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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="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>;<br/><b>let </b><font color="Maroon">b</font> be   <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>;<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>let </b><font color="Maroon">K</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>assume </b><b>that </b><br/><a NAME="E1:51"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">a</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A</font> &amp; <font color="Maroon">a</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">K</font> )
 <b>and </b><br/><a NAME="E2:51"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">b</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">K</font> )
 ;<br/>

<b>assume </b><a NAME="E3:51"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</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="E5:51"/><i><font color="Green">E31</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/>
<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="E7:51"/><i><font color="Green">E33</font></i>: 
( ( <font color="Maroon">K</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">K</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">E34</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:51_1"/>
<font color="Maroon">a</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>
 ;<br/><b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">a</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font> ;<br/><a NAME="E3:51_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 ;<br/><a NAME="E4:51_1"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X'</font> )
 <b>by </b><i><a class="ref" href="afproj.html#T6" target="_self">Th6</a>, , , <a class="txt" href="afproj.html#E2:51"><i><font color="Green">E26</font></i></a>, </i>;<br/><div><i><font color="Green">E38</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:51_1_1"/>
<font color="Maroon">b</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>
 ;<br/><b>then reconsider </b><font color="Maroon">y</font> = <font color="Maroon">b</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font> ;<br/><a NAME="E3:51_1_1"/>
( <font color="Maroon">y</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>by </b><i>, , , <a class="txt" href="afproj.html#E2:51"><i><font color="Green">E26</font></i></a>, </i>;<br/><b>hence </b><a NAME="E4:51_1_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">K</font>
 <b>by </b><i><a class="ref" href="afproj.html#T6" target="_self">Th6</a>, , , , , , <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_2"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">Y</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AS</font><b> such that </b><a NAME="E1:51_1_2"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="afproj.html#K5" target="_self">LDir</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">Y</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  )
 ;<br/><a NAME="E2:51_1_2"/>
( <font color="Maroon">Y</font> <a href="aff_4.html#R1">'||'</a> <font color="Maroon">X</font> &amp; <font color="Maroon">Y</font> <a href="aff_4.html#R1">'||'</a> <font color="Maroon">X'</font> )
 <b>by </b><i><a class="ref" href="afproj.html#T6" target="_self">Th6</a>, , , , , , , </i>;<br/><a NAME="E3:51_1_2"/><b>then </b>
( <font color="Maroon">Y</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">X</font> &amp; <font color="Maroon">Y</font> <a href="aff_1.html#R4">//</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#T40">AFF_4:40</a></i>;<br/><a NAME="E4:51_1_2"/><b>then </b>
<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#T58">AFF_1:58</a></i>;<br/><b>hence </b><a NAME="E5:51_1_2"/>
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">K</font>
 <b>by </b><i><a class="ref" href="afproj.html#T6" target="_self">Th6</a>, , , , , <a class="ref" href="aff_1.html#T59">AFF_1:59</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E7:51_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">K</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_2"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">Y</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AS</font><b> such that </b><a NAME="E1:51_2"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="afproj.html#K5" target="_self">LDir</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">Y</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  )
 ;<br/><div><i><font color="Green">E49</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="51_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:51_2_1"/>
<font color="Maroon">b</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>
 ;<br/><b>then reconsider </b><font color="Maroon">y</font> = <font color="Maroon">b</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font> ;<br/><a NAME="E3:51_2_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 ;<br/><a NAME="E4:51_2_1"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">y</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>by </b><i>, , , <a class="txt" href="afproj.html#E2:51"><i><font color="Green">E26</font></i></a>, </i>;<br/><a NAME="E5:51_2_1"/>
( <font color="Maroon">Y</font> <a href="aff_4.html#R1">'||'</a> <font color="Maroon">X</font> &amp; <font color="Maroon">Y</font> <a href="aff_4.html#R1">'||'</a> <font color="Maroon">X'</font> )
 <b>by </b><i><a class="ref" href="afproj.html#T6" target="_self">Th6</a>, , , , <a class="txt" href="afproj.html#E1:51_2"><i><font color="Green">E48</font></i></a>, <a class="txt" href="afproj.html#E3:51_2_1"><i><font color="Green">E50</font></i></a>, , </i>;<br/><a NAME="E6:51_2_1"/><b>then </b>
( <font color="Maroon">Y</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">X</font> &amp; <font color="Maroon">Y</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">X'</font> )
 <b>by </b><i>, , , <a class="txt" href="afproj.html#E1:51_2"><i><font color="Green">E48</font></i></a>, <a class="txt" href="afproj.html#E3:51_2_1"><i><font color="Green">E50</font></i></a>, , <a class="ref" href="aff_4.html#T40">AFF_4:40</a></i>;<br/><a NAME="E7:51_2_1"/><b>then </b>
<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#T58">AFF_1:58</a></i>;<br/><b>hence </b><a NAME="E8:51_2_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">K</font>
 <b>by </b><i>, , , , , <a class="ref" href="aff_1.html#T59">AFF_1:59</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_2_2"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">Y'</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AS</font><b> such that </b><a NAME="E1:51_2_2"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="afproj.html#K5" target="_self">LDir</a> <font color="Maroon">Y'</font> &amp; <font color="Maroon">Y'</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  )
 ;<br/><a NAME="E2:51_2_2"/><i><font color="Green">E54</font></i>: 
for <font color="Olive">Z</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AS</font><br/> for <font color="Olive">M</font> being  <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>  st <font color="Olive">M</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">Z</font>,1</span><a href="tarski.html#K4">]</a></span> &amp; <font color="Olive">Z</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">a</font> <a href="incsp_1.html#R1">on</a> <font color="Olive">M</font> holds <br/> not <font color="Maroon">b</font> <a href="incsp_1.html#R1">on</a> <font color="Olive">M</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_2_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">Z</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AS</font>;<br/><b>let </b><font color="Maroon">M</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>assume </b><a NAME="E1:51_2_2_1"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">M</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">Z</font>,1</span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">Z</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  )
 ;<br/>

