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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>; <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>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font>, <font color="Olive" title="b4">d</font>, <font color="Olive" title="b5">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="b6">M</font>, <font color="Olive" title="b7">N</font>, <font color="Olive" title="b8">P</font>, <font color="Olive" title="b9">Q</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="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b6">M</font> &amp; <font color="Olive" title="b2">b</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b6">M</font> &amp; <font color="Olive" title="b3">c</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b7">N</font> &amp; <font color="Olive" title="b4">d</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b7">N</font> &amp; <font color="Olive" title="b5">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b6">M</font> &amp; <font color="Olive" title="b5">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b7">N</font> &amp; <font color="Olive" title="b1">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b8">P</font> &amp; <font color="Olive" title="b3">c</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b8">P</font> &amp; <font color="Olive" title="b2">b</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b9">Q</font> &amp; <font color="Olive" title="b4">d</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b9">Q</font> &amp; not <font color="Olive" title="b5">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b8">P</font> &amp; not <font color="Olive" title="b5">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b9">Q</font> &amp; <font color="Olive" title="b6">M</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b7">N</font> &amp; <font color="Olive" title="b5">p</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font> holds <br/> ex <font color="Olive" title="b10">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="b10">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b8">P</font> &amp; <font color="Olive" title="b10">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b9">Q</font> )</span><br/><b>let </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font>, <font color="Maroon" title="c4">c</font>, <font color="Maroon" title="c5">d</font>, <font color="Maroon" title="c6">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">M</font>, <font color="Olive" title="b2">N</font>, <font color="Olive" title="b3">P</font>, <font color="Olive" title="b4">Q</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="Maroon" title="c2">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b1">M</font> &amp; <font color="Maroon" title="c3">b</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b1">M</font> &amp; <font color="Maroon" title="c4">c</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b2">N</font> &amp; <font color="Maroon" title="c5">d</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b2">N</font> &amp; <font color="Maroon" title="c6">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b1">M</font> &amp; <font color="Maroon" title="c6">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b2">N</font> &amp; <font color="Maroon" title="c2">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b3">P</font> &amp; <font color="Maroon" title="c4">c</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b3">P</font> &amp; <font color="Maroon" title="c3">b</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b4">Q</font> &amp; <font color="Maroon" title="c5">d</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b4">Q</font> &amp; not <font color="Maroon" title="c6">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b3">P</font> &amp; not <font color="Maroon" title="c6">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b4">Q</font> &amp; <font color="Olive" title="b1">M</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">N</font> &amp; <font color="Maroon" title="c6">p</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font> holds <br/> ex <font color="Olive" title="b5">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="b5">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b3">P</font> &amp; <font color="Olive" title="b5">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b4">Q</font> )</span><br/><b>let </b><font color="Maroon" title="c7">M</font>, <font color="Maroon" title="c8">N</font>, <font color="Maroon" title="c9">P</font>, <font color="Maroon" title="c10">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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">M</font> &amp; <font color="Maroon" title="c3">b</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">M</font> &amp; <font color="Maroon" title="c4">c</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c8">N</font> &amp; <font color="Maroon" title="c5">d</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c8">N</font> &amp; <font