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<div class="add">

<b>let </b><font color="Maroon" title="c1">AS</font> be   <a href="diraf.html#NM1" title="DIRAF:NM.1">AffinSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font>, <font color="Maroon" title="c4">c</font>, <font color="Maroon" title="c5">d</font>, <font color="Maroon" title="c6">p</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <span class="p1">(<span class="default"><a href="afproj.html#K14" title="AFPROJ:func.14">ProjHorizon</a> <font color="Maroon" title="c1">AS</font></span>)</span>;<br/><b>let </b><font color="Maroon" title="c7">M</font>, <font color="Maroon" title="c8">N</font>, <font color="Maroon" title="c9">P</font>, <font color="Maroon" title="c10">Q</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="incsp_1.html#U2" title="INCSP_1:sel.2">Lines</a> of <span class="p1">(<span class="default"><a href="afproj.html#K14" title="AFPROJ:func.14">ProjHorizon</a> <font color="Maroon" title="c1">AS</font></span>)</span>;<br/>





<b>assume </b><a NAME="E1:65"/><i><font color="Green" title="E54">A1</font></i>: 
( <font color="Maroon" title="c2">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">M</font> &amp; <font color="Maroon" title="c3">b</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">M</font> &amp; <font color="Maroon" title="c4">c</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c8">N</font> &amp; <font color="Maroon" title="c5">d</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c8">N</font> &amp; <font color="Maroon" title="c6">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">M</font> &amp; <font color="Maroon" title="c6">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c8">N</font> &amp; <font color="Maroon" title="c2">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c9">P</font> &amp; <font color="Maroon" title="c4">c</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c9">P</font> &amp; <font color="Maroon" title="c3">b</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c10">Q</font> &amp; <font color="Maroon" title="c5">d</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c10">Q</font> &amp; not <font color="Maroon" title="c6">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c9">P</font> &amp; not <font color="Maroon" title="c6">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c10">Q</font> &amp; <font color="Maroon" title="c7">M</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">N</font> )
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c11">a'</font> = <font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c12">b'</font> = <font color="Maroon" title="c3">b</font>, <font color="Maroon" title="c13">c'</font> = <font color="Maroon" title="c4">c</font>, <font color="Maroon" title="c14">d'</font> = <font color="Maroon" title="c5">d</font>, <font color="Maroon" title="c15">p'</font> = <font color="Maroon" title="c6">p</font> as   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <span class="p1">(<span class="default"><a href="afproj.html#K13" title="AFPROJ:func.13">IncProjSp_of</a> <font color="Maroon" title="c1">AS</font></span>)</span> <b>by </b><i><a class="ref" href="afproj.html#T22" target="_self" title="AFPROJ:th.22">Th22</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c16">M'</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">M</font>,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>, <font color="Maroon" title="c17">N'</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">N</font>,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>, <font color="Maroon" title="c18">P'</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">P</font>,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>, <font color="Maroon" title="c19">Q'</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">Q</font>,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> as   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <span class="p1">(<span class="default"><a href="afproj.html#K13" title="AFPROJ:func.13">IncProjSp_of</a> <font color="Maroon" title="c1">AS</font></span>)</span> <b>by </b><i><a class="ref" href="afproj.html#T25" target="_self" title="AFPROJ:th.25">Th25</a></i>;<br/>
<a NAME="E4:65"/><i><font color="Green" title="E55">A2</font></i>: 
( <font color="Maroon" title="c11">a'</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c16">M'</font> &amp; <font color="Maroon" title="c12">b'</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c16">M'</font> &amp; <font color="Maroon" title="c13">c'</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c17">N'</font> &amp; <font color="Maroon" title="c14">d'</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c17">N'</font> &amp; <font color="Maroon" title="c15">p'</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c16">M'</font> &amp; <font color="Maroon" title="c15">p'</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c17">N'</font> &amp; <font color="Maroon" title="c11">a'</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c18">P'</font> &amp; <font color="Maroon" title="c13">c'</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c18">P'</font> &amp; <font color="Maroon" title="c12">b'</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c19">Q'</font> &amp; <font color="Maroon" title="c14">d'</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c19">Q'</font> &amp; not <font color="Maroon" title="c15">p'</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c18">P'</font> &amp; not <font color="Maroon" title="c15">p'</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c19">Q'</font> )
 
<b>by </b><i><a class="txt" href="afproj.html#E1:65"><i><font color="Green" title="E54">A1</font></i></a>, <a class="ref" href="afproj.html#T37" target="_self" title="AFPROJ:th.37">Th37</a></i>;<br/>
<a NAME="E5:65"/>
<font color="Maroon" title="c16">M'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c17">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" title="c16">M'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">N'</font>
 ;<br/>

<b>then </b><font color="Maroon" title="c7">M</font> = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">N</font>,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
<b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
<br/>.= 
<font color="Maroon" title="c8">N</font>
<b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
;<br/>
<b>hence </b><a NAME="E3:65_1"/>
contradiction
 <b>by </b><i><a class="txt" href="afproj.html#E1:65"><i><font color="Green" title="E54">A1</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon" title="c20">q'</font> being   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <span class="p1">(<span class="default"><a href="afproj.html#K13" title="AFPROJ:func.13">IncProjSp_of</a> <font color="Maroon" title="c1">AS</font></span>)</span><b> such that </b><br/><a NAME="E7:65"/><i><font color="Green" title="E56">A3</font></i>: 
( <font color="Maroon" title="c20">q'</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c18">P'</font> &amp; <font color="Maroon" title="c20">q'</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c19">Q'</font> )
 <b>by </b><i><a class="txt" href="afproj.html#E4:65"><i><font color="Green" title="E55">A2</font></i></a>, <a class="txt" href="afproj.html#E69"><i><font color="Green" title="E52">Lm12</font></i></a></i>;<br/>
<a NAME="E8:65"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c20">q'</font>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">P</font>,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="incsp_1.html#U3" title="INCSP_1:sel.3">Inc</a> of <span class="p1">(<span class="default"><a href="afproj.html#K13" title="AFPROJ:func.13">IncProjSp_of</a> <font color="Maroon" title="c1">AS</font></span>)</span>
 
<b>by </b><i><a class="txt" href="afproj.html#E7:65"><i><font color="Green" title="E56">A3</font></i></a>, <a class="ref" href="incsp_1.html#D1" title="INCSP_1:def.1">INCSP_1:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c21">q</font> = <font color="Maroon" title="c20">q'</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <span class="p1">(<span class="default"><a href="afproj.html#K14" title="AFPROJ:func.14">ProjHorizon</a> <font color="Maroon" title="c1">AS</font></span>)</span> <b>by </b><i><a class="ref" href="afproj.html#T42" target="_self" title="AFPROJ:th.42">Th42</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c21">q</font>
;<br/>

<b>thus </b><a NAME="E10:65"/>
( <font color="Maroon" title="c21">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c9">P</font> &amp; <font color="Maroon" title="c21">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c10">Q</font> )
 <b>by </b><i><a class="txt" href="afproj.html#E7:65"><i><font color="Green" title="E56">A3</font></i></a>, <a class="ref" href="afproj.html#T37" target="_self" title="AFPROJ:th.37">Th37</a></i>;<br/>


</div>
