<?xml version="1.0"?>
<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> 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">A</font>, <font color="Maroon" title="c5">K</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><b>that </b><br/><a NAME="E1:47"/><i><font color="Green" title="E38">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="c4">A</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="c5">K</font> )
 <b>and </b><br/><a NAME="E2:47"/><i><font color="Green" title="E39">A2</font></i>: 
( <font color="Maroon" title="c3">b</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c4">A</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="c5">K</font> )
 ;<br/>

<b>assume </b><a NAME="E3:47"/><i><font color="Green" title="E40">A3</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c6">Y</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="E5:47"/><i><font color="Green" title="E41">A4</font></i>: 
( <font color="Maroon" title="c2">a</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="c6">Y</font> &amp; <font color="Maroon" title="c6">Y</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#T21" target="_self" title="AFPROJ:th.21">Th21</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">Y'</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="E7:47"/><i><font color="Green" title="E42">A5</font></i>: 
( <font color="Maroon" title="c3">b</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">Y'</font> &amp; <font color="Maroon" title="c7">Y'</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#T21" target="_self" title="AFPROJ:th.21">Th21</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">X</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="E9:47"/><i><font color="Green" title="E43">A6</font></i>: 
( <font color="Maroon" title="c4">A</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="c8">X</font> &amp; <font color="Maroon" title="c8">X</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#T24" target="_self" title="AFPROJ:th.24">Th24</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">X'</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="E11:47"/><i><font color="Green" title="E44">A7</font></i>: 
( <font color="Maroon" title="c5">K</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="c9">X'</font> &amp; <font color="Maroon" title="c9">X'</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#T24" target="_self" title="AFPROJ:th.24">Th24</a></i>;<br/>
<a NAME="E12:47"/><i><font color="Green" title="E45">A8</font></i>: 
not <font color="Maroon" title="c6">Y</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c7">Y'</font>
 
<b>by </b><i><a class="txt" href="afproj.html#E3:47"><i><font color="Green" title="E40">A3</font></i></a>, <a class="txt" href="afproj.html#E5:47"><i><font color="Green" title="E41">A4</font></i></a>, <a class="txt" href="afproj.html#E7:47"><i><font color="Green" title="E42">A5</font></i></a>, <a class="ref" href="afproj.html#T11" target="_self" title="AFPROJ:th.11">Th11</a></i>;<br/>
<a NAME="E13:47"/>
( <font color="Maroon" title="c6">Y</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c8">X</font> &amp; <font color="Maroon" title="c6">Y</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c9">X'</font> &amp; <font color="Maroon" title="c7">Y'</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c8">X</font> &amp; <font color="Maroon" title="c7">Y'</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c9">X'</font> )
 
<b>by </b><i><a class="txt" href="afproj.html#E1:47"><i><font color="Green" title="E38">A1</font></i></a>, <a class="txt" href="afproj.html#E2:47"><i><font color="Green" title="E39">A2</font></i></a>, <a class="txt" href="afproj.html#E5:47"><i><font color="Green" title="E41">A4</font></i></a>, <a class="txt" href="afproj.html#E7:47"><i><font color="Green" title="E42">A5</font></i></a>, <a class="txt" href="afproj.html#E9:47"><i><font color="Green" title="E43">A6</font></i></a>, <a class="txt" href="afproj.html#E11:47"><i><font color="Green" title="E44">A7</font></i></a>, <a class="ref" href="afproj.html#T36" target="_self" title="AFPROJ:th.36">Th36</a></i>;<br/>
<a NAME="E14:47"/><b>then </b>
<font color="Maroon" title="c8">X</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c9">X'</font>
 
<b>by </b><i><a class="txt" href="afproj.html#E5:47"><i><font color="Green" title="E41">A4</font></i></a>, <a class="txt" href="afproj.html#E7:47"><i><font color="Green" title="E42">A5</font></i></a>, <a class="txt" href="afproj.html#E9:47"><i><font color="Green" title="E43">A6</font></i></a>, <a class="txt" href="afproj.html#E11:47"><i><font color="Green" title="E44">A7</font></i></a>, <a class="txt" href="afproj.html#E12:47"><i><font color="Green" title="E45">A8</font></i></a>, <a class="ref" href="afproj.html#T5" target="_self" title="AFPROJ:th.5">Th5</a></i>;<br/>
<b>hence </b><a NAME="E15:47"/>
<font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">K</font>
 <b>by </b><i><a class="txt" href="afproj.html#E9:47"><i><font color="Green" title="E43">A6</font></i></a>, <a class="txt" href="afproj.html#E11:47"><i><font color="Green" title="E44">A7</font></i></a>, <a class="ref" href="afproj.html#T13" target="_self" title="AFPROJ:th.13">Th13</a></i>;<br/>


</div>