color="Maroon" title="c6">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">M</font> &amp; <font color="Maroon" title="c6">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c8">N</font> &amp; <font color="Maroon" title="c2">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c9">P</font> &amp; <font color="Maroon" title="c4">c</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c9">P</font> &amp; <font color="Maroon" title="c3">b</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c10">Q</font> &amp; <font color="Maroon" title="c5">d</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c10">Q</font> &amp; not <font color="Maroon" title="c6">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c9">P</font> &amp; not <font color="Maroon" title="c6">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c10">Q</font> &amp; <font color="Maroon" title="c7">M</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">N</font> &amp; <font color="Maroon" title="c6">p</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font> implies  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="c9">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="c10">Q</font> ) )</span><br/>



















<b>assume </b><b>that </b><br/><a NAME="E1:60"/><i><font color="Green" title="E49">A1</font></i>: 
( <font color="Maroon" title="c2">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">M</font> &amp; <font color="Maroon" title="c3">b</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">M</font> &amp; <font color="Maroon" title="c4">c</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c8">N</font> &amp; <font color="Maroon" title="c5">d</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c8">N</font> )
 <b>and </b><br/><a NAME="E2:60"/><i><font color="Green" title="E50">A2</font></i>: 
( <font color="Maroon" title="c6">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">M</font> &amp; <font color="Maroon" title="c6">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c8">N</font> )
 <b>and </b><br/><a NAME="E3:60"/><i><font color="Green" title="E51">A3</font></i>: 
( <font color="Maroon" title="c2">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c9">P</font> &amp; <font color="Maroon" title="c4">c</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c9">P</font> &amp; <font color="Maroon" title="c3">b</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c10">Q</font> &amp; <font color="Maroon" title="c5">d</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c10">Q</font> )
 <b>and </b><br/><a NAME="E4:60"/><i><font color="Green" title="E52">A4</font></i>: 
( not <font color="Maroon" title="c6">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c9">P</font> &amp; not <font color="Maroon" title="c6">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c10">Q</font> )
 <b>and </b><br/><a NAME="E5:60"/><i><font color="Green" title="E53">A5</font></i>: 
<font color="Maroon" title="c7">M</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">N</font>
 <b>and </b><br/><a NAME="E6:60"/><i><font color="Green" title="E54">A6</font></i>: 
<font color="Maroon" title="c6">p</font> is   <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">  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="c9">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="c10">Q</font> )</span><br/>

<a NAME="E7:60"/><i><font color="Green" title="E55">A7</font></i>: 
( <font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">c</font> &amp; <font color="Maroon" title="c3">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">d</font> )
 
<b>by </b><i><a class="txt" href="#E1:60"><i><font color="Green" title="E49">A1</font></i></a>, <a class="txt" href="#E2:60"><i><font color="Green" title="E50">A2</font></i></a>, <a class="txt" href="#E3:60"><i><font color="Green" title="E51">A3</font></i></a>, <a class="txt" href="#E4:60"><i><font color="Green" title="E52">A4</font></i></a>, <a class="txt" href="#E5:60"><i><font color="Green" title="E53">A5</font></i></a>, <a class="txt" href="afproj.html#E56"><i><font color="Green" title="E38">Lm2</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c11">XM</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="E9:60"/><i><font color="Green" title="E56">A8</font></i>: 
( ( <font color="Maroon" title="c7">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"><font color="Maroon" title="c11">XM</font>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c11">XM</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> ) or ( <font color="Maroon" title="c7">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="c11">XM</font></span>)</span>,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c11">XM</font> is <a href="aff_4.html#V1" title="AFF_4:attr.1">being_plane</a> ) )
 <b>by </b><i><a class="ref" href="afproj.html#T23" target="_self" title="AFPROJ:th.23">Th23</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">XN</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="E11:60"/><i><font color="Green" title="E57">A9</font></i>: 
