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<b>let </b><font color="Maroon">AS</font> be   <a href="diraf.html#NM1">AffinSpace</a>;<br/>

<b>consider </b><font color="Maroon">x</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>, <font color="Maroon">y</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>, <font color="Maroon">z</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font><b> such that </b><br/><a NAME="E2:55"/><i><font color="Green">E25</font></i>: 
not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font>
 <b>by </b><i><a class="ref" href="aff_1.html#T21">AFF_1:21</a></i>;<br/>
<a NAME="E3:55"/>
<font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">z</font>
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T16">AFF_1:16</a></i>;<br/>
<a NAME="E4:55"/><b>then </b><i><font color="Green">E26</font></i>: 
 <a href="aff_1.html#K2">Line</a> <font color="Maroon">y</font>,<font color="Maroon">z</font> <a href="aff_1.html#NR2" target="_self">is_line</a> 
 
<b>by </b><i><a class="ref" href="aff_1.html#D3">AFF_1:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">x</font> as   <a href="incsp_1.html#NM1">POINT</a> of <span class="p1">(<span class="default"><a href="afproj.html#K13" target="_self">IncProjSp_of</a> <font color="Maroon">AS</font></span>)</span> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">A</font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="aff_1.html#K2">Line</a> <font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span>,1</span><a href="tarski.html#K4">]</a></span> as   <a href="incsp_1.html#NM2">LINE</a> of <span class="p1">(<span class="default"><a href="afproj.html#K13" target="_self">IncProjSp_of</a> <font color="Maroon">AS</font></span>)</span> <b>by </b><i>, </i>;<br/>
<b>take </b>
<font color="Maroon">a</font>
;<br/>

<b>take </b>
<font color="Maroon">A</font>
;<br/>

<b>thus </b><a NAME="E7:55"/>
not <font color="Maroon">a</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:55_1"/>
<font color="Maroon">a</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A</font>
 ;<br/>

<a NAME="E2:55_1"/><b>then </b><i><font color="Green">E30</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">y</font>,<font color="Maroon">z</font>
 
<b>by </b><i><a class="txt" href="afproj.html#E4:55"><i><font color="Green">E26</font></i></a></i>;<br/>
<a NAME="E3:55_1"/>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">y</font>,<font color="Maroon">z</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">y</font>,<font color="Maroon">z</font> )
 
<b>by </b><i><a class="ref" href="aff_1.html#T26">AFF_1:26</a></i>;<br/>
<b>hence </b><a NAME="E4:55_1"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="aff_1.html#T33">AFF_1:33</a></i>;<br/>


</div><b>end;</b></div>


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