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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">K</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="c1">AS</font> is   <a href="diraf.html#NM2" title="DIRAF:NM.2">AffinPlane</a> holds <br/> ex <font color="Olive" title="b3">a</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="b3">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b1">A</font> &amp; <font color="Olive" title="b3">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b2">K</font> )</span><br/><b>let </b><font color="Maroon" title="c2">A</font>, <font color="Maroon" title="c3">K</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="c1">AS</font> is   <a href="diraf.html#NM2" title="DIRAF:NM.2">AffinPlane</a> implies  ex <font color="Olive" title="b1">a</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">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b1">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c3">K</font> ) )</span><br/>





<b>assume </b><a NAME="E1:53"/><i><font color="Green" title="E42">A1</font></i>: 
<font color="Maroon" title="c1">AS</font> is   <a href="diraf.html#NM2" title="DIRAF:NM.2">AffinPlane</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">a</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">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b1">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c3">K</font> )</span><br/>

<b>consider </b><font color="Maroon" title="c4">X</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font><b> such that </b><br/><a NAME="E3:53"/><i><font color="Green" title="E43">A2</font></i>: 
( ( <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">X</font>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c4">X</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  ) or ( <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="afproj.html#K6" title="AFPROJ:func.6">PDir</a> <font color="Maroon" title="c4">X</font></span>)</span>,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c4">X</font> <a href="aff_4.html#NR1" title="AFF_4:NR.1">is_plane</a>  ) )
 <b>by </b><i><a class="ref" href="afproj.html#T23" target="_self" title="AFPROJ:th.23">Th23</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">X'</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font><b> such that </b><br/><a NAME="E5:53"/><i><font color="Green" title="E44">A3</font></i>: 
( ( <font color="Maroon" title="c3">K</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="c5">X'</font>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c5">X'</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  ) or ( <font color="Maroon" title="c3">K</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="c5">X'</font></span>)</span>,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c5">X'</font> <a href="aff_4.html#NR1" title="AFF_4:NR.1">is_plane</a>  ) )
 <b>by </b><i><a class="ref" href="afproj.html#T23" target="_self" title="AFPROJ:th.23">Th23</a></i>;<br/>
<div><i><font color="Green" title="E45">A4</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:53_1"/><i><font color="Green" title="E46">A5</font></i>: 
( <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="afproj.html#K6" title="AFPROJ:func.6">PDir</a> <font color="Maroon" title="c4">X</font></span>)</span>,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c4">X</font> <a href="aff_4.html#NR1" title="AFF_4:NR.1">is_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">a</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">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b1">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c3">K</font> )</span><br/><a NAME="E2:53_1"/><b>then </b><i><font color="Green" title="E47">A6</font></i>: 
<font color="Maroon" title="c4">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">AS</font>
 <b>by </b><i><a class="txt" href="afproj.html#E1:53"><i><font color="Green" title="E42">A1</font></i></a>, <a class="ref" href="afproj.html#T2" target="_self" title="AFPROJ:th.2">Th2</a></i>;<br/><div><i><font color="Green" title="E48">A7</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:53_1_1"/><i><font color="Green" title="E49">A8</font></i>: 
( <font color="Maroon" title="c3">K</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="c5">X'</font></span>)</span>,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c5">X'</font> <a href="aff_4.html#NR1" title="AFF_4:NR.1">is_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">a</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">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b1">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c3">K</font> )</span><br/><b>consider </b><font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c7">b</font>, <font color="Maroon" title="c8">c</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="E3:53_1_1"/><i><font color="Green" title="E50">A9</font></i>: 
<font color="Maroon" title="c6">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c2">A</font>
 <b>and </b><br/><a NAME="E4:53_1_1"/>
