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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> 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> 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> 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="E1:63"/><i><font color="Green" title="E52">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> &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> )
 <b>and </b><br/><a NAME="E2:63"/><i><font color="Green" title="E53">A2</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="E3:63"/><i><font color="Green" title="E54">A3</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>
 ; <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/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:63_1"/><i><font color="Green" title="E55">A4</font></i>: 
<font color="Maroon" title="c6">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">  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/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:63_1_1"/><i><font color="Green" title="E56">A5</font></i>: 
( <font color="Maroon" title="c2">a</font> is  not   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font> &amp; <font color="Maroon" title="c3">b</font> is  not   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font> &amp; <font color="Maroon" title="c4">c</font> is  not   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font> &amp; <font color="Maroon" title="c5">d</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">  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>consider </b><font color="Maroon" title="c11">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="E3:63_1_1"/><i><font color="Green" title="E57">A6</font></i>: 
( <font color="Maroon" title="c6">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="c11">Xp</font> &amp; <font color="Maroon" title="c11">Xp</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> )
 <b>by </b><i><a class="txt" href="#E1:63_1"><i><font color="Green" title="E55">A4</font></i></a>, <a class="ref" href="afproj.html#T20" target="_self" title="AFPROJ:th.20">Th20</a></i>;<br/><b>consider </b><font color="Maroon" title="c12">Xa</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="E5:63_1_1"/><i><font color="Green" title="E58">A7</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="c12">Xa</font> &amp; <font color="Maroon" title="c12">Xa</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> )
 <b>by </b><i><a class="txt" href="#E1:63_1_1"><i><font color="Green" title="E56">A5</font></i></a>, <a class="ref" href="afproj.html#T20" target="_self" title="AFPROJ:th.20">Th20</a></i>;<br/><b>consider </b><font color="Maroon" title="c13">Xb</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="E7:63_1_1"/><i><font color="Green" title="E59">A8</font></i>: 
( <font color="Maroon" title="c3">b</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="c13">Xb</font> &amp; <font color="Maroon" title="c13">Xb</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> )
 <b>by </b><i><a class="txt" href="#E1:63_1_1"><i><font color="Green" title="E56">A5</font></i></a>, <a class="ref" href="afproj.html#T20" target="_self" title="AFPROJ:th.20">Th20</a></i>;<br/><b>consider </b><font color="Maroon" title="c14">Xc</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="E9:63_1_1"/><i><font color="Green" title="E60">A9</font></i>: 
( <font color="Maroon" title="c4">c</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="c14">Xc</font> &amp; <font color="Maroon" title="c14">Xc</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> )
 <b>by </b><i><a class="txt" href="#E1:63_1_1"><i><font color="Green" title="E56">A5</font></i></a>, <a class="ref" href="afproj.html#T20" target="_self" title="AFPROJ:th.20">Th20</a></i>;<br/><b>consider </b><font color="Maroon" title="c15">Xd</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="E11:63_1_1"/><i><font color="Green" title="E61">A10</font></i>: 
( <font color="Maroon" title="c5">d</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="c15">Xd</font> &amp; <font color="Maroon" title="c15">Xd</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> )
 <b>by </b><i><a class="txt" href="#E1:63_1_1"><i><font color="Green" title="E56">A5</font></i></a>, <a class="ref" href="afproj.html#T20" target="_self" title="AFPROJ:th.20">Th20</a></i>;<br/><a NAME="E12:63_1_1"/><i><font color="Green" title="E62">A11</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="#E1:63"><i><font color="Green" title="E52">A1</font></i></a>, <a class="txt" href="#E2:63"><i><font color="Green" title="E53">A2</font></i></a>, <a class="txt" href="#E3:63"><i><font color="Green" title="E54">A3</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="c16">x</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>set </b><font color="Maroon" title="c17">Xa'</font> = <font color="Maroon" title="c16">x</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c12">Xa</font>;<br/><b>set </b><font color="Maroon" title="c18">Xb'</font> = <font color="Maroon" title="c16">x</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c13">Xb</font>;<br/><b>set </b><font color="Maroon" title="c19">Xc'</font> = <font color="Maroon" title="c16">x</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c14">Xc</font>;<br/><b>set </b><font color="Maroon" title="c20">Xd'</font> = <font color="Maroon" title="c16">x</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c15">Xd</font>;<br/><b>set </b><font color="Maroon" title="c21">Xp'</font> = <font color="Maroon" title="c16">x</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c11">Xp</font>;<br/><a NAME="E14:63_1_1"/><i><font color="Green" title="E63">A12</font></i>: 
