<?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">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font>;<br/><b>let </b><font color="Maroon" title="c3">X</font>, <font color="Maroon" title="c4">X'</font>, <font color="Maroon" title="c5">Y</font>, <font color="Maroon" title="c6">Y'</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font>;<br/><b>let </b><font color="Maroon" title="c7">p</font>, <font color="Maroon" title="c8">b</font>, <font color="Maroon" title="c9">c</font> be   <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>;<br/><b>let </b><font color="Maroon" title="c10">A</font>, <font color="Maroon" title="c11">K</font>, <font color="Maroon" title="c12">M</font> be   <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>;<br/>























<b>assume </b><b>that </b><br/><a NAME="E1:59"/><i><font color="Green" title="E48">A1</font></i>: 
( <font color="Maroon" title="c3">X</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <font color="Maroon" title="c4">X'</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <font color="Maroon" title="c5">Y</font> <a href="aff_4.html#NR1" title="AFF_4:NR.1">is_plane</a>  )
 <b>and </b><br/><a NAME="E2:59"/><i><font color="Green" title="E49">A2</font></i>: 
( <font color="Maroon" title="c3">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">Y</font> &amp; <font color="Maroon" title="c4">X'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">Y</font> )
 <b>and </b><br/><a NAME="E3:59"/><i><font color="Green" title="E50">A3</font></i>: 
( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p</font> &amp; <font color="Maroon" title="c10">A</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="c3">X</font>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c11">K</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="c4">X'</font>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c10">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">K</font> )
 <b>and </b><br/><a NAME="E4:59"/><i><font color="Green" title="E51">A4</font></i>: 
( <font color="Maroon" title="c8">b</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c10">A</font> &amp; <font color="Maroon" title="c9">c</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c11">K</font> &amp; <font color="Maroon" title="c8">b</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c12">M</font> &amp; <font color="Maroon" title="c9">c</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c12">M</font> &amp; <font color="Maroon" title="c8">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">c</font> )
 <b>and </b><br/><a NAME="E5:59"/><i><font color="Green" title="E52">A5</font></i>: 
( <font color="Maroon" title="c12">M</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"><span class="p2">(<span class="default"><a href="afproj.html#K6" title="AFPROJ:func.6">PDir</a> <font color="Maroon" title="c6">Y'</font></span>)</span>,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c6">Y'</font> <a href="aff_4.html#NR1" title="AFF_4:NR.1">is_plane</a>  )
 ;<br/>

<a NAME="E6:59"/><i><font color="Green" title="E53">A6</font></i>: 
( <font color="Maroon" title="c8">b</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font> or  ex <font color="Olive" title="b1">Xb</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font> st <br/>( <font color="Maroon" title="c8">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="Olive" title="b1">Xb</font> &amp; <font color="Olive" title="b1">Xb</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#T20" target="_self" title="AFPROJ:th.20">Th20</a></i>;<br/>
<a NAME="E7:59"/><i><font color="Green" title="E54">A7</font></i>: 
( <font color="Maroon" title="c9">c</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font> or  ex <font color="Olive" title="b1">Xc</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font> st <br/>( <font color="Maroon" title="c9">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afproj.html#K5" title="AFPROJ:func.5">LDir</a> <font color="Olive" title="b1">Xc</font> &amp; <font color="Olive" title="b1">Xc</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#T20" target="_self" title="AFPROJ:th.20">Th20</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c13">Xb</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="E9:59"/><i><font color="Green" title="E55">A8</font></i>: 
( <font color="Maroon" title="c8">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="c13">Xb</font> &amp; <font color="Maroon" title="c13">Xb</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  )
 <b>by </b><i><a class="txt" href="afproj.html#E4:59"><i><font color="Green" title="E51">A4</font></i></a>, <a class="txt" href="afproj.html#E5:59"><i><font color="Green" title="E52">A5</font></i></a>, <a class="txt" href="afproj.html#E6:59"><i><font color="Green" title="E53">A6</font></i></a>, <a class="ref" href="afproj.html#T27" target="_self" title="AFPROJ:th.27">Th27</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c14">Xc</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="E11:59"/><i><font color="Green" title="E56">A9</font></i>: 
( <font color="Maroon" title="c9">c</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="c14">Xc</font> &amp; <font color="Maroon" title="c14">Xc</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  )
 <b>by </b><i><a class="txt" href="afproj.html#E4:59"><i><font color="Green" title="E51">A4</font></i></a>, <a class="txt" href="afproj.html#E5:59"><i><font color="Green" title="E52">A5</font></i></a>, <a class="txt" href="afproj.html#E7:59"><i><font color="Green" title="E54">A7</font></i></a>, <a class="ref" href="afproj.html#T27" target="_self" title="AFPROJ:th.27">Th27</a></i>;<br/>
<a NAME="E12:59"/><i><font color="Green" title="E57">A10</font></i>: 
not <font color="Maroon" title="c13">Xb</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c14">Xc</font>
 
