<?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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">X</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font><br/> for <font color="Olive" title="b2">p</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><br/> for <font color="Olive" title="b3">A</font> being  <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>  st <font color="Olive" title="b3">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="Olive" title="b1">X</font>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b1">X</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <font color="Olive" title="b2">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b3">A</font> &amp; <font color="Olive" title="b2">p</font> is  not   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font> holds <br/><font color="Olive" title="b2">p</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">X</font></span><br/><b>let </b><font color="Maroon" title="c2">X</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">p</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><br/> for <font color="Olive" title="b2">A</font> being  <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>  st <font color="Olive" title="b2">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="c2">X</font>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c2">X</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <font color="Olive" title="b1">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b2">A</font> &amp; <font color="Olive" title="b1">p</font> is  not   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font> holds <br/><font color="Olive" title="b1">p</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="c2">X</font></span><br/><b>let </b><font color="Maroon" title="c3">p</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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">A</font> being  <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>  st <font color="Olive" title="b1">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="c2">X</font>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c2">X</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <font color="Maroon" title="c3">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b1">A</font> &amp; <font color="Maroon" title="c3">p</font> is  not   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font> holds <br/><font color="Maroon" title="c3">p</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="c2">X</font></span><br/><b>let </b><font color="Maroon" title="c4">A</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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">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="c2">X</font>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c2">X</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <font color="Maroon" title="c3">p</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">p</font> is  not   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font> implies <font color="Maroon" title="c3">p</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="c2">X</font> )</span><br/>







<b>assume </b><b>that </b><br/><a NAME="E1:42"/><i><font color="Green" title="E31">A1</font></i>: 
( <font color="Maroon" title="c4">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="c2">X</font>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c2">X</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <font color="Maroon" title="c3">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c4">A</font> )
 <b>and </b><br/><a NAME="E2:42"/><i><font color="Green" title="E32">A2</font></i>: 
<font color="Maroon" title="c3">p</font> is  not   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">p</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="c2">X</font></span><br/>

<b>consider </b><font color="Maroon" title="c5">Xp</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="E4:42"/><i><font color="Green" title="E33">A3</font></i>: 
( <font color="Maroon" title="c3">p</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="c5">Xp</font> &amp; <font color="Maroon" title="c5">Xp</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#E2:42"><i><font color="Green" title="E32">A2</font></i></a>, <a class="ref" href="afproj.html#T20" target="_self" title="AFPROJ:th.20">Th20</a></i>;<br/>
<a NAME="E5:42"/>
<font color="Maroon" title="c5">Xp</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c2">X</font>
 
<b>by </b><i><a class="txt" href="afproj.html#E1:42"><i><font color="Green" title="E31">A1</font></i></a>, <a class="txt" href="afproj.html#E4:42"><i><font color="Green" title="E33">A3</font></i></a>, <a class="ref" href="afproj.html#T28" target="_self" title="AFPROJ:th.28">Th28</a></i>;<br/>
<b>hence </b><a NAME="E6:42"/>
<font color="Maroon" title="c3">p</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="c2">X</font>
 <b>by </b><i><a class="txt" href="afproj.html#E1:42"><i><font color="Green" title="E31">A1</font></i></a>, <a class="txt" href="afproj.html#E4:42"><i><font color="Green" title="E33">A3</font></i></a>, <a class="ref" href="afproj.html#T12" target="_self" title="AFPROJ:th.12">Th12</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