( <font color="Maroon" title="c16">x</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c12">Xa</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Maroon" title="c16">x</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c13">Xb</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Maroon" title="c16">x</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c14">Xc</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Maroon" title="c16">x</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c15">Xd</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Maroon" title="c16">x</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c11">Xp</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">x</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c12">Xa</font> &amp; <font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">x</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c13">Xb</font> &amp; <font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">x</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c14">Xc</font> &amp; <font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">x</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c15">Xd</font> &amp; <font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">x</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c11">Xp</font> &amp; <font color="Maroon" title="c12">Xa</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c16">x</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c12">Xa</font> &amp; <font color="Maroon" title="c13">Xb</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c16">x</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c13">Xb</font> &amp; <font color="Maroon" title="c14">Xc</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c16">x</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c14">Xc</font> &amp; <font color="Maroon" title="c15">Xd</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c16">x</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c15">Xd</font> &amp; <font color="Maroon" title="c11">Xp</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c16">x</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c11">Xp</font> )
 <b>by </b><i><a class="txt" href="#E3:63_1_1"><i><font color="Green" title="E57">A6</font></i></a>, <a class="txt" href="#E5:63_1_1"><i><font color="Green" title="E58">A7</font></i></a>, <a class="txt" href="#E7:63_1_1"><i><font color="Green" title="E59">A8</font></i></a>, <a class="txt" href="#E9:63_1_1"><i><font color="Green" title="E60">A9</font></i></a>, <a class="txt" href="#E11:63_1_1"><i><font color="Green" title="E61">A10</font></i></a>, <a class="ref" href="aff_4.html#T27" title="AFF_4:th.27">AFF_4:27</a>, <a class="ref" href="aff_4.html#D3" title="AFF_4:def.3">AFF_4:def 3</a></i>;<br/><b>then consider </b><font color="Maroon" title="c22">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><br/><a NAME="E16:63_1_1"/><i><font color="Green" title="E64">A13</font></i>: 
( <font color="Maroon" title="c16">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c22">y</font> &amp; <font color="Maroon" title="c22">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">x</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c12">Xa</font> )
 <b>by </b><i><a class="ref" href="aff_1.html#T32" title="AFF_1:th.32">AFF_1:32</a></i>;<br/><b>consider </b><font color="Maroon" title="c23">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="E18:63_1_1"/><i><font color="Green" title="E65">A14</font></i>: 
( <font color="Maroon" title="c16">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c23">t</font> &amp; <font color="Maroon" title="c23">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">x</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c14">Xc</font> )
 <b>by </b><i><a class="txt" href="#E14:63_1_1"><i><font color="Green" title="E63">A12</font></i></a>, <a class="ref" href="aff_1.html#T32" title="AFF_1:th.32">AFF_1:32</a></i>;<br/><b>set </b><font color="Maroon" title="c24">Y1</font> = <font color="Maroon" title="c22">y</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c11">Xp</font></span>)</span>;<br/><b>set </b><font color="Maroon" title="c25">Y2</font> = <font color="Maroon" title="c23">t</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c11">Xp</font></span>)</span>;<br/><a NAME="E19:63_1_1"/><i><font color="Green" title="E66">A15</font></i>: 
( <font color="Maroon" title="c22">y</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c11">Xp</font></span>)</span> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Maroon" title="c23">t</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c11">Xp</font></span>)</span> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Maroon" title="c22">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">y</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c11">Xp</font></span>)</span> &amp; <font color="Maroon" title="c23">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">t</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c11">Xp</font></span>)</span> &amp; <font color="Maroon" title="c16">x</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c11">Xp</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c22">y</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c11">Xp</font></span>)</span> &amp; <font color="Maroon" title="c16">x</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c11">Xp</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c23">t</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c11">Xp</font></span>)</span> )
 <b>by </b><i><a class="txt" href="#E14:63_1_1"><i><font color="Green" title="E63">A12</font></i></a>, <a class="ref" href="aff_4.html#T27" title="AFF_4:th.27">AFF_4:27</a>, <a class="ref" href="aff_4.html#D3" title="AFF_4:def.3">AFF_4:def 3</a></i>;<br/><a NAME="E20:63_1_1"/><b>then </b><i><font color="Green" title="E67">A16</font></i>: 
<font color="Maroon" title="c22">y</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c11">Xp</font></span>)</span> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c23">t</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c11">Xp</font></span>)</span>
 <b>by </b><i><a class="ref" href="aff_1.html#T58" title="AFF_1:th.58">AFF_1:58</a></i>;<br/><b>consider </b><font color="Maroon" title="c26">X1</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">AS</font><b> such that </b><br/><a NAME="E22:63_1_1"/><i><font color="Green" title="E68">A17</font></i>: 
( <font color="Maroon" title="c16">x</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c12">Xa</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c26">X1</font> &amp; <font color="Maroon" title="c16">x</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c11">Xp</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c26">X1</font> &amp; <font color="Maroon" title="c26">X1</font> is <a href="aff_4.html#V1" title="AFF_4:attr.1">being_plane</a> )
 <b>by </b><i><a class="txt" href="#E14:63_1_1"><i><font color="Green" title="E63">A12</font></i></a>, <a class="ref" href="aff_4.html#T38" title="AFF_4:th.38">AFF_4:38</a></i>;<br/><b>consider </b><font color="Maroon" title="c27">X2</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="E24:63_1_1"/><i><font color="Green" title="E69">A18</font></i>: 
