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<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">x</font>, <font color="Maroon" title="c3">y</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="c4">X</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#K13" title="AFPROJ:func.13">IncProjSp_of</a> <font color="Maroon" title="c1">AS</font></span>)</span>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:55"/><i><font color="Green" title="E44">A1</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">y</font>
 <b>and </b><br/><a NAME="E2:55"/><i><font color="Green" title="E45">A2</font></i>: 
( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">X</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</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> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c4">X</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</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> )
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c5">a</font> = <font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c6">b</font> = <font color="Maroon" title="c3">y</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/>
<a NAME="E4:55"/><i><font color="Green" title="E46">A3</font></i>: 
( <font color="Maroon" title="c5">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c4">X</font> &amp; <font color="Maroon" title="c6">b</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c4">X</font> )
 
<b>by </b><i><a class="txt" href="afproj.html#E2:55"><i><font color="Green" title="E45">A2</font></i></a>, <a class="ref" href="incsp_1.html#D1" title="INCSP_1:def.1">INCSP_1:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">X1</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font><b> such that </b><br/><a NAME="E6:55"/><i><font color="Green" title="E47">A4</font></i>: 
( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afproj.html#K5" title="AFPROJ:func.5">LDir</a> <font color="Maroon" title="c7">X1</font> &amp; <font color="Maroon" title="c7">X1</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  )
 <b>by </b><i><a class="ref" href="afproj.html#T14" target="_self" title="AFPROJ:th.14">Th14</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">X2</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font><b> such that </b><br/><a NAME="E8:55"/><i><font color="Green" title="E48">A5</font></i>: 
( <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afproj.html#K5" title="AFPROJ:func.5">LDir</a> <font color="Maroon" title="c8">X2</font> &amp; <font color="Maroon" title="c8">X2</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  )
 <b>by </b><i><a class="ref" href="afproj.html#T14" target="_self" title="AFPROJ:th.14">Th14</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">Y</font> being    <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><b> such that </b><br/><a NAME="E10:55"/><i><font color="Green" title="E49">A6</font></i>: 
( <font color="Maroon" title="c2">x</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c9">Y</font> &amp; <font color="Maroon" title="c3">y</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c9">Y</font> )
 <b>by </b><i><a class="ref" href="afproj.html#T40" target="_self" title="AFPROJ:th.40">Th40</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c9">Y</font>
;<br/>

<b>consider </b><font color="Maroon" title="c10">Z</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font><b> such that </b><br/><a NAME="E12:55"/><i><font color="Green" title="E50">A7</font></i>: 
( <font color="Maroon" title="c9">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afproj.html#K6" title="AFPROJ:func.6">PDir</a> <font color="Maroon" title="c10">Z</font> &amp; <font color="Maroon" title="c10">Z</font> <a href="aff_4.html#NR1" title="AFF_4:NR.1">is_plane</a>  )
 <b>by </b><i><a class="ref" href="afproj.html#T15" target="_self" title="AFPROJ:th.15">Th15</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">A</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">Y</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="E14:55"/>
( <font color="Maroon" title="c7">X1</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c10">Z</font> &amp; <font color="Maroon" title="c8">X2</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c10">Z</font> )
 
<b>by </b><i><a class="txt" href="afproj.html#E6:55"><i><font color="Green" title="E47">A4</font></i></a>, <a class="txt" href="afproj.html#E8:55"><i><font color="Green" title="E48">A5</font></i></a>, <a class="txt" href="afproj.html#E10:55"><i><font color="Green" title="E49">A6</font></i></a>, <a class="txt" href="afproj.html#E12:55"><i><font color="Green" title="E50">A7</font></i></a>, <a class="ref" href="afproj.html#T36" target="_self" title="AFPROJ:th.36">Th36</a></i>;<br/>
<a NAME="E15:55"/><b>then </b>
( <font color="Maroon" title="c5">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c11">A</font> &amp; <font color="Maroon" title="c6">b</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c11">A</font> )
 
<b>by </b><i><a class="txt" href="afproj.html#E6:55"><i><font color="Green" title="E47">A4</font></i></a>, <a class="txt" href="afproj.html#E8:55"><i><font color="Green" title="E48">A5</font></i></a>, <a class="txt" href="afproj.html#E12:55"><i><font color="Green" title="E50">A7</font></i></a>, <a class="ref" href="afproj.html#T29" target="_self" title="AFPROJ:th.29">Th29</a></i>;<br/>
<b>hence </b><a NAME="E16:55"/>
<font color="Maroon" title="c4">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">Y</font>,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="afproj.html#E1:55"><i><font color="Green" title="E44">A1</font></i></a>, <a class="txt" href="afproj.html#E4:55"><i><font color="Green" title="E46">A3</font></i></a>, <a class="txt" href="afproj.html#E56"><i><font color="Green" title="E38">Lm2</font></i></a></i>;<br/>


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