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

<b>assume </b><a NAME="E1:50"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">AS</font> is  not   <a href="diraf.html#NM2">AffinPlane</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>;<br/>
<b>consider </b><font color="Maroon">X</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AS</font><b> such that </b><br/><a NAME="E4:50"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>and </b><br/><a NAME="E5:50"/>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 <b>and </b><br/><a NAME="E6:50"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">X</font> <a href="aff_4.html#NR1" target="_self">is_plane</a> 
 <b>by </b><i><a class="ref" href="aff_4.html#T37">AFF_4:37</a></i>;<br/>
<b>consider </b><font color="Maroon">t</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font><b> such that </b><br/><a NAME="E8:50"/><i><font color="Green">E31</font></i>: 
not <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="afproj.html#T5" target="_self">Th5</a>, , <a class="ref" href="aff_4.html#T48">AFF_4:48</a></i>;<br/>
<b>set </b><font color="Maroon">Y</font> =  <a href="aff_1.html#K2">Line</a> <font color="Maroon">x</font>,<font color="Maroon">t</font>;<br/>
<a NAME="E9:50"/><i><font color="Green">E33</font></i>: 
(  <a href="aff_1.html#K2">Line</a> <font color="Maroon">x</font>,<font color="Maroon">t</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">x</font>,<font color="Maroon">t</font> &amp; <font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">x</font>,<font color="Maroon">t</font> )
 
<b>by </b><i><a class="ref" href="afproj.html#T6" target="_self">Th6</a>, , <a class="ref" href="aff_1.html#T26">AFF_1:26</a>, <a class="ref" href="aff_1.html#D3">AFF_1:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">a</font> =  <a href="afproj.html#K5" target="_self">LDir</a> <span class="p1">(<span class="default"><a href="aff_1.html#K2">Line</a> <font color="Maroon">x</font>,<font color="Maroon">t</font></span>)</span> as    <a href="subset_1.html#M1">Element</a> of the <a href="incsp_1.html#U1">Points</a> of <span class="p1">(<span class="default"><a href="afproj.html#K14" target="_self">ProjHorizon</a> <font color="Maroon">AS</font></span>)</span> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">A</font> =  <a href="afproj.html#K6" target="_self">PDir</a> <font color="Maroon">X</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="incsp_1.html#U2">Lines</a> of <span class="p1">(<span class="default"><a href="afproj.html#K14" target="_self">ProjHorizon</a> <font color="Maroon">AS</font></span>)</span> <b>by </b><i>, <a class="ref" href="afproj.html#T1" target="_self">Th1</a></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="E12:50"/>
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="50_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:50_1"/>
<font color="Maroon">a</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A</font>
 ;<br/>

<a NAME="E2:50_1"/><b>then </b>
 <a href="aff_1.html#K2">Line</a> <font color="Maroon">x</font>,<font color="Maroon">t</font> <a href="aff_4.html#R1">'||'</a> <font color="Maroon">X</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E3:50_1"/><b>then </b>
 <a href="aff_1.html#K2">Line</a> <font color="Maroon">x</font>,<font color="Maroon">t</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="afproj.html#T6" target="_self">Th6</a>, , , <a class="ref" href="aff_4.html#T43">AFF_4:43</a></i>;<br/>
<b>hence </b><a NAME="E4:50_1"/>
contradiction
 <b>by </b><i>, </i>;<br/>


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


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