( <font color="Maroon" title="c16">x</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c14">Xc</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c27">X2</font> &amp; <font color="Maroon" title="c16">x</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c11">Xp</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c27">X2</font> &amp; <font color="Maroon" title="c27">X2</font> is <a href="aff_4.html#V1" title="AFF_4:attr.1">being_plane</a> )
 <b>by </b><i><a class="txt" href="#E14:63_1_1"><i><font color="Green" title="E63">A12</font></i></a>, <a class="ref" href="aff_4.html#T38" title="AFF_4:th.38">AFF_4:38</a></i>;<br/><b>consider </b><font color="Maroon" title="c28">X3</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="E26:63_1_1"/><i><font color="Green" title="E70">A19</font></i>: 
( <font color="Maroon" title="c22">y</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c11">Xp</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c28">X3</font> &amp; <font color="Maroon" title="c23">t</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c11">Xp</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c28">X3</font> &amp; <font color="Maroon" title="c28">X3</font> is <a href="aff_4.html#V1" title="AFF_4:attr.1">being_plane</a> )
 <b>by </b><i><a class="txt" href="#E20:63_1_1"><i><font color="Green" title="E67">A16</font></i></a>, <a class="ref" href="aff_4.html#T39" title="AFF_4:th.39">AFF_4:39</a></i>;<br/><a NAME="E27:63_1_1"/><i><font color="Green" title="E71">A20</font></i>: 
( <font color="Maroon" title="c22">y</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c11">Xp</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c26">X1</font> &amp; <font color="Maroon" title="c23">t</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c11">Xp</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c27">X2</font> )
 <b>by </b><i><a class="txt" href="#E14:63_1_1"><i><font color="Green" title="E63">A12</font></i></a>, <a class="txt" href="#E16:63_1_1"><i><font color="Green" title="E64">A13</font></i></a>, <a class="txt" href="#E18:63_1_1"><i><font color="Green" title="E65">A14</font></i></a>, <a class="txt" href="#E22:63_1_1"><i><font color="Green" title="E68">A17</font></i></a>, <a class="txt" href="#E24:63_1_1"><i><font color="Green" title="E69">A18</font></i></a>, <a class="ref" href="aff_4.html#T28" title="AFF_4:th.28">AFF_4:28</a></i>;<br/><a NAME="E28:63_1_1"/><i><font color="Green" title="E72">A21</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"><span class="p2">(<span class="default"><a href="afproj.html#K6" title="AFPROJ:func.6">PDir</a> <font color="Maroon" title="c26">X1</font></span>)</span>,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1_1_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon" title="c29">M'</font> = <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="c26">X1</font></span>)</span>,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> 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#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="#E22:63_1_1"><i><font color="Green" title="E68">A17</font></i></a>, <a class="ref" href="afproj.html#T23" target="_self" title="AFPROJ:th.23">Th23</a></i>;<br/>
<a NAME="E2:63_1_1_1"/>
( <font color="Maroon" title="c6">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c29">M'</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="c29">M'</font> )
 
<b>by </b><i><a class="txt" href="#E3:63_1_1"><i><font color="Green" title="E57">A6</font></i></a>, <a class="txt" href="#E5:63_1_1"><i><font color="Green" title="E58">A7</font></i></a>, <a class="txt" href="#E14:63_1_1"><i><font color="Green" title="E63">A12</font></i></a>, <a class="txt" href="#E22:63_1_1"><i><font color="Green" title="E68">A17</font></i></a>, <a class="ref" href="afproj.html#T32" target="_self" title="AFPROJ:th.32">Th32</a></i>;<br/>
<b>hence </b><a NAME="E3:63_1_1_1"/>
<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="c26">X1</font></span>)</span>,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="#E1:63"><i><font color="Green" title="E52">A1</font></i></a>, <a class="txt" href="#E2:63"><i><font color="Green" title="E53">A2</font></i></a>, <a class="txt" href="afproj.html#E56"><i><font color="Green" title="E38">Lm2</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><a NAME="E29:63_1_1"/><i><font color="Green" title="E73">A22</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"><span class="p2">(<span class="default"><a href="afproj.html#K6" title="AFPROJ:func.6">PDir</a> <font color="Maroon" title="c27">X2</font></span>)</span>,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1_1_2"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon" title="c29">N'</font> = <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="c27">X2</font></span>)</span>,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> 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#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="#E24:63_1_1"><i><font color="Green" title="E69">A18</font></i></a>, <a class="ref" href="afproj.html#T23" target="_self" title="AFPROJ:th.23">Th23</a></i>;<br/>
<a NAME="E2:63_1_1_2"/>
( <font color="Maroon" title="c6">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c29">N'</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="c29">N'</font> )
 
<b>by </b><i><a class="txt" href="#E3:63_1_1"><i><font color="Green" title="E57">A6</font></i></a>, <a class="txt" href="#E9:63_1_1"><i><font color="Green" title="E60">A9</font></i></a>, <a class="txt" href="#E14:63_1_1"><i><font color="Green" title="E63">A12</font></i></a>, <a class="txt" href="#E24:63_1_1"><i><font color="Green" title="E69">A18</font></i></a>, <a class="ref" href="afproj.html#T32" target="_self" title="AFPROJ:th.32">Th32</a></i>;<br/>
<b>hence </b><a NAME="E3:63_1_1_2"/>
<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="c27">X2</font></span>)</span>,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="#E1:63"><i><font color="Green" title="E52">A1</font></i></a>, <a class="txt" href="#E2:63"><i><font color="Green" title="E53">A2</font></i></a>, <a class="txt" href="afproj.html#E56"><i><font color="Green" title="E38">Lm2</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><a NAME="E30:63_1_1"/><i><font color="Green" title="E74">A23</font></i>: 
( <font color="Maroon" title="c16">x</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c13">Xb</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c26">X1</font> &amp; <font color="Maroon" title="c16">x</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c15">Xd</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c27">X2</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1_1_3"><b>proof </b></a><div class="add">

<a NAME="E1:63_1_1_3"/>