( ( <font color="Maroon" title="c8">N</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="c12">XN</font>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c12">XN</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> ) or ( <font color="Maroon" title="c8">N</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="c12">XN</font></span>)</span>,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c12">XN</font> is <a href="aff_4.html#V1" title="AFF_4:attr.1">being_plane</a> ) )
 <b>by </b><i><a class="ref" href="afproj.html#T23" target="_self" title="AFPROJ:th.23">Th23</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c13">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="E13:60"/><i><font color="Green" title="E58">A10</font></i>: 
( ( <font color="Maroon" title="c9">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="c13">XP</font>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c13">XP</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> ) or ( <font color="Maroon" title="c9">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"><span class="p2">(<span class="default"><a href="afproj.html#K6" title="AFPROJ:func.6">PDir</a> <font color="Maroon" title="c13">XP</font></span>)</span>,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c13">XP</font> is <a href="aff_4.html#V1" title="AFF_4:attr.1">being_plane</a> ) )
 <b>by </b><i><a class="ref" href="afproj.html#T23" target="_self" title="AFPROJ:th.23">Th23</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c14">XQ</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="E15:60"/><i><font color="Green" title="E59">A11</font></i>: 
( ( <font color="Maroon" title="c10">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="c14">XQ</font>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c14">XQ</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> ) or ( <font color="Maroon" title="c10">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"><span class="p2">(<span class="default"><a href="afproj.html#K6" title="AFPROJ:func.6">PDir</a> <font color="Maroon" title="c14">XQ</font></span>)</span>,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c14">XQ</font> is <a href="aff_4.html#V1" title="AFF_4:attr.1">being_plane</a> ) )
 <b>by </b><i><a class="ref" href="afproj.html#T23" target="_self" title="AFPROJ:th.23">Th23</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c15">x</font> = <font color="Maroon" title="c6">p</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font> <b>by </b><i><a class="txt" href="#E6:60"><i><font color="Green" title="E54">A6</font></i></a></i>;<br/>
<a NAME="E17:60"/><i><font color="Green" title="E60">A12</font></i>: 
( <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">XM</font> &amp; <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">XN</font> )
 
<b>by </b><i><a class="txt" href="#E2:60"><i><font color="Green" title="E50">A2</font></i></a>, <a class="txt" href="#E9:60"><i><font color="Green" title="E56">A8</font></i></a>, <a class="txt" href="#E11:60"><i><font color="Green" title="E57">A9</font></i></a>, <a class="ref" href="afproj.html#T26" target="_self" title="AFPROJ:th.26">Th26</a>, <a class="ref" href="afproj.html#T27" target="_self" title="AFPROJ:th.27">Th27</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c16">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="E19:60"/><i><font color="Green" title="E61">A13</font></i>: 
( <font color="Maroon" title="c11">XM</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c16">Y</font> &amp; <font color="Maroon" title="c12">XN</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c16">Y</font> )
 <b>and </b><br/><a NAME="E20:60"/><i><font color="Green" title="E62">A14</font></i>: 
<font color="Maroon" title="c16">Y</font> is <a href="aff_4.html#V1" title="AFF_4:attr.1">being_plane</a>
 <b>by </b><i><a class="txt" href="#E2:60"><i><font color="Green" title="E50">A2</font></i></a>, <a class="txt" href="#E9:60"><i><font color="Green" title="E56">A8</font></i></a>, <a class="txt" href="#E11:60"><i><font color="Green" title="E57">A9</font></i></a>, <a class="ref" href="afproj.html#T27" target="_self" title="AFPROJ:th.27">Th27</a>, <a class="ref" href="aff_4.html#T38" title="AFF_4:th.38">AFF_4:38</a></i>;<br/>
<div><i><font color="Green" title="E63">A15</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"/><i><font color="Green" title="E64">A16</font></i>: 
( <font color="Maroon" title="c9">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="c13">XP</font>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c13">XP</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  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="c9">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="c10">Q</font> )</span><br/><a NAME="E2:60_1"/><b>then </b><i><font color="Green" title="E65">A17</font></i>: 
<font color="Maroon" title="c13">XP</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c16">Y</font>
