</font>,<font color="Maroon">p</font>,<font color="Maroon">b</font> <a href="semi_af1.html#R1" target="_self">is_collinear</a>  implies ( <font color="Maroon">o</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font> <a href="semi_af1.html#R1" target="_self">is_collinear</a>  &amp; <font color="Maroon">p</font>,<font color="Maroon">c</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b</font>,<font color="Maroon">d</font> ) )
 <b>by </b><i><a class="ref" href="semi_af1.html#T69" target="_self">Th69</a>, <a class="ref" href="semi_af1.html#T56" target="_self">Th56</a></i>;<br/>
<a NAME="E10:109"/>
( <font color="Maroon">o</font>,<font color="Maroon">o</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o</font>,<font color="Maroon">b</font> implies ( <font color="Maroon">o</font>,<font color="Maroon">c</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o</font>,<font color="Maroon">d</font> &amp; <font color="Maroon">o</font>,<font color="Maroon">c</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b</font>,<font color="Maroon">d</font> ) )
 
<b>by </b><i><a class="ref" href="semi_af1.html#T70" target="_self">Th70</a>, , </i>;<br/>
<a NAME="E11:109"/><b>then </b><i><font color="Green">E90</font></i>: 
( ( <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> &amp; not <font color="Maroon">o</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o</font>,<font color="Maroon">c</font> &amp; <font color="Maroon">o</font>,<font color="Maroon">c</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o</font>,<font color="Maroon">d</font> ) or ( <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">d</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c</font> &amp; not <font color="Maroon">o</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o</font>,<font color="Maroon">c</font> &amp; <font color="Maroon">o</font>,<font color="Maroon">c</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o</font>,<font color="Maroon">d</font> &amp; <font color="Maroon">o</font>,<font color="Maroon">c</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b</font>,<font color="Maroon">d</font> ) )
 
<b>by </b><i><a class="txt" href="semi_af1.html#E11:109"><i><font color="Green">E90</font></i></a>, , </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:109_2"/><i><font color="Green">E91</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">d</font> &amp; not <font color="Maroon">o</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o</font>,<font color="Maroon">c</font> &amp; <font color="Maroon">o</font>,<font color="Maroon">d</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b</font>,<font color="Maroon">d</font> &amp; <font color="Maroon">o</font>,<font color="Maroon">c</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b</font>,<font color="Maroon">d</font> )
 ;<br/><a NAME="E2:109_2"/><b>then </b>
<font color="Maroon">d</font>,<font color="Maroon">o</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">d</font>,<font color="Maroon">b</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:109_2"/>
( <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">d</font> &amp; not <font color="Maroon">o</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o</font>,<font color="Maroon">c</font> &amp; <font color="Maroon">b</font>,<font color="Maroon">d</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o</font>,<font color="Maroon">b</font> &amp; <font color="Maroon">b</font>,<font color="Maroon">d</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o</font>,<font color="Maroon">c</font> )
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E13:109"/>
contradiction
 <b>by </b><i>, , </i>;<br/>


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