( <font color="Maroon" title="c13">Xb</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c26">X1</font> &amp; <font color="Maroon" title="c15">Xd</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c27">X2</font> )
 
<b>by </b><i><a class="txt" href="#E1:63"><i><font color="Green" title="E52">A1</font></i></a>, <a class="txt" href="#E7:63_1_1"><i><font color="Green" title="E59">A8</font></i></a>, <a class="txt" href="#E11:63_1_1"><i><font color="Green" title="E61">A10</font></i></a>, <a class="txt" href="#E22:63_1_1"><i><font color="Green" title="E68">A17</font></i></a>, <a class="txt" href="#E24:63_1_1"><i><font color="Green" title="E69">A18</font></i></a>, <a class="txt" href="#E28:63_1_1"><i><font color="Green" title="E72">A21</font></i></a>, <a class="txt" href="#E29:63_1_1"><i><font color="Green" title="E73">A22</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="E2:63_1_1_3"/>
( <font color="Maroon" title="c16">x</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c13">Xb</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c26">X1</font> &amp; <font color="Maroon" title="c16">x</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c15">Xd</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c27">X2</font> )
 <b>by </b><i><a class="txt" href="#E14:63_1_1"><i><font color="Green" title="E63">A12</font></i></a>, <a class="txt" href="#E22:63_1_1"><i><font color="Green" title="E68">A17</font></i></a>, <a class="txt" href="#E24:63_1_1"><i><font color="Green" title="E69">A18</font></i></a>, <a class="ref" href="aff_4.html#T43" title="AFF_4:th.43">AFF_4:43</a>, <a class="ref" href="aff_4.html#T56" title="AFF_4:th.56">AFF_4:56</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><a NAME="E31:63_1_1"/><i><font color="Green" title="E75">A24</font></i>: 
not <font color="Maroon" title="c16">x</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c13">Xb</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c22">y</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c11">Xp</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1_1_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:63_1_1_4"/>
<font color="Maroon" title="c16">x</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c13">Xb</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c22">y</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c11">Xp</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:63_1_1_4"/><b>then </b>
<font color="Maroon" title="c13">Xb</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c22">y</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c11">Xp</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E14:63_1_1"><i><font color="Green" title="E63">A12</font></i></a>, <a class="ref" href="aff_1.html#T58" title="AFF_1:th.58">AFF_1:58</a></i>;<br/>
<a NAME="E3:63_1_1_4"/><b>then </b>
<font color="Maroon" title="c13">Xb</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c16">x</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c11">Xp</font>
 
<b>by </b><i><a class="txt" href="#E19:63_1_1"><i><font color="Green" title="E66">A15</font></i></a>, <a class="ref" href="aff_1.html#T58" title="AFF_1:th.58">AFF_1:58</a></i>;<br/>
<a NAME="E4:63_1_1_4"/><b>then </b>
<font color="Maroon" title="c13">Xb</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c11">Xp</font>
 
<b>by </b><i><a class="txt" href="#E14:63_1_1"><i><font color="Green" title="E63">A12</font></i></a>, <a class="ref" href="aff_1.html#T58" title="AFF_1:th.58">AFF_1:58</a></i>;<br/>
<b>hence </b><a NAME="E5:63_1_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:63"><i><font color="Green" title="E52">A1</font></i></a>, <a class="txt" href="#E2:63"><i><font color="Green" title="E53">A2</font></i></a>, <a class="txt" href="#E3:63_1_1"><i><font color="Green" title="E57">A6</font></i></a>, <a class="txt" href="#E7:63_1_1"><i><font color="Green" title="E59">A8</font></i></a>, <a class="ref" href="afproj.html#T11" target="_self" title="AFPROJ:th.11">Th11</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><a NAME="E32:63_1_1"/><i><font color="Green" title="E76">A25</font></i>: 
not <font color="Maroon" title="c16">x</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c15">Xd</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c23">t</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c11">Xp</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1_1_5"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:63_1_1_5"/>
<font color="Maroon" title="c16">x</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c15">Xd</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c23">t</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c11">Xp</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:63_1_1_5"/><b>then </b>
<font color="Maroon" title="c15">Xd</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c23">t</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c11">Xp</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E14:63_1_1"><i><font color="Green" title="E63">A12</font></i></a>, <a class="ref" href="aff_1.html#T58" title="AFF_1:th.58">AFF_1:58</a></i>;<br/>
<a NAME="E3:63_1_1_5"/><b>then </b>
<font color="Maroon" title="c15">Xd</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c16">x</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c11">Xp</font>
 
<b>by </b><i><a class="txt" href="#E19:63_1_1"><i><font color="Green" title="E66">A15</font></i></a>, <a class="ref" href="aff_1.html#T58" title="AFF_1:th.58">AFF_1:58</a></i>;<br/>
<a NAME="E4:63_1_1_5"/><b>then </b>
<font color="Maroon" title="c15">Xd</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c11">Xp</font>
 
<b>by </b><i><a class="txt" href="#E14:63_1_1"><i><font color="Green" title="E63">A12</font></i></a>, <a class="ref" href="aff_1.html#T58" title="AFF_1:th.58">AFF_1:58</a></i>;<br/>
<b>hence </b><a NAME="E5:63_1_1_5"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:63"><i><font color="Green" title="E52">A1</font></i></a>, <a class="txt" href="#E2:63"><i><font color="Green" title="E53">A2</font></i></a>, <a class="txt" href="#E3:63_1_1"><i><font color="Green" title="E57">A6</font></i></a>, <a class="txt" href="#E11:63_1_1"><i><font color="Green" title="E61">A10</font></i></a>, <a class="ref" href="afproj.html#T11" target="_self" title="AFPROJ:th.11">Th11</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>consider </b><font color="Maroon" title="c29">z</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="E34:63_1_1"/><i><font color="Green" title="E77">A26</font></i>: 