 <b>by </b><i><a class="txt" href="#E1:60"><i><font color="Green" title="E49">A1</font></i></a>, <a class="txt" href="#E2:60"><i><font color="Green" title="E50">A2</font></i></a>, <a class="txt" href="#E3:60"><i><font color="Green" title="E51">A3</font></i></a>, <a class="txt" href="#E5:60"><i><font color="Green" title="E53">A5</font></i></a>, <a class="txt" href="#E7:60"><i><font color="Green" title="E55">A7</font></i></a>, <a class="txt" href="#E9:60"><i><font color="Green" title="E56">A8</font></i></a>, <a class="txt" href="#E11:60"><i><font color="Green" title="E57">A9</font></i></a>, <a class="txt" href="#E17:60"><i><font color="Green" title="E60">A12</font></i></a>, <a class="txt" href="#E19:60"><i><font color="Green" title="E61">A13</font></i></a>, <a class="txt" href="#E20:60"><i><font color="Green" title="E62">A14</font></i></a>, <a class="txt" href="afproj.html#E63"><i><font color="Green" title="E45">Lm7</font></i></a>, <a class="ref" href="afproj.html#T27" target="_self" title="AFPROJ:th.27">Th27</a></i>;<br/><div><i><font color="Green" title="E66">A18</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:60_1_1"/><i><font color="Green" title="E67">A19</font></i>: 
( <font color="Maroon" title="c10">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="c14">XQ</font>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c14">XQ</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  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="c9">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="c10">Q</font> )</span><br/><a NAME="E2:60_1_1"/><b>then </b>
<font color="Maroon" title="c14">XQ</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c16">Y</font>
 <b>by </b><i><a class="txt" href="#E1:60"><i><font color="Green" title="E49">A1</font></i></a>, <a class="txt" href="#E2:60"><i><font color="Green" title="E50">A2</font></i></a>, <a class="txt" href="#E3:60"><i><font color="Green" title="E51">A3</font></i></a>, <a class="txt" href="#E5:60"><i><font color="Green" title="E53">A5</font></i></a>, <a class="txt" href="#E7:60"><i><font color="Green" title="E55">A7</font></i></a>, <a class="txt" href="#E9:60"><i><font color="Green" title="E56">A8</font></i></a>, <a class="txt" href="#E11:60"><i><font color="Green" title="E57">A9</font></i></a>, <a class="txt" href="#E17:60"><i><font color="Green" title="E60">A12</font></i></a>, <a class="txt" href="#E19:60"><i><font color="Green" title="E61">A13</font></i></a>, <a class="txt" href="#E20:60"><i><font color="Green" title="E62">A14</font></i></a>, <a class="txt" href="afproj.html#E63"><i><font color="Green" title="E45">Lm7</font></i></a>, <a class="ref" href="afproj.html#T27" target="_self" title="AFPROJ:th.27">Th27</a></i>;<br/><b>hence </b><a NAME="E3:60_1_1"/>
 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="c9">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="c10">Q</font> )
 <b>by </b><i><a class="txt" href="#E20:60"><i><font color="Green" title="E62">A14</font></i></a>, <a class="txt" href="#E1:60_1"><i><font color="Green" title="E64">A16</font></i></a>, <a class="txt" href="#E2:60_1"><i><font color="Green" title="E65">A17</font></i></a>, <a class="txt" href="#E1:60_1_1"><i><font color="Green" title="E67">A19</font></i></a>, <a class="ref" href="afproj.html#T43" target="_self" title="AFPROJ:th.43">Th43</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><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:60_1_2"/><i><font color="Green" title="E67">A20</font></i>: 
( <font color="Maroon" title="c10">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"><span class="p2">(<span class="default"><a href="afproj.html#K6" title="AFPROJ:func.6">PDir</a> <font color="Maroon" title="c14">XQ</font></span>)</span>,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c14">XQ</font> is <a href="aff_4.html#V1" title="AFF_4:attr.1">being_plane</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  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="c9">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="c10">Q</font> )</span><br/><a NAME="E2:60_1_2"/><b>then </b><i><font color="Green" title="E68">A21</font></i>: 
<font color="Maroon" title="c16">Y</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c14">XQ</font>
 <b>by </b><i><a class="txt" href="#E1:60"><i><font color="Green" title="E49">A1</font></i></a>, <a class="txt" href="#E2:60"><i><font color="Green" title="E50">A2</font></i></a>, <a class="txt" href="#E3:60"><i><font color="Green" title="E51">A3</font></i></a>, <a class="txt" href="#E5:60"><i><font color="Green" title="E53">A5</font></i></a>, <a class="txt" href="#E7:60"><i><font color="Green" title="E55">A7</font></i></a>, <a class="txt" href="#E9:60"><i><font color="Green" title="E56">A8</font></i></a>, <a class="txt" href="#E11:60"><i><font color="Green" title="E57">A9</font></i></a>, <a class="txt" href="#E17:60"><i><font color="Green" title="E60">A12</font></i></a>, <a class="txt" href="#E19:60"><i><font color="Green" title="E61">A13</font></i></a>, <a