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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>consider </b><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y</font>, <font color="Maroon" title="c4">z</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font><b> such that </b><br/><a NAME="E2:55"/><i><font color="Green" title="E46">A1</font></i>: 
not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c4">z</font>
 <b>by </b><i><a class="ref" href="aff_1.html#T21" title="AFF_1:th.21">AFF_1:21</a></i>;<br/>
<a NAME="E3:55"/>
<font color="Maroon" title="c3">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">z</font>
 
<b>by </b><i><a class="txt" href="afproj.html#E2:55"><i><font color="Green" title="E46">A1</font></i></a>, <a class="ref" href="aff_1.html#T16" title="AFF_1:th.16">AFF_1:16</a></i>;<br/>
<a NAME="E4:55"/><b>then </b><i><font color="Green" title="E47">A2</font></i>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c4">z</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a> 
 
<b>by </b><i><a class="ref" href="aff_1.html#D3" title="AFF_1:def.3">AFF_1:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">a</font> = <font color="Maroon" title="c2">x</font> as   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <span class="p1">(<span class="default"><a href="afproj.html#K13" title="AFPROJ:func.13">IncProjSp_of</a> <font color="Maroon" title="c1">AS</font></span>)</span> <b>by </b><i><a class="ref" href="afproj.html#T20" target="_self" title="AFPROJ:th.20">Th20</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">A</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c4">z</font></span>)</span>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> as   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <span class="p1">(<span class="default"><a href="afproj.html#K13" title="AFPROJ:func.13">IncProjSp_of</a> <font color="Maroon" title="c1">AS</font></span>)</span> <b>by </b><i><a class="txt" href="afproj.html#E4:55"><i><font color="Green" title="E47">A2</font></i></a>, <a class="ref" href="afproj.html#T23" target="_self" title="AFPROJ:th.23">Th23</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c5">a</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c6">A</font>
;<br/>

<b>thus </b><a NAME="E7:55"/>
not <font color="Maroon" title="c5">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c6">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" title="c5">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c6">A</font>
 ;<br/>

<a NAME="E2:55_1"/><b>then </b><i><font color="Green" title="E48">A3</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c4">z</font>
 
<b>by </b><i><a class="ref" href="afproj.html#T26" target="_self" title="AFPROJ:th.26">Th26</a></i>;<br/>
<a NAME="E3:55_1"/>
( <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c4">z</font> &amp; <font color="Maroon" title="c4">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c4">z</font> )
 
<b>by </b><i><a class="ref" href="aff_1.html#T26" title="AFF_1:th.26">AFF_1:26</a></i>;<br/>
<b>hence </b><a NAME="E4:55_1"/>
contradiction
 <b>by </b><i><a class="txt" href="afproj.html#E2:55"><i><font color="Green" title="E46">A1</font></i></a>, <a class="txt" href="afproj.html#E4:55"><i><font color="Green" title="E47">A2</font></i></a>, <a class="txt" href="afproj.html#E2:55_1"><i><font color="Green" title="E48">A3</font></i></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></i>;<br/>


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


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