<b>by </b><i><a class="txt" href="afproj.html#E4:59"><i><font color="Green" title="E51">A4</font></i></a>, <a class="txt" href="afproj.html#E9:59"><i><font color="Green" title="E55">A8</font></i></a>, <a class="txt" href="afproj.html#E11:59"><i><font color="Green" title="E56">A9</font></i></a>, <a class="ref" href="afproj.html#T11" target="_self" title="AFPROJ:th.11">Th11</a></i>;<br/>
<a NAME="E13:59"/><i><font color="Green" title="E58">A11</font></i>: 
( <font color="Maroon" title="c13">Xb</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c6">Y'</font> &amp; <font color="Maroon" title="c14">Xc</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c6">Y'</font> )
 
<b>by </b><i><a class="txt" href="afproj.html#E4:59"><i><font color="Green" title="E51">A4</font></i></a>, <a class="txt" href="afproj.html#E5:59"><i><font color="Green" title="E52">A5</font></i></a>, <a class="txt" href="afproj.html#E9:59"><i><font color="Green" title="E55">A8</font></i></a>, <a class="txt" href="afproj.html#E11:59"><i><font color="Green" title="E56">A9</font></i></a>, <a class="ref" href="afproj.html#T29" target="_self" title="AFPROJ:th.29">Th29</a></i>;<br/>
<a NAME="E14:59"/><i><font color="Green" title="E59">A12</font></i>: 
( <font color="Maroon" title="c3">X</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c5">Y</font> &amp; <font color="Maroon" title="c4">X'</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c5">Y</font> )
 
<b>by </b><i><a class="txt" href="afproj.html#E1:59"><i><font color="Green" title="E48">A1</font></i></a>, <a class="txt" href="afproj.html#E2:59"><i><font color="Green" title="E49">A2</font></i></a>, <a class="ref" href="aff_4.html#T42" title="AFF_4:th.42">AFF_4:42</a></i>;<br/>
<a NAME="E15:59"/>
( <font color="Maroon" title="c13">Xb</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c14">Xc</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c4">X'</font> )
 
<b>by </b><i><a class="txt" href="afproj.html#E1:59"><i><font color="Green" title="E48">A1</font></i></a>, <a class="txt" href="afproj.html#E3:59"><i><font color="Green" title="E50">A3</font></i></a>, <a class="txt" href="afproj.html#E4:59"><i><font color="Green" title="E51">A4</font></i></a>, <a class="txt" href="afproj.html#E9:59"><i><font color="Green" title="E55">A8</font></i></a>, <a class="txt" href="afproj.html#E11:59"><i><font color="Green" title="E56">A9</font></i></a>, <a class="ref" href="afproj.html#T28" target="_self" title="AFPROJ:th.28">Th28</a></i>;<br/>
<a NAME="E16:59"/><b>then </b>
( <font color="Maroon" title="c13">Xb</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c14">Xc</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c4">X'</font> )
 
<b>by </b><i><a class="txt" href="afproj.html#E1:59"><i><font color="Green" title="E48">A1</font></i></a>, <a class="txt" href="afproj.html#E9:59"><i><font color="Green" title="E55">A8</font></i></a>, <a class="txt" href="afproj.html#E11:59"><i><font color="Green" title="E56">A9</font></i></a>, <a class="ref" href="aff_4.html#T40" title="AFF_4:th.40">AFF_4:40</a></i>;<br/>
<a NAME="E17:59"/><b>then </b>
( <font color="Maroon" title="c13">Xb</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c5">Y</font> &amp; <font color="Maroon" title="c14">Xc</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c5">Y</font> )
 
<b>by </b><i><a class="txt" href="afproj.html#E14:59"><i><font color="Green" title="E59">A12</font></i></a>, <a class="ref" href="aff_4.html#T56" title="AFF_4:th.56">AFF_4:56</a></i>;<br/>
<b>hence </b><a NAME="E18:59"/>
( <font color="Maroon" title="c6">Y'</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c5">Y</font> &amp; <font color="Maroon" title="c5">Y</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c6">Y'</font> )
 <b>by </b><i><a class="txt" href="afproj.html#E1:59"><i><font color="Green" title="E48">A1</font></i></a>, <a class="txt" href="afproj.html#E5:59"><i><font color="Green" title="E52">A5</font></i></a>, <a class="txt" href="afproj.html#E9:59"><i><font color="Green" title="E55">A8</font></i></a>, <a class="txt" href="afproj.html#E11:59"><i><font color="Green" title="E56">A9</font></i></a>, <a class="txt" href="afproj.html#E12:59"><i><font color="Green" title="E57">A10</font></i></a>, <a class="txt" href="afproj.html#E13:59"><i><font color="Green" title="E58">A11</font></i></a>, <a class="ref" href="afproj.html#T5" target="_self" title="AFPROJ:th.5">Th5</a></i>;<br/>


</div>