class="txt" href="#E20:60"><i><font color="Green" title="E62">A14</font></i></a>, <a class="txt" href="afproj.html#E64"><i><font color="Green" title="E46">Lm8</font></i></a>, <a class="ref" href="afproj.html#T27" target="_self" title="AFPROJ:th.27">Th27</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c17">q</font> =  <a href="afproj.html#K5" title="AFPROJ:func.5">LDir</a> <font color="Maroon" title="c13">XP</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="#E1:60_1"><i><font color="Green" title="E64">A16</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="c18">q</font> = <font color="Maroon" title="c17">q</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c18">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c9">P</font> &amp; <font color="Maroon" title="c18">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c10">Q</font> )</span><br/><b>thus </b><a NAME="E4:60_1_2"/>
<font color="Maroon" title="c18">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c9">P</font>
 <b>by </b><i><a class="txt" href="#E1:60_1"><i><font color="Green" title="E64">A16</font></i></a>, <a class="ref" href="afproj.html#T30" target="_self" title="AFPROJ:th.30">Th30</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c18">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c10">Q</font></span><br/><a NAME="E5:60_1_2"/>
<font color="Maroon" title="c13">XP</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c16">Y</font>
 <b>by </b><i><a class="txt" href="#E20:60"><i><font color="Green" title="E62">A14</font></i></a>, <a class="txt" href="#E1:60_1"><i><font color="Green" title="E64">A16</font></i></a>, <a class="txt" href="#E2:60_1"><i><font color="Green" title="E65">A17</font></i></a>, <a class="ref" href="aff_4.html#T42" title="AFF_4:th.42">AFF_4:42</a></i>;<br/><a NAME="E6:60_1_2"/><b>then </b>
<font color="Maroon" title="c13">XP</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c14">XQ</font>
 <b>by </b><i><a class="txt" href="#E20:60"><i><font color="Green" title="E62">A14</font></i></a>, <a class="txt" href="#E2:60_1_2"><i><font color="Green" title="E68">A21</font></i></a>, <a class="ref" href="aff_4.html#T59" title="AFF_4:th.59">AFF_4:59</a>, <a class="ref" href="aff_4.html#T60" title="AFF_4:th.60">AFF_4:60</a></i>;<br/><b>hence </b><a NAME="E7:60_1_2"/>
<font color="Maroon" title="c18">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c10">Q</font>
 <b>by </b><i><a class="txt" href="#E1:60_1"><i><font color="Green" title="E64">A16</font></i></a>, <a class="txt" href="#E1:60_1_2"><i><font color="Green" title="E67">A20</font></i></a>, <a class="ref" href="afproj.html#T29" target="_self" title="AFPROJ:th.29">Th29</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><b>end;</b></div><b>hence </b><a NAME="E5:60_1"/>
 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="c9">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="c10">Q</font> )
 <b>by </b><i><a class="txt" href="#E15:60"><i><font color="Green" title="E59">A11</font></i></a>, <a class="txt" href="#E3:60_1"><i><font color="Green" title="E66">A18</font></i></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><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:60_2"/><i><font color="Green" title="E64">A22</font></i>: 
( <font color="Maroon" title="c9">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"><span class="p2">(<span class="default"><a href="afproj.html#K6" title="AFPROJ:func.6">PDir</a> <font color="Maroon" title="c13">XP</font></span>)</span>,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c13">XP</font> is <a href="aff_4.html#V1" title="AFF_4:attr.1">being_plane</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  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="c9">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="c10">Q</font> )</span><br/><a NAME="E2:60_2"/><b>then </b><i><font color="Green" title="E65">A23</font></i>: 
<font color="Maroon" title="c16">Y</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c13">XP</font>
 <b>by </b><i><a class="txt" href="#E1:60"><i><font color="Green" title="E49">A1</font></i></a>, <a class="txt" href="#E2:60"><i><font color="Green" title="E50">A2</font></i></a>, <a class="txt" href="#E3:60"><i><font color="Green" title="E51">A3</font></i></a>, <a class="txt" href="#E5:60"><i><font color="Green" title="E53">A5</font></i></a>, <a class="txt" href="#E7:60"><i><font color="Green" title="E55">A7</font></i></a>, <a class="txt" href="#E9:60"><i><font color="Green" title="E56">A8</font></i></a>, <a class="txt" href="#E11:60"><i><font color="Green" title="E57">A9</font></i></a>, <a class="txt" href="#E17:60"><i><font color="Green" title="E60">A12</font></i></a>, <a class="txt" href="#E19:60"><i><font color="Green" title="E61">A13</font></i></a>, <a class="txt" href="#E20:60"><i><font color="Green" title="E62">A14</font></i></a>, <a class="txt" href="afproj.html#E64"><i><font color="Green" title="E46">Lm8</font></i></a>, <a class="ref" href="afproj.html#T27" target="_self" title="AFPROJ:th.27">Th27</a></i>;<br/><div><i><font color="Green" title="E66">A24</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="60_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:60_2_1"/><i><font color="Green" title="E67">A25</font></i>: 