( <font color="Maroon" title="c29">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">x</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c13">Xb</font> &amp; <font color="Maroon" title="c29">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">y</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c11">Xp</font></span>)</span> )
 <b>by </b><i><a class="txt" href="#E14:63_1_1"><i><font color="Green" title="E63">A12</font></i></a>, <a class="txt" href="#E19:63_1_1"><i><font color="Green" title="E66">A15</font></i></a>, <a class="txt" href="#E22:63_1_1"><i><font color="Green" title="E68">A17</font></i></a>, <a class="txt" href="#E27:63_1_1"><i><font color="Green" title="E71">A20</font></i></a>, <a class="txt" href="#E30:63_1_1"><i><font color="Green" title="E74">A23</font></i></a>, <a class="txt" href="#E31:63_1_1"><i><font color="Green" title="E75">A24</font></i></a>, <a class="ref" href="aff_4.html#T22" title="AFF_4:th.22">AFF_4:22</a></i>;<br/><b>consider </b><font color="Maroon" title="c30">u</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="E36:63_1_1"/><i><font color="Green" title="E78">A27</font></i>: 
( <font color="Maroon" title="c30">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">x</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c15">Xd</font> &amp; <font color="Maroon" title="c30">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">t</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c11">Xp</font></span>)</span> )
 <b>by </b><i><a class="txt" href="#E14:63_1_1"><i><font color="Green" title="E63">A12</font></i></a>, <a class="txt" href="#E19:63_1_1"><i><font color="Green" title="E66">A15</font></i></a>, <a class="txt" href="#E24:63_1_1"><i><font color="Green" title="E69">A18</font></i></a>, <a class="txt" href="#E27:63_1_1"><i><font color="Green" title="E71">A20</font></i></a>, <a class="txt" href="#E30:63_1_1"><i><font color="Green" title="E74">A23</font></i></a>, <a class="txt" href="#E32:63_1_1"><i><font color="Green" title="E76">A25</font></i></a>, <a class="ref" href="aff_4.html#T22" title="AFF_4:th.22">AFF_4:22</a></i>;<br/><b>set </b><font color="Maroon" title="c31">AC</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c22">y</font>,<font color="Maroon" title="c23">t</font>;<br/><b>set </b><font color="Maroon" title="c32">BD</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c29">z</font>,<font color="Maroon" title="c30">u</font>;<br/><a NAME="E37:63_1_1"/><i><font color="Green" title="E79">A28</font></i>: 
<font color="Maroon" title="c16">x</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c12">Xa</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c16">x</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c14">Xc</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1_1_6"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:63_1_1_6"/>
<font color="Maroon" title="c16">x</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c12">Xa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">x</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c14">Xc</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<b>then </b><font color="Maroon" title="c2">a</font> = 
 <a href="afproj.html#K5" title="AFPROJ:func.5">LDir</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c14">Xc</font></span>)</span>
<b>by </b><i><a class="txt" href="#E5:63_1_1"><i><font color="Green" title="E58">A7</font></i></a>, <a class="txt" href="#E14:63_1_1"><i><font color="Green" title="E63">A12</font></i></a>, <a class="ref" href="afproj.html#T11" target="_self" title="AFPROJ:th.11">Th11</a></i>
<br/>.= 
<font color="Maroon" title="c4">c</font>
<b>by </b><i><a class="txt" href="#E9:63_1_1"><i><font color="Green" title="E60">A9</font></i></a>, <a class="txt" href="#E14:63_1_1"><i><font color="Green" title="E63">A12</font></i></a>, <a class="ref" href="afproj.html#T11" target="_self" title="AFPROJ:th.11">Th11</a></i>
;<br/>
<b>hence </b><a NAME="E3:63_1_1_6"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:63"><i><font color="Green" title="E52">A1</font></i></a>, <a class="txt" href="#E2:63"><i><font color="Green" title="E53">A2</font></i></a>, <a class="txt" href="#E3:63"><i><font color="Green" title="E54">A3</font></i></a>, <a class="txt" href="afproj.html#E56"><i><font color="Green" title="E38">Lm2</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><a NAME="E38:63_1_1"/><b>then </b><i><font color="Green" title="E80">A29</font></i>: 
<font color="Maroon" title="c22">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c23">t</font>
 <b>by </b><i><a class="txt" href="#E14:63_1_1"><i><font color="Green" title="E63">A12</font></i></a>, <a class="txt" href="#E16:63_1_1"><i><font color="Green" title="E64">A13</font></i></a>, <a class="txt" href="#E18:63_1_1"><i><font color="Green" title="E65">A14</font></i></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></i>;<br/><a NAME="E39:63_1_1"/><i><font color="Green" title="E81">A30</font></i>: 
<font color="Maroon" title="c16">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c29">z</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1_1_7"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:63_1_1_7"/><i><font color="Green" title="E82">A31</font></i>: 
not <font color="Maroon" title="c16">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c29">z</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<font color="Maroon" title="c2">a</font> = 
 <a href="afproj.html#K5" title="AFPROJ:func.5">LDir</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c12">Xa</font></span>)</span>
<b>by </b><i><a class="txt" href="#E5:63_1_1"><i><font color="Green" title="E58">A7</font></i></a>, <a class="txt" href="#E14:63_1_1"><i><font color="Green" title="E63">A12</font></i></a>, <a class="ref" href="afproj.html#T11" target="_self" title="AFPROJ:th.11">Th11</a></i>
<br/>.= 
 <a href="afproj.html#K5" title="AFPROJ:func.5">LDir</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">y</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c11">Xp</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E14:63_1_1"><i><font color="Green" title="E63">A12</font></i></a>, <a class="txt" href="#E16:63_1_1"><i><font color="Green" title="E64">A13</font></i></a>, <a class="txt" href="#E19:63_1_1"><i><font color="Green" title="E66">A15</font></i></a>, <a class="txt" href="#E34:63_1_1"><i><font color="Green" title="E77">A26</font></i></a>, <a class="txt" href="#E1:63_1_1_7"><i><font color="Green" title="E82">A31</font></i></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></i>
<br/>.= 
 <a href="afproj.html#K5" title="AFPROJ:func.5">LDir</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c11">Xp</font></span>)</span>
