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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">Y</font>, <font color="Maroon" title="c3">X</font>, <font color="Maroon" title="c4">X'</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="c5">P</font>, <font color="Maroon" title="c6">Q</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="E1:60"/><i><font color="Green" title="E51">A1</font></i>: 
( <font color="Maroon" title="c2">Y</font> <a href="aff_4.html#NR1" title="AFF_4:NR.1">is_plane</a>  &amp; <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="c3">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">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="c2">Y</font> )
 <b>and </b><br/><a NAME="E2:60"/><i><font color="Green" title="E52">A2</font></i>: 
( <font color="Maroon" title="c5">P</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="c6">Q</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> )
 ;<br/>

<div><i><font color="Green" title="E53">A3</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:60_1"/>
<font color="Maroon" title="c3">X</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c4">X'</font>
 ;<br/><a NAME="E2:60_1"/><b>then </b><i><font color="Green" title="E54">A4</font></i>: 
 <a href="afproj.html#K5" title="AFPROJ:func.5">LDir</a> <font color="Maroon" title="c3">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="c4">X'</font>
 <b>by </b><i><a class="txt" href="afproj.html#E1:60"><i><font color="Green" title="E51">A1</font></i></a>, <a class="ref" href="afproj.html#T11" target="_self" title="AFPROJ:th.11">Th11</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c7">q</font> =  <a href="afproj.html#K5" title="AFPROJ:func.5">LDir</a> <font color="Maroon" title="c3">X</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="txt" href="afproj.html#E1:60"><i><font color="Green" title="E51">A1</font></i></a>, <a class="ref" href="afproj.html#T20" target="_self" title="AFPROJ:th.20">Th20</a></i>;<br/><b>take </b><font color="Maroon" title="c8">q</font> = <font color="Maroon" title="c7">q</font>;<br/><b>thus </b><a NAME="E4:60_1"/>
( <font color="Maroon" title="c8">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">P</font> &amp; <font color="Maroon" title="c8">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c6">Q</font> )
 <b>by </b><i><a class="txt" href="afproj.html#E1:60"><i><font color="Green" title="E51">A1</font></i></a>, <a class="txt" href="afproj.html#E2:60"><i><font color="Green" title="E52">A2</font></i></a>, <a class="txt" href="afproj.html#E2:60_1"><i><font color="Green" title="E54">A4</font></i></a>, <a class="ref" href="afproj.html#T30" target="_self" title="AFPROJ:th.30">Th30</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_2"><b>now </b></a><div class="add"><b>given </b><font color="Maroon" title="c7">y</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><a NAME="E1:60_2"/><i><font color="Green" title="E54">A5</font></i>: 
( <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X'</font> )
 ;<br/><b>reconsider </b><font color="Maroon" title="c8">q</font> = <font color="Maroon" title="c7">y</font> 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#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#T20" target="_self" title="AFPROJ:th.20">Th20</a></i>;<br/><b>take </b><font color="Maroon" title="c9">q</font> = <font color="Maroon" title="c8">q</font>;<br/><b>thus </b><a NAME="E3:60_2"/>
( <font color="Maroon" title="c9">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">P</font> &amp; <font color="Maroon" title="c9">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c6">Q</font> )
 <b>by </b><i><a class="txt" href="afproj.html#E1:60"><i><font color="Green" title="E51">A1</font></i></a>, <a class="txt" href="afproj.html#E2:60"><i><font color="Green" title="E52">A2</font></i></a>, <a class="txt" href="afproj.html#E1:60_2"><i><font color="Green" title="E54">A5</font></i></a>, <a class="ref" href="afproj.html#T26" target="_self" title="AFPROJ:th.26">Th26</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:60"/>
 ex <font color="Olive" title="b1">q</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> st <br/>( <font color="Olive" title="b1">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">P</font> &amp; <font color="Olive" title="b1">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c6">Q</font> )
 <b>by </b><i><a class="txt" href="afproj.html#E1:60"><i><font color="Green" title="E51">A1</font></i></a>, <a class="txt" href="afproj.html#E3:60"><i><font color="Green" title="E53">A3</font></i></a>, <a class="ref" href="aff_4.html#T22" title="AFF_4:th.22">AFF_4:22</a></i>;<br/>


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