<?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>assume </b><a NAME="E1:49"/><i><font color="Green" title="E38">A1</font></i>: 
<font color="Maroon" title="c1">AS</font> is  not   <a href="diraf.html#NM2" title="DIRAF:NM.2">AffinPlane</a>
 ;<br/>

<b>consider </b><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y</font>, <font color="Maroon" title="c4">z</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font>;<br/>
<b>consider </b><font color="Maroon" title="c5">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="E4:49"/><i><font color="Green" title="E39">A2</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X</font>
 <b>and </b><br/><a NAME="E5:49"/>
( <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X</font> &amp; <font color="Maroon" title="c4">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X</font> )
 <b>and </b><br/><a NAME="E6:49"/><i><font color="Green" title="E40">A3</font></i>: 
<font color="Maroon" title="c5">X</font> <a href="aff_4.html#NR1" title="AFF_4:NR.1">is_plane</a> 
 <b>by </b><i><a class="ref" href="aff_4.html#T37" title="AFF_4:th.37">AFF_4:37</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">t</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font><b> such that </b><br/><a NAME="E8:49"/><i><font color="Green" title="E41">A4</font></i>: 
not <font color="Maroon" title="c6">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X</font>
 <b>by </b><i><a class="txt" href="afproj.html#E1:49"><i><font color="Green" title="E38">A1</font></i></a>, <a class="txt" href="afproj.html#E6:49"><i><font color="Green" title="E40">A3</font></i></a>, <a class="ref" href="aff_4.html#T48" title="AFF_4:th.48">AFF_4:48</a></i>;<br/>
<b>set </b><font color="Maroon" title="c7">Y</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c6">t</font>;<br/>
<a NAME="E9:49"/><i><font color="Green" title="E42">A5</font></i>: 
(  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c6">t</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c6">t</font> &amp; <font color="Maroon" title="c6">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c6">t</font> )
 
<b>by </b><i><a class="txt" href="afproj.html#E4:49"><i><font color="Green" title="E39">A2</font></i></a>, <a class="txt" href="afproj.html#E8:49"><i><font color="Green" title="E41">A4</font></i></a>, <a class="ref" href="aff_1.html#T26" title="AFF_1:th.26">AFF_1:26</a>, <a class="ref" href="aff_1.html#D3" title="AFF_1:def.3">AFF_1:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c8">a</font> =  <a href="afproj.html#K5" title="AFPROJ:func.5">LDir</a> <span class="p1">(<span class="default"><a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c6">t</font></span>)</span> 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#T14" target="_self" title="AFPROJ:th.14">Th14</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">A</font> =  <a href="afproj.html#K6" title="AFPROJ:func.6">PDir</a> <font color="Maroon" title="c5">X</font> as    <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>by </b><i><a class="txt" href="afproj.html#E6:49"><i><font color="Green" title="E40">A3</font></i></a>, <a class="ref" href="afproj.html#T15" target="_self" title="AFPROJ:th.15">Th15</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c8">a</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c9">A</font>
;<br/>

<b>thus </b><a NAME="E12:49"/>
not <font color="Maroon" title="c8">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c9">A</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:49_1"/>
<font color="Maroon" title="c8">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c9">A</font>
 ;<br/>

<a NAME="E2:49_1"/><b>then </b>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c6">t</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c5">X</font>
 
<b>by </b><i><a class="txt" href="afproj.html#E6:49"><i><font color="Green" title="E40">A3</font></i></a>, <a class="txt" href="afproj.html#E9:49"><i><font color="Green" title="E42">A5</font></i></a>, <a class="ref" href="afproj.html#T36" target="_self" title="AFPROJ:th.36">Th36</a></i>;<br/>
<a NAME="E3:49_1"/><b>then </b>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c6">t</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">X</font>
 
<b>by </b><i><a class="txt" href="afproj.html#E4:49"><i><font color="Green" title="E39">A2</font></i></a>, <a class="txt" href="afproj.html#E6:49"><i><font color="Green" title="E40">A3</font></i></a>, <a class="txt" href="afproj.html#E9:49"><i><font color="Green" title="E42">A5</font></i></a>, <a class="ref" href="aff_4.html#T43" title="AFF_4:th.43">AFF_4:43</a></i>;<br/>
<b>hence </b><a NAME="E4:49_1"/>
contradiction
 <b>by </b><i><a class="txt" href="afproj.html#E8:49"><i><font color="Green" title="E41">A4</font></i></a>, <a class="txt" href="afproj.html#E9:49"><i><font color="Green" title="E42">A5</font></i></a></i>;<br/>


</div><b>end;</b></div>


</div>