<b>by </b><i><a class="txt" href="#E14:63_1_1"><i><font color="Green" title="E63">A12</font></i></a>, <a class="txt" href="#E19:63_1_1"><i><font color="Green" title="E66">A15</font></i></a>, <a class="ref" href="afproj.html#T11" target="_self" title="AFPROJ:th.11">Th11</a></i>
<br/>.= 
<font color="Maroon" title="c6">p</font>
<b>by </b><i><a class="txt" href="#E3:63_1_1"><i><font color="Green" title="E57">A6</font></i></a>, <a class="txt" href="#E14:63_1_1"><i><font color="Green" title="E63">A12</font></i></a>, <a class="ref" href="afproj.html#T11" target="_self" title="AFPROJ:th.11">Th11</a></i>
;<br/>
<b>hence </b><a NAME="E3:63_1_1_7"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:63"><i><font color="Green" title="E52">A1</font></i></a>, <a class="txt" href="#E2:63"><i><font color="Green" title="E53">A2</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><a NAME="E40:63_1_1"/><i><font color="Green" title="E82">A32</font></i>: 
<font color="Maroon" title="c29">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c30">u</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1_1_8"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:63_1_1_8"/><i><font color="Green" title="E83">A33</font></i>: 
not <font color="Maroon" title="c29">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c30">u</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<font color="Maroon" title="c3">b</font> = 
 <a href="afproj.html#K5" title="AFPROJ:func.5">LDir</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c13">Xb</font></span>)</span>
<b>by </b><i><a class="txt" href="#E7:63_1_1"><i><font color="Green" title="E59">A8</font></i></a>, <a class="txt" href="#E14:63_1_1"><i><font color="Green" title="E63">A12</font></i></a>, <a class="ref" href="afproj.html#T11" target="_self" title="AFPROJ:th.11">Th11</a></i>
<br/>.= 
 <a href="afproj.html#K5" title="AFPROJ:func.5">LDir</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c15">Xd</font></span>)</span>
<b>by </b><i><a class="txt" href="#E14:63_1_1"><i><font color="Green" title="E63">A12</font></i></a>, <a class="txt" href="#E34:63_1_1"><i><font color="Green" title="E77">A26</font></i></a>, <a class="txt" href="#E36:63_1_1"><i><font color="Green" title="E78">A27</font></i></a>, <a class="txt" href="#E39:63_1_1"><i><font color="Green" title="E81">A30</font></i></a>, <a class="txt" href="#E1:63_1_1_8"><i><font color="Green" title="E83">A33</font></i></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></i>
<br/>.= 
<font color="Maroon" title="c5">d</font>
<b>by </b><i><a class="txt" href="#E11:63_1_1"><i><font color="Green" title="E61">A10</font></i></a>, <a class="txt" href="#E14:63_1_1"><i><font color="Green" title="E63">A12</font></i></a>, <a class="ref" href="afproj.html#T11" target="_self" title="AFPROJ:th.11">Th11</a></i>
;<br/>
<b>hence </b><a NAME="E3:63_1_1_8"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:63"><i><font color="Green" title="E52">A1</font></i></a>, <a class="txt" href="#E2:63"><i><font color="Green" title="E53">A2</font></i></a>, <a class="txt" href="#E3:63"><i><font color="Green" title="E54">A3</font></i></a>, <a class="txt" href="afproj.html#E56"><i><font color="Green" title="E38">Lm2</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><a NAME="E41:63_1_1"/><b>then </b><i><font color="Green" title="E83">A34</font></i>: 
(  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c22">y</font>,<font color="Maroon" title="c23">t</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp;  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c29">z</font>,<font color="Maroon" title="c30">u</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> )
 <b>by </b><i><a class="txt" href="#E38:63_1_1"><i><font color="Green" title="E80">A29</font></i></a>, <a class="ref" href="aff_1.html#D3" title="AFF_1:def.3">AFF_1:def 3</a></i>;<br/><a NAME="E42:63_1_1"/><i><font color="Green" title="E84">A35</font></i>: 
(  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c22">y</font>,<font color="Maroon" title="c23">t</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c28">X3</font> &amp;  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c29">z</font>,<font color="Maroon" title="c30">u</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c28">X3</font> )
 <b>by </b><i><a class="txt" href="#E19:63_1_1"><i><font color="Green" title="E66">A15</font></i></a>, <a class="txt" href="#E26:63_1_1"><i><font color="Green" title="E70">A19</font></i></a>, <a class="txt" href="#E34:63_1_1"><i><font color="Green" title="E77">A26</font></i></a>, <a class="txt" href="#E36:63_1_1"><i><font color="Green" title="E78">A27</font></i></a>, <a class="txt" href="#E38:63_1_1"><i><font color="Green" title="E80">A29</font></i></a>, <a class="txt" href="#E40:63_1_1"><i><font color="Green" title="E82">A32</font></i></a>, <a class="ref" href="aff_4.html#T19" title="AFF_4:th.19">AFF_4:19</a></i>;<br/><b>consider </b><font color="Maroon" title="c33">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="E44:63_1_1"/><i><font color="Green" title="E85">A36</font></i>: 
( <font color="Maroon" title="c16">x</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c12">Xa</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c33">XP</font> &amp; <font color="Maroon" title="c16">x</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c14">Xc</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c33">XP</font> &amp; <font color="Maroon" title="c33">XP</font> is <a href="aff_4.html#V1" title="AFF_4:attr.1">being_plane</a> )
 <b>by </b><i><a class="txt" href="#E14:63_1_1"><i><font color="Green" title="E63">A12</font></i></a>, <a class="ref" href="aff_4.html#T38" title="AFF_4:th.38">AFF_4:38</a></i>;<br/><b>consider </b><font color="Maroon" title="c34">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="E46:63_1_1"/><i><font color="Green" title="E86">A37</font></i>: 
( <font color="Maroon" title="c16">x</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c13">Xb</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c34">XQ</font> &amp; <font color="Maroon" title="c16">x</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c15">Xd</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c34">XQ</font> &amp; <font color="Maroon" title="c34">XQ</font> is <a href="aff_4.html#V1" title="AFF_4:attr.1">being_plane</a> )
 <b>by </b><i><a class="txt" href="#E14:63_1_1"><i><font color="Green" title="E63">A12</font></i></a>, <a class="ref" href="aff_4.html#T38" title="AFF_4:th.38">AFF_4:38</a></i>;<br/><a NAME="E47:63_1_1"/><i><font color="Green" title="E87">A38</font></i>: 
