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<b>let </b><font color="Maroon">AS</font> be   <a href="diraf.html#NM1">AffinSpace</a>;<br/><b>let </b><font color="Maroon">A</font> be    <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>;<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="E2:48"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="afproj.html#K6" target="_self">PDir</a> <font color="Maroon">X</font>
 <b>and </b><br/><a NAME="E3:48"/><i><font color="Green">E26</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="afproj.html#T1" target="_self">Th1</a></i>;<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="E5:48"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <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="E6:48"/><i><font color="Green">E31</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="afproj.html#T3" target="_self">Th3</a>, <a class="ref" href="aff_4.html#T34">AFF_4:34</a></i>;<br/>
<a NAME="E7:48"/><i><font color="Green">E33</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="afproj.html#T6" target="_self">Th6</a>, <a class="ref" href="aff_1.html#T16">AFF_1:16</a></i>;<br/>
<a NAME="E8:48"/><b>then </b><i><font color="Green">E34</font></i>: 
(  <a href="aff_1.html#K2">Line</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp;  <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>  &amp;  <a href="aff_1.html#K2">Line</a> <font color="Maroon">x</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>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">y</font></span>)</span>, <font color="Maroon">b</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">y</font>,<font color="Maroon">z</font></span>)</span>, <font color="Maroon">c</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">z</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>take </b>
<font color="Maroon">a</font>
;<br/>

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

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

<a NAME="E10:48"/>
(  <a href="aff_1.html#K2">Line</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> &amp;  <a href="aff_1.html#K2">Line</a> <font color="Maroon">y</font>,<font color="Maroon">z</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> &amp;  <a href="aff_1.html#K2">Line</a> <font color="Maroon">x</font>,<font color="Maroon">z</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> )
 
<b>by </b><i><a class="ref" href="afproj.html#T3" target="_self">Th3</a>, <a class="ref" href="afproj.html#T5" target="_self">Th5</a>, , <a class="ref" href="aff_4.html#T19">AFF_4:19</a></i>;<br/>
<a NAME="E11:48"/><b>then </b>
(  <a href="aff_1.html#K2">Line</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="aff_4.html#R1">'||'</a> <font color="Maroon">X</font> &amp;  <a href="aff_1.html#K2">Line</a> <font color="Maroon">y</font>,<font color="Maroon">z</font> <a href="aff_4.html#R1">'||'</a> <font color="Maroon">X</font> &amp;  <a href="aff_1.html#K2">Line</a> <font color="Maroon">x</font>,<font color="Maroon">z</font> <a href="aff_4.html#R1">'||'</a> <font color="Maroon">X</font> )
 
<b>by </b><i><a class="ref" href="afproj.html#T3" target="_self">Th3</a>, , <a class="ref" href="aff_4.html#T42">AFF_4:42</a></i>;<br/>
<b>hence </b><a NAME="E12:48"/>
( <font color="Maroon">a</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A</font> &amp; <font color="Maroon">b</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A</font> &amp; <font color="Maroon">c</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A</font> )
 <b>by </b><i><a class="ref" href="afproj.html#T2" target="_self">Th2</a>, <a class="ref" href="afproj.html#T3" target="_self">Th3</a>, , <a class="txt" href="afproj.html#E13:48"><i><font color="Green">E35</font></i></a></i>;<br/>

<a NAME="E13:48"/><i><font color="Green">E35</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">x</font>,<font color="Maroon">y</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> &amp; <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> &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">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">x</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>thus </b><a NAME="E14:48"/>
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:48_1"/>
not <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font>
 ;<br/>

<a NAME="E2:48_1"/><b>then </b>
 <a href="aff_1.html#K2">Line</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="aff_1.html#R4">//</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">y</font>,<font color="Maroon">z</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E3:48_1"/><b>then </b>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T59">AFF_1:59</a></i>;<br/>
<b>hence </b><a NAME="E4:48_1"/>
contradiction
 <b>by </b><i><a class="ref" href="afproj.html#T6" target="_self">Th6</a>, , , <a class="ref" href="aff_1.html#T33">AFF_1:33</a></i>;<br/>


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

<b>thus </b><a NAME="E15:48"/>
<font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:48_2"/>
not <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c</font>
 ;<br/>

<a NAME="E2:48_2"/><b>then </b>
 <a href="aff_1.html#K2">Line</a> <font color="Maroon">y</font>,<font color="Maroon">z</font> <a href="aff_1.html#R4">//</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">x</font>,<font color="Maroon">z</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E3:48_2"/><b>then </b>
<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="ref" href="aff_1.html#T59">AFF_1:59</a></i>;<br/>
<b>hence </b><a NAME="E4:48_2"/>
contradiction
 <b>by </b><i><a class="ref" href="afproj.html#T6" target="_self">Th6</a>, , , <a class="ref" href="aff_1.html#T33">AFF_1:33</a></i>;<br/>


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

<b>thus </b><a NAME="E16:48"/>
<font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:48_3"/>
not <font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font>
 ;<br/>

<a NAME="E2:48_3"/><b>then </b>
 <a href="aff_1.html#K2">Line</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="aff_1.html#R4">//</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">x</font>,<font color="Maroon">z</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E3:48_3"/><b>then </b>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T59">AFF_1:59</a></i>;<br/>
<b>hence </b><a NAME="E4:48_3"/>
contradiction
 <b>by </b><i><a class="ref" href="afproj.html#T6" target="_self">Th6</a>, , , <a class="ref" href="aff_1.html#T33">AFF_1:33</a></i>;<br/>


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


</div>
