<?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="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>, <font color="Maroon">b</font> be    <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>, <font color="Maroon">c</font> be    <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>, <font color="Maroon">d</font> be    <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>, <font color="Maroon">p</font> be    <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>;<br/><b>let </b><font color="Maroon">M</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>, <font color="Maroon">N</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>, <font color="Maroon">P</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>, <font color="Maroon">Q</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>assume </b><a NAME="E1:65"/><i><font color="Green">E25</font></i>: 
( <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> &amp; <font color="Maroon">c</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">N</font> &amp; <font color="Maroon">d</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">N</font> &amp; <font color="Maroon">p</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">M</font> &amp; <font color="Maroon">p</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">N</font> &amp; <font color="Maroon">a</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">P</font> &amp; <font color="Maroon">c</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">P</font> &amp; <font color="Maroon">b</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">Q</font> &amp; <font color="Maroon">d</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">Q</font> &amp; not <font color="Maroon">p</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">P</font> &amp; not <font color="Maroon">p</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">Q</font> &amp; <font color="Maroon">M</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">N</font> )
 ;<br/>

<b>reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">a</font>, <font color="Maroon">b'</font> = <font color="Maroon">b</font>, <font color="Maroon">c'</font> = <font color="Maroon">c</font>, <font color="Maroon">d'</font> = <font color="Maroon">d</font>, <font color="Maroon">p'</font> = <font color="Maroon">p</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/>;<br/>
<b>reconsider </b><font color="Maroon">M'</font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">M</font>,2</span><a href="tarski.html#K4">]</a></span>, <font color="Maroon">N'</font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">N</font>,2</span><a href="tarski.html#K4">]</a></span>, <font color="Maroon">P'</font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">P</font>,2</span><a href="tarski.html#K4">]</a></span>, <font color="Maroon">Q'</font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">Q</font>,2</span><a href="tarski.html#K4">]</a></span> as   <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> <b>by </b><i><a class="txt" href="afproj.html#E1:65"><i><font color="Green">E25</font></i></a></i>;<br/>
<a NAME="E4:65"/><i><font color="Green">E26</font></i>: 
( <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> &amp; <font color="Maroon">c'</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">N'</font> &amp; <font color="Maroon">d'</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">N'</font> &amp; <font color="Maroon">p'</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">M'</font> &amp; <font color="Maroon">p'</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">N'</font> &amp; <font color="Maroon">a'</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">P'</font> &amp; <font color="Maroon">c'</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">P'</font> &amp; <font color="Maroon">b'</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">Q'</font> &amp; <font color="Maroon">d'</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">Q'</font> &amp; not <font color="Maroon">p'</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">P'</font> &amp; not <font color="Maroon">p'</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">Q'</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:65"/>
<font color="Maroon">M'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">N'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:65_1"/>
<font color="Maroon">M'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">N'</font>
 ;<br/>

<b>then </b><font color="Maroon">M</font> = 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">N</font>,2</span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
<font color="Maroon">N</font>
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
;<br/>
<b>hence </b><a NAME="E3:65_1"/>
contradiction
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">q'</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><b> such that </b><br/><a NAME="E7:65"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">q'</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">P'</font> &amp; <font color="Maroon">q'</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">Q'</font> )
 <b>by </b><i>, </i>;<br/>
<a NAME="E8:65"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">q'</font>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">P</font>,2</span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="incsp_1.html#U3">Inc</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>, <a class="ref" href="incsp_1.html#D1">INCSP_1:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">q</font> = <font color="Maroon">q'</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#K14" target="_self">ProjHorizon</a> <font color="Maroon">AS</font></span>)</span> <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">q</font>
;<br/>

<b>thus </b><a NAME="E10:65"/>
( <font color="Maroon">q</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">P</font> &amp; <font color="Maroon">q</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">Q</font> )
 <b>by </b><i>, </i>;<br/>


</div>