( <font color="Maroon" title="c22">y</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="c22">y</font>,<font color="Maroon" title="c23">t</font> &amp; <font color="Maroon" title="c23">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="c22">y</font>,<font color="Maroon" title="c23">t</font> )
 <b>by </b><i><a class="ref" href="aff_1.html#T26" title="AFF_1:th.26">AFF_1:26</a></i>;<br/><a NAME="E48:63_1_1"/><i><font color="Green" title="E88">A39</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="c33">XP</font></span>)</span>,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <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="c34">XQ</font></span>)</span>,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1_1_9"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon" title="c35">P'</font> = <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="c33">XP</font></span>)</span>,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>, <font color="Maroon" title="c36">Q'</font> = <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="c34">XQ</font></span>)</span>,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> as   <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> <b>by </b><i><a class="txt" href="#E44:63_1_1"><i><font color="Green" title="E85">A36</font></i></a>, <a class="txt" href="#E46:63_1_1"><i><font color="Green" title="E86">A37</font></i></a>, <a class="ref" href="afproj.html#T23" target="_self" title="AFPROJ:th.23">Th23</a></i>;<br/>
<a NAME="E2:63_1_1_9"/>
( <font color="Maroon" title="c2">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c35">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="c35">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="c36">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="c36">Q'</font> )
 
<b>by </b><i><a class="txt" href="#E5:63_1_1"><i><font color="Green" title="E58">A7</font></i></a>, <a class="txt" href="#E7:63_1_1"><i><font color="Green" title="E59">A8</font></i></a>, <a class="txt" href="#E9:63_1_1"><i><font color="Green" title="E60">A9</font></i></a>, <a class="txt" href="#E11:63_1_1"><i><font color="Green" title="E61">A10</font></i></a>, <a class="txt" href="#E14:63_1_1"><i><font color="Green" title="E63">A12</font></i></a>, <a class="txt" href="#E44:63_1_1"><i><font color="Green" title="E85">A36</font></i></a>, <a class="txt" href="#E46:63_1_1"><i><font color="Green" title="E86">A37</font></i></a>, <a class="ref" href="afproj.html#T32" target="_self" title="AFPROJ:th.32">Th32</a></i>;<br/>
<b>hence </b><a NAME="E3:63_1_1_9"/>
( <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="c33">XP</font></span>)</span>,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <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="c34">XQ</font></span>)</span>,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="txt" href="#E1:63"><i><font color="Green" title="E52">A1</font></i></a>, <a class="txt" href="#E12:63_1_1"><i><font color="Green" title="E62">A11</font></i></a>, <a class="txt" href="afproj.html#E56"><i><font color="Green" title="E38">Lm2</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><a NAME="E49:63_1_1"/><i><font color="Green" title="E89">A40</font></i>: 
(  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c22">y</font>,<font color="Maroon" title="c23">t</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c33">XP</font> &amp;  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c29">z</font>,<font color="Maroon" title="c30">u</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c34">XQ</font> )
 <b>by </b><i><a class="txt" href="#E16:63_1_1"><i><font color="Green" title="E64">A13</font></i></a>, <a class="txt" href="#E18:63_1_1"><i><font color="Green" title="E65">A14</font></i></a>, <a class="txt" href="#E34:63_1_1"><i><font color="Green" title="E77">A26</font></i></a>, <a class="txt" href="#E36:63_1_1"><i><font color="Green" title="E78">A27</font></i></a>, <a class="txt" href="#E38:63_1_1"><i><font color="Green" title="E80">A29</font></i></a>, <a class="txt" href="#E40:63_1_1"><i><font color="Green" title="E82">A32</font></i></a>, <a class="txt" href="#E44:63_1_1"><i><font color="Green" title="E85">A36</font></i></a>, <a class="txt" href="#E46:63_1_1"><i><font color="Green" title="E86">A37</font></i></a>, <a class="ref" href="aff_4.html#T19" title="AFF_4:th.19">AFF_4:19</a></i>;<br/><div><i><font color="Green" title="E90">A41</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1_1_10"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:63_1_1_10"/><i><font color="Green" title="E91">A42</font></i>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c22">y</font>,<font color="Maroon" title="c23">t</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c29">z</font>,<font color="Maroon" title="c30">u</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="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> 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>reconsider </b><font color="Maroon" title="c35">q</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="c22">y</font>,<font color="Maroon" title="c23">t</font></span>)</span>, <font color="Maroon" title="c36">q'</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="c29">z</font>,<font color="Maroon" title="c30">u</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#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="#E41:63_1_1"><i><font color="Green" title="E83">A34</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="c37">q</font> = <font color="Maroon" title="c35">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="c37">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="c37">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="E3:63_1_1_10"/>
<font color="Maroon" title="c37">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c36">q'</font>
 <b>by </b><i><a class="txt" href="#E41:63_1_1"><i><font color="Green" title="E83">A34</font></i></a>, <a class="txt" href="#E1:63_1_1_10"><i><font color="Green" title="E91">A42</font></i></a>, <a class="ref" href="afproj.html#T11" target="_self" title="AFPROJ:th.11">Th11</a></i>;<br/><b>hence </b><a NAME="E4:63_1_1_10"/>
