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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">a</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="c3">a'</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#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="c4">A</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>let </b><font color="Maroon" title="c5">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>;<br/>









<b>assume </b><a NAME="E1:46"/><i><font color="Green" title="E35">A1</font></i>: 
( <font color="Maroon" title="c3">a'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c5">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="c4">A</font>,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 ;<br/>

<b>consider </b><font color="Maroon" title="c6">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="E3:46"/><i><font color="Green" title="E36">A2</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">X</font> &amp; <font color="Maroon" title="c6">X</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#T14" target="_self" title="AFPROJ:th.14">Th14</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="E5:46"/><i><font color="Green" title="E37">A3</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="c7">Y</font> &amp; <font color="Maroon" title="c7">Y</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#T15" target="_self" title="AFPROJ:th.15">Th15</a></i>;<br/>
<div><i><font color="Green" title="E38">A4</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:46_1"/>
<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>
 ;<br/><a NAME="E2:46_1"/><b>then </b>
<font color="Maroon" title="c6">X</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c7">Y</font>
 <b>by </b><i><a class="txt" href="afproj.html#E3:46"><i><font color="Green" title="E36">A2</font></i></a>, <a class="txt" href="afproj.html#E5:46"><i><font color="Green" title="E37">A3</font></i></a>, <a class="ref" href="afproj.html#T36" target="_self" title="AFPROJ:th.36">Th36</a></i>;<br/><b>hence </b><a NAME="E3:46_1"/>
<font color="Maroon" title="c3">a'</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">A'</font>
 <b>by </b><i><a class="txt" href="afproj.html#E1:46"><i><font color="Green" title="E35">A1</font></i></a>, <a class="txt" href="afproj.html#E3:46"><i><font color="Green" title="E36">A2</font></i></a>, <a class="txt" href="afproj.html#E5:46"><i><font color="Green" title="E37">A3</font></i></a>, <a class="ref" href="afproj.html#T29" target="_self" title="AFPROJ:th.29">Th29</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:46_2"/>
<font color="Maroon" title="c3">a'</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">A'</font>
 ;<br/><a NAME="E2:46_2"/><b>then </b>
<font color="Maroon" title="c6">X</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c7">Y</font>
 <b>by </b><i><a class="txt" href="afproj.html#E1:46"><i><font color="Green" title="E35">A1</font></i></a>, <a class="txt" href="afproj.html#E3:46"><i><font color="Green" title="E36">A2</font></i></a>, <a class="txt" href="afproj.html#E5:46"><i><font color="Green" title="E37">A3</font></i></a>, <a class="ref" href="afproj.html#T29" target="_self" title="AFPROJ:th.29">Th29</a></i>;<br/><b>hence </b><a NAME="E3:46_2"/>
<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>
 <b>by </b><i><a class="txt" href="afproj.html#E3:46"><i><font color="Green" title="E36">A2</font></i></a>, <a class="txt" href="afproj.html#E5:46"><i><font color="Green" title="E37">A3</font></i></a>, <a class="ref" href="afproj.html#T36" target="_self" title="AFPROJ:th.36">Th36</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:46"/>
( <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> iff <font color="Maroon" title="c3">a'</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">A'</font> )
 <b>by </b><i><a class="txt" href="afproj.html#E6:46"><i><font color="Green" title="E38">A4</font></i></a></i>;<br/>


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