<?xml version="1.0"?>
<div class="add">

<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="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>;<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="E2:52"/><i><font color="Green">E25</font></i>: 
( ( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">X</font>,1</span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">X</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  ) or ( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="afproj.html#K6" target="_self">PDir</a> <font color="Maroon">X</font></span>)</span>,2</span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">X</font> <a href="aff_4.html#NR1" target="_self">is_plane</a>  ) )
 <b>by </b><i/>;<br/>
<div><i><font color="Green">E26</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1"><b>now </b></a><div class="add"><b>assume </b><b>that </b><br/><a NAME="E1:52_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">X</font>,1</span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E2:52_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">X</font> <a href="aff_1.html#NR2" target="_self">is_line</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><b> such that </b><br/><a NAME="E4:52_1"/><i><font color="Green">E33</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> )
 <b>and </b><br/><a NAME="E5:52_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font>
 <b>by </b><i>, <a class="ref" href="aff_1.html#T31">AFF_1:31</a></i>;<br/><b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">x</font>, <font color="Maroon">b</font> = <font color="Maroon">y</font> 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#K13" target="_self">IncProjSp_of</a> <font color="Maroon">AS</font></span>)</span> <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">c</font> =  <a href="afproj.html#K5" target="_self">LDir</a> <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>, </i>;<br/><b>take </b><font color="Maroon">a</font> = <font color="Maroon">a</font>;<br/><b>take </b><font color="Maroon">b</font> = <font color="Maroon">b</font>;<br/><b>take </b><font color="Maroon">c</font> = <font color="Maroon">c</font>;<br/><a NAME="E8:52_1"/>
<font color="Maroon">X</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="ref" href="aff_1.html#T55">AFF_1:55</a></i>;<br/><a NAME="E9:52_1"/><b>then </b>
<font color="Maroon">X</font> <a href="aff_4.html#R1">'||'</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="ref" href="aff_4.html#T40">AFF_4:40</a></i>;<br/><b>hence </b><a NAME="E10:52_1"/>
( <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="txt" href="afproj.html#E3:52"><i><font color="Green">E26</font></i></a>, </i>;<br/><b>thus </b><a NAME="E11:52_1"/>
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font>
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E12:52_1"/>
( <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c</font> &amp; <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="52_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:52_1_1"/>
(  not <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c</font> or  not <font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font> )
 ;<br/>

<a NAME="E2:52_1_1"/><b>then </b>
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">AS</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="afproj.html#K7" target="_self">Dir_of_Lines</a> <font color="Maroon">AS</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E3:52_1_1"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">AS</font> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="afproj.html#K7" target="_self">Dir_of_Lines</a> <font color="Maroon">AS</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E4:52_1_1"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">AS</font> <a href="subset_1.html#R2">meets</a>  <a href="afproj.html#K7" target="_self">Dir_of_Lines</a> <font color="Maroon">AS</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<b>hence </b><a NAME="E5:52_1_1"/>
contradiction
 <b>by </b><i/>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_2"><b>now </b></a><div class="add"><b>assume </b><b>that </b><br/><a NAME="E1:52_2"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="afproj.html#K6" target="_self">PDir</a> <font color="Maroon">X</font></span>)</span>,2</span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E2:52_2"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">X</font> <a href="aff_4.html#NR1" target="_self">is_plane</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="E4:52_2"/><i><font color="Green">E38</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="E5:52_2"/><i><font color="Green">E47</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_4.html#T34">AFF_4:34</a></i>;<br/><a NAME="E6:52_2"/><i><font color="Green">E48</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="txt" href="afproj.html#E6:52_2"><i><font color="Green">E48</font></i></a>, <a class="ref" href="aff_1.html#T16">AFF_1:16</a></i>;<br/><a NAME="E7:52_2"/><b>then </b><i><font color="Green">E49</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="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>take </b><font color="Maroon">a</font> = <font color="Maroon">a</font>;<br/><b>take </b><font color="Maroon">b</font> = <font color="Maroon">b</font>;<br/><b>take </b><font color="Maroon">c</font> = <font color="Maroon">c</font>;<br/><a NAME="E9:52_2"/>
(  <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="txt" href="afproj.html#E5:52_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="afproj.html#E7:52_2"><i><font color="Green">E49</font></i></a>, <a class="ref" href="aff_4.html#T19">AFF_4:19</a></i>;<br/><a NAME="E10:52_2"/><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="txt" href="afproj.html#E12:52_2"><i><font color="Green">E50</font></i></a>, <a class="ref" href="aff_4.html#T42">AFF_4:42</a></i>;<br/><b>hence </b><a NAME="E11:52_2"/>
( <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="txt" href="afproj.html#E12:52_2"><i><font color="Green">E50</font></i></a>, </i>;<br/><a NAME="E12:52_2"/><i><font color="Green">E50</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="E13:52_2"/>
<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="52_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:52_2_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="E2:52_2_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><a class="txt" href="afproj.html#E12:52_2"><i><font color="Green">E50</font></i></a>, </i>;<br/>
<a NAME="E3:52_2_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="E4:52_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="afproj.html#E6:52_2"><i><font color="Green">E48</font></i></a>, <a class="txt" href="afproj.html#E12:52_2"><i><font color="Green">E50</font></i></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="E14:52_2"/>
<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="52_2_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:52_2_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="E2:52_2_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><a class="txt" href="afproj.html#E12:52_2"><i><font color="Green">E50</font></i></a>, </i>;<br/>
<a NAME="E3:52_2_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="E4:52_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="afproj.html#E6:52_2"><i><font color="Green">E48</font></i></a>, <a class="txt" href="afproj.html#E12:52_2"><i><font color="Green">E50</font></i></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="E15:52_2"/>
<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="52_2_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:52_2_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="E2:52_2_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><a class="txt" href="afproj.html#E12:52_2"><i><font color="Green">E50</font></i></a>, </i>;<br/>
<a NAME="E3:52_2_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="E4:52_2_3"/>
contradiction
 <b>by </b><i><a class="txt" href="afproj.html#E6:52_2"><i><font color="Green">E48</font></i></a>, <a class="txt" href="afproj.html#E12:52_2"><i><font color="Green">E50</font></i></a>, , <a class="ref" href="aff_1.html#T33">AFF_1:33</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E5:52"/>
 ex <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being  <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> st <br/>( <font color="Olive">a</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A</font> &amp; <font color="Olive">b</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A</font> &amp; <font color="Olive">c</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A</font> &amp; <font color="Olive">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b</font> &amp; <font color="Olive">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">c</font> &amp; <font color="Olive">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">a</font> )
 <b>by </b><i>, </i>;<br/>


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