( <font color="Maroon" title="c37">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="c37">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="#E16:63_1_1"><i><font color="Green" title="E64">A13</font></i></a>, <a class="txt" href="#E18:63_1_1"><i><font color="Green" title="E65">A14</font></i></a>, <a class="txt" href="#E34:63_1_1"><i><font color="Green" title="E77">A26</font></i></a>, <a class="txt" href="#E36:63_1_1"><i><font color="Green" title="E78">A27</font></i></a>, <a class="txt" href="#E38:63_1_1"><i><font color="Green" title="E80">A29</font></i></a>, <a class="txt" href="#E40:63_1_1"><i><font color="Green" title="E82">A32</font></i></a>, <a class="txt" href="#E41:63_1_1"><i><font color="Green" title="E83">A34</font></i></a>, <a class="txt" href="#E44:63_1_1"><i><font color="Green" title="E85">A36</font></i></a>, <a class="txt" href="#E46:63_1_1"><i><font color="Green" title="E86">A37</font></i></a>, <a class="txt" href="#E48:63_1_1"><i><font color="Green" title="E88">A39</font></i></a>, <a class="ref" href="afproj.html#T31" target="_self" title="AFPROJ:th.31">Th31</a>, <a class="ref" href="aff_4.html#T19" title="AFF_4:th.19">AFF_4:19</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="63_1_1_11"><b>now </b></a><div class="add"><b>given </b><font color="Maroon" title="c35">w</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="E2:63_1_1_11"/><i><font color="Green" title="E91">A43</font></i>: 
( <font color="Maroon" title="c35">w</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="c22">y</font>,<font color="Maroon" title="c23">t</font> &amp; <font color="Maroon" title="c35">w</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="c29">z</font>,<font color="Maroon" title="c30">u</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="E3:63_1_1_11"/><i><font color="Green" title="E92">A44</font></i>: 
<font color="Maroon" title="c16">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c35">w</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1_1_11_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:63_1_1_11_1"/>
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c35">w</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:63_1_1_11_1"/><b>then </b>
( <font color="Maroon" title="c16">x</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c12">Xa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c22">y</font>,<font color="Maroon" title="c23">t</font> &amp; <font color="Maroon" title="c16">x</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c14">Xc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c22">y</font>,<font color="Maroon" title="c23">t</font> )
 
<b>by </b><i><a class="txt" href="#E14:63_1_1"><i><font color="Green" title="E63">A12</font></i></a>, <a class="txt" href="#E16:63_1_1"><i><font color="Green" title="E64">A13</font></i></a>, <a class="txt" href="#E18:63_1_1"><i><font color="Green" title="E65">A14</font></i></a>, <a class="txt" href="#E41:63_1_1"><i><font color="Green" title="E83">A34</font></i></a>, <a class="txt" href="#E47:63_1_1"><i><font color="Green" title="E87">A38</font></i></a>, <a class="txt" href="#E2:63_1_1_11"><i><font color="Green" title="E91">A43</font></i></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></i>;<br/>
<b>hence </b><a NAME="E3:63_1_1_11_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E37:63_1_1"><i><font color="Green" title="E79">A28</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>set </b><font color="Maroon" title="c36">R</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c16">x</font>,<font color="Maroon" title="c35">w</font>;<br/><a NAME="E4:63_1_1_11"/><i><font color="Green" title="E93">A45</font></i>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c16">x</font>,<font color="Maroon" title="c35">w</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>
 <b>by </b><i><a class="txt" href="#E3:63_1_1_11"><i><font color="Green" title="E92">A44</font></i></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="c37">q</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="c16">x</font>,<font color="Maroon" title="c35">w</font></span>)</span> 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="ref" href="afproj.html#T20" target="_self" title="AFPROJ:th.20">Th20</a></i>;<br/><b>take </b><font color="Maroon" title="c38">q</font> = <font color="Maroon" title="c37">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="c38">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="c38">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="E6:63_1_1_11"/>
( <font color="Maroon" title="c38">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="c38">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="#E14:63_1_1"><i><font color="Green" title="E63">A12</font></i></a>, <a class="txt" href="#E44:63_1_1"><i><font color="Green" title="E85">A36</font></i></a>, <a class="txt" href="#E46:63_1_1"><i><font color="Green" title="E86">A37</font></i></a>, <a class="txt" href="#E48:63_1_1"><i><font color="Green" title="E88">A39</font></i></a>, <a class="txt" href="#E49:63_1_1"><i><font color="Green" title="E89">A40</font></i></a>, <a class="txt" href="#E2:63_1_1_11"><i><font color="Green" title="E91">A43</font></i></a>, <a class="txt" href="#E3:63_1_1_11"><i><font color="Green" title="E92">A44</font></i></a>, <a class="txt" href="#E4:63_1_1_11"><i><font color="Green" title="E93">A45</font></i></a>, <a class="ref" href="afproj.html#T31" target="_self" title="AFPROJ:th.31">Th31</a>, <a class="ref" href="aff_4.html#T19" title="AFF_4:th.19">AFF_4:19</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="E52:63_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="#E26:63_1_1"><i><font color="Green" title="E70">A19</font></i></a>, <a class="txt" href="#E41:63_1_1"><i><font color="Green" title="E83">A34</font></i></a>, <a class="txt" href="#E42:63_1_1"><i><font color="Green" title="E84">A35</font></i></a>, <a class="txt" href="#E50:63_1_1"><i><font color="Green" title="E90">A41</font></i></a>, <a class="ref" href="aff_4.html#T22" title="AFF_4:th.22">AFF_4:22</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="E3:63_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="#E1:63"><i><font color="Green" title="E52">A1</font></i></a>, <a class="txt" href="#E2:63"><i><font color="Green" title="E53">A2</font></i></a>, <a class="txt" href="#E3:63"><i><font color="Green" title="E54">A3</font></i></a>, <a class="txt" href="#E1:63_1"><i><font color="Green" title="E55">A4</font></i></a>, <a class="txt" href="afproj.html#E68"><i><font color="Green" title="E50">Lm11</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="E5:63"/>
 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="#E1:63"><i><font color="Green" title="E52">A1</font></i></a>, <a class="txt" href="#E2:63"><i><font color="Green" title="E53">A2</font></i></a>, <a class="txt" href="#E3:63"><i><font color="Green" title="E54">A3</font></i></a>, <a class="txt" href="afproj.html#E66"><i><font color="Green" title="E48">Lm9</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/>


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