<b>assume </b><a NAME="E2:51_2_2_1"/>
( <font color="Maroon">a</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">M</font> &amp; <font color="Maroon">b</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">M</font> )
 ;<br/>

<a NAME="E3:51_2_2_1"/><b>then </b>
( <font color="Maroon">Y</font> <a href="aff_4.html#R1">'||'</a> <font color="Maroon">Z</font> &amp; <font color="Maroon">Y'</font> <a href="aff_4.html#R1">'||'</a> <font color="Maroon">Z</font> )
 
<b>by </b><i><a class="txt" href="afproj.html#E1:51_2"><i><font color="Green">E48</font></i></a>, , , </i>;<br/>
<a NAME="E4:51_2_2_1"/><b>then </b>
( <font color="Maroon">Y</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">Z</font> &amp; <font color="Maroon">Y'</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">Z</font> )
 
<b>by </b><i><a class="txt" href="afproj.html#E1:51_2"><i><font color="Green">E48</font></i></a>, , , <a class="ref" href="aff_4.html#T40">AFF_4:40</a></i>;<br/>
<a NAME="E5:51_2_2_1"/><b>then </b>
<font color="Maroon">Y</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">Y'</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T58">AFF_1:58</a></i>;<br/>
<b>hence </b><a NAME="E6:51_2_2_1"/>
contradiction
 <b>by </b><i>, <a class="txt" href="afproj.html#E1:51_2"><i><font color="Green">E48</font></i></a>, , </i>;<br/>


</div><b>end;</b></div><a NAME="E3:51_2_2"/><i><font color="Green">E56</font></i>: 
not <font color="Maroon">Y</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">Y'</font>
 <b>by </b><i>, <a class="txt" href="afproj.html#E1:51_2"><i><font color="Green">E48</font></i></a>, , </i>;<br/><a NAME="E4:51_2_2"/>
( <font color="Maroon">Y</font> <a href="aff_4.html#R1">'||'</a> <font color="Maroon">X</font> &amp; <font color="Maroon">Y</font> <a href="aff_4.html#R1">'||'</a> <font color="Maroon">X'</font> &amp; <font color="Maroon">Y'</font> <a href="aff_4.html#R1">'||'</a> <font color="Maroon">X</font> &amp; <font color="Maroon">Y'</font> <a href="aff_4.html#R1">'||'</a> <font color="Maroon">X'</font> )
 <b>by </b><i><a class="ref" href="afproj.html#T6" target="_self">Th6</a>, , , , <a class="txt" href="afproj.html#E1:51_2"><i><font color="Green">E48</font></i></a>, , , </i>;<br/><a NAME="E5:51_2_2"/><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="afproj.html#T6" target="_self">Th6</a>, , , , <a class="txt" href="afproj.html#E1:51_2"><i><font color="Green">E48</font></i></a>, , , , </i>;<br/><b>hence </b><a NAME="E6:51_2_2"/>
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">K</font>
 <b>by </b><i><a class="ref" href="afproj.html#T6" target="_self">Th6</a>, , , , , </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:51_2"/>
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">K</font>
 <b>by </b><i><a class="txt" href="afproj.html#E2:51_2"><i><font color="Green">E49</font></i></a>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E10:51"/>
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">K</font>
 <b>by </b><i>, </i>;<br/>


</div>