( <font color="Maroon" title="c10">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"><span class="p2">(<span class="default"><a href="afproj.html#K6" title="AFPROJ:func.6">PDir</a> <font color="Maroon" title="c14">XQ</font></span>)</span>,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c14">XQ</font> is <a href="aff_4.html#V1" title="AFF_4:attr.1">being_plane</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  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="c9">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="c10">Q</font> )</span><br/><a NAME="E2:60_2_1"/><b>then </b>
<font color="Maroon" title="c16">Y</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c14">XQ</font>
 <b>by </b><i><a class="txt" href="#E1:60"><i><font color="Green" title="E49">A1</font></i></a>, <a class="txt" href="#E2:60"><i><font color="Green" title="E50">A2</font></i></a>, <a class="txt" href="#E3:60"><i><font color="Green" title="E51">A3</font></i></a>, <a class="txt" href="#E5:60"><i><font color="Green" title="E53">A5</font></i></a>, <a class="txt" href="#E7:60"><i><font color="Green" title="E55">A7</font></i></a>, <a class="txt" href="#E9:60"><i><font color="Green" title="E56">A8</font></i></a>, <a class="txt" href="#E11:60"><i><font color="Green" title="E57">A9</font></i></a>, <a class="txt" href="#E17:60"><i><font color="Green" title="E60">A12</font></i></a>, <a class="txt" href="#E19:60"><i><font color="Green" title="E61">A13</font></i></a>, <a class="txt" href="#E20:60"><i><font color="Green" title="E62">A14</font></i></a>, <a class="txt" href="afproj.html#E64"><i><font color="Green" title="E46">Lm8</font></i></a>, <a class="ref" href="afproj.html#T27" target="_self" title="AFPROJ:th.27">Th27</a></i>;<br/><a NAME="E3:60_2_1"/><b>then </b><i><font color="Green" title="E68">A26</font></i>: 
<font color="Maroon" title="c13">XP</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c14">XQ</font>
 <b>by </b><i><a class="txt" href="#E20:60"><i><font color="Green" title="E62">A14</font></i></a>, <a class="txt" href="#E1:60_2"><i><font color="Green" title="E64">A22</font></i></a>, <a class="txt" href="#E2:60_2"><i><font color="Green" title="E65">A23</font></i></a>, <a class="txt" href="#E1:60_2_1"><i><font color="Green" title="E67">A25</font></i></a>, <a class="ref" href="aff_4.html#T61" title="AFF_4:th.61">AFF_4:61</a></i>;<br/><b>consider </b><font color="Maroon" title="c17">q</font>, <font color="Maroon" title="c18">r</font>, <font color="Maroon" title="c19">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><b> such that </b><br/><a NAME="E5:60_2_1"/><i><font color="Green" title="E69">A27</font></i>: 
<font color="Maroon" title="c17">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c9">P</font>
 <b>and </b><br/><a NAME="E6:60_2_1"/>
( <font color="Maroon" title="c18">r</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c9">P</font> &amp; <font color="Maroon" title="c19">p'</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c9">P</font> &amp; <font color="Maroon" title="c17">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c18">r</font> &amp; <font color="Maroon" title="c18">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c19">p'</font> &amp; <font color="Maroon" title="c19">p'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c17">q</font> )
 <b>by </b><i><a class="txt" href="afproj.html#E57"><i><font color="Green" title="E39">Lm3</font></i></a></i>;<br/><b>take </b><font color="Maroon" title="c20">q</font> = <font color="Maroon" title="c17">q</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c20">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c9">P</font> &amp; <font color="Maroon" title="c20">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c10">Q</font> )</span><br/><b>thus </b><a NAME="E7:60_2_1"/>
<font color="Maroon" title="c20">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c9">P</font>
 <b>by </b><i><a class="txt" href="#E5:60_2_1"><i><font color="Green" title="E69">A27</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c20">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c10">Q</font></span><br/><b>thus </b><a NAME="E8:60_2_1"/>
<font color="Maroon" title="c20">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c10">Q</font>
 <b>by </b><i><a class="txt" href="#E1:60_2"><i><font color="Green" title="E64">A22</font></i></a>, <a class="txt" href="#E1:60_2_1"><i><font color="Green" title="E67">A25</font></i></a>, <a class="txt" href="#E3:60_2_1"><i><font color="Green" title="E68">A26</font></i></a>, <a class="txt" href="#E5:60_2_1"><i><font color="Green" title="E69">A27</font></i></a>, <a class="ref" href="afproj.html#T13" target="_self" title="AFPROJ:th.13">Th13</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><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:60_2_2"/><i><font color="Green" title="E67">A28</font></i>: 
( <font color="Maroon" title="c10">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="c14">XQ</font>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c14">XQ</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  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="c9">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="c10">Q</font> )</span><br/><a NAME="E2:60_2_2"/><b>then </b><i><font color="Green" title="E68">A29</font></i>: 
<font color="Maroon" title="c14">XQ</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c16">Y</font>
 <b>by </b><i><a class="txt" href="#E1:60"><i><font color="Green" title="E49">A1</font></i></a>, <a class="txt" href="#E2:60"><i><font color="Green" title="E50">A2</font></i></a>, <a class="txt" href="#E3:60"><i><font color="Green" title="E51">A3</font></i></a>, <a class="txt" href="#E5:60"><i><font color="Green" title="E53">A5</font></i></a>, <a class="txt" href="#E7:60"><i><font color="Green" title="E55">A7</font></i></a>, <a class="txt" href="#E9:60"><i><font color="Green" title="E56">A8</font></i></a>, <a class="txt" href="#E11:60"><i><font color="Green" title="E57">A9</font></i></a>, <a class="txt" href="#E17:60"><i><font color="Green" title="E60">A12</font></i></a>, <a class="txt" href="#E19:60"><i><font color="Green" title="E61">A13</font></i></a>, <a class="txt" href="#E20:60"><i><font color="Green" title="E62">A14</font></i></a>, <a class="txt" href="afproj.html#E63"><i><font color="Green" title="E45">Lm7</font></i></a>, <a class="ref" href="afproj.html#T27" target="_self" title="AFPROJ:th.27">Th27</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c17">q</font> =  <a href="afproj.html#K5" title="AFPROJ:func.5">LDir</a> <font color="Maroon" title="c14">XQ</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="#E1:60_2_2"><i><font color="Green" title="E67">A28</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="c18">q</font> = <font color="Maroon" title="c17">q</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c18">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c9">P</font> &amp; <font color="Maroon" title="c18">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c10">Q</font> )</span><br/><a NAME="E4:60_2_2"/>
<font color="Maroon" title="c14">XQ</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c16">Y</font>
 <b>by </b><i><a class="txt" href="#E20:60"><i><font color="Green" title="E62">A14</font></i></a>, <a class="txt" href="#E1:60_2_2"><i><font color="Green" title="E67">A28</font></i></a>, <a class="txt" href="#E2:60_2_2"><i><font color="Green" title="E68">A29</font></i></a>, <a class="ref" href="aff_4.html#T42" title="AFF_4:th.42">AFF_4:42</a></i>;<br/><a NAME="E5:60_2_2"/><b>then </b>
<font color="Maroon" title="c14">XQ</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c13">XP</font>
 <b>by </b><i><a class="txt" href="#E20:60"><i><font color="Green" title="E62">A14</font></i></a>, <a class="txt" href="#E2:60_2"><i><font color="Green" title="E65">A23</font></i></a>, <a class="ref" href="aff_4.html#T59" title="AFF_4:th.59">AFF_4:59</a>, <a class="ref" href="aff_4.html#T60" title="AFF_4:th.60">AFF_4:60</a></i>;<br/><b>hence </b><a NAME="E6:60_2_2"/>
<font color="Maroon" title="c18">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c9">P</font>
 <b>by </b><i><a class="txt" href="#E1:60_2"><i><font color="Green" title="E64">A22</font></i></a>, <a class="txt" href="#E1:60_2_2"><i><font color="Green" title="E67">A28</font></i></a>, <a class="ref" href="afproj.html#T29" target="_self" title="AFPROJ:th.29">Th29</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c18">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c10">Q</font></span><br/><b>thus </b><a NAME="E7:60_2_2"/>
<font color="Maroon" title="c18">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c10">Q</font>
 <b>by </b><i><a class="txt" href="#E1:60_2_2"><i><font color="Green" title="E67">A28</font></i></a>, <a class="ref" href="afproj.html#T30" target="_self" title="AFPROJ:th.30">Th30</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><b>end;</b></div><b>hence </b><a NAME="E5:60_2"/>
 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="c9">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="c10">Q</font> )
 <b>by </b><i><a class="txt" href="#E15:60"><i><font color="Green" title="E59">A11</font></i></a>, <a class="txt" href="#E3:60_2"><i><font color="Green" title="E66">A24</font></i></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><b>end;</b></div>
<b>hence </b><a NAME="E23:60"/>
 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="c9">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="c10">Q</font> )
 <b>by </b><i><a class="txt" href="#E13:60"><i><font color="Green" title="E58">A10</font></i></a>, <a class="txt" href="#E21:60"><i><font color="Green" title="E63">A15</font></i></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>