( <font color="Maroon" title="c7">b</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c8">c</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c6">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">b</font> &amp; <font color="Maroon" title="c7">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">c</font> &amp; <font color="Maroon" title="c8">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">a</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="c9">a</font> = <font color="Maroon" title="c6">a</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c9">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c9">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c3">K</font> )</span><br/><b>thus </b><a NAME="E5:53_1_1"/>
( <font color="Maroon" title="c9">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c9">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c3">K</font> )
 <b>by </b><i><a class="txt" href="afproj.html#E1:53"><i><font color="Green" title="E42">A1</font></i></a>, <a class="txt" href="afproj.html#E1:53_1"><i><font color="Green" title="E46">A5</font></i></a>, <a class="txt" href="afproj.html#E1:53_1_1"><i><font color="Green" title="E49">A8</font></i></a>, <a class="txt" href="afproj.html#E3:53_1_1"><i><font color="Green" title="E50">A9</font></i></a>, <a class="ref" href="afproj.html#T3" target="_self" title="AFPROJ:th.3">Th3</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="53_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:53_1_2"/><i><font color="Green" title="E49">A10</font></i>: 
( <font color="Maroon" title="c3">K</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="c5">X'</font>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c5">X'</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_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">a</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">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b1">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c3">K</font> )</span><br/><a NAME="E2:53_1_2"/><b>then </b><i><font color="Green" title="E50">A11</font></i>: 
<font color="Maroon" title="c5">X'</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c4">X</font>
 <b>by </b><i><a class="txt" href="afproj.html#E1:53_1"><i><font color="Green" title="E46">A5</font></i></a>, <a class="txt" href="afproj.html#E2:53_1"><i><font color="Green" title="E47">A6</font></i></a>, <a class="ref" href="aff_4.html#T42" title="AFF_4:th.42">AFF_4:42</a></i>;<br/><a NAME="E3:53_1_2"/>
<font color="Maroon" title="c5">X'</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c5">X'</font>
 <b>by </b><i><a class="txt" href="afproj.html#E1:53_1_2"><i><font color="Green" title="E49">A10</font></i></a>, <a class="ref" href="aff_1.html#T55" title="AFF_1:th.55">AFF_1:55</a></i>;<br/><a NAME="E4:53_1_2"/><b>then </b><i><font color="Green" title="E51">A12</font></i>: 
<font color="Maroon" title="c5">X'</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c5">X'</font>
 <b>by </b><i><a class="txt" href="afproj.html#E1:53_1_2"><i><font color="Green" title="E49">A10</font></i></a>, <a class="ref" href="aff_4.html#T40" title="AFF_4:th.40">AFF_4:40</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c6">a</font> =  <a href="afproj.html#K5" title="AFPROJ:func.5">LDir</a> <font color="Maroon" title="c5">X'</font> as   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <span class="p1">(<span class="default"><a href="afproj.html#K13" title="AFPROJ:func.13">IncProjSp_of</a> <font color="Maroon" title="c1">AS</font></span>)</span> <b>by </b><i><a class="txt" href="afproj.html#E1:53_1_2"><i><font color="Green" title="E49">A10</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="c7">a</font> = <font color="Maroon" title="c6">a</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c7">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c3">K</font> )</span><br/><b>thus </b><a NAME="E6:53_1_2"/>
( <font color="Maroon" title="c7">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c7">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c3">K</font> )
 <b>by </b><i><a class="txt" href="afproj.html#E1:53_1"><i><font color="Green" title="E46">A5</font></i></a>, <a class="txt" href="afproj.html#E1:53_1_2"><i><font color="Green" title="E49">A10</font></i></a>, <a class="txt" href="afproj.html#E2:53_1_2"><i><font color="Green" title="E50">A11</font></i></a>, <a class="txt" href="afproj.html#E4:53_1_2"><i><font color="Green" title="E51">A12</font></i></a>, <a class="ref" href="afproj.html#T28" target="_self" title="AFPROJ:th.28">Th28</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="E5:53_1"/>
 ex <font color="Olive" title="b1">a</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">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b1">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c3">K</font> )
 <b>by </b><i><a class="txt" href="afproj.html#E5:53"><i><font color="Green" title="E44">A3</font></i></a>, <a class="txt" href="afproj.html#E3:53_1"><i><font color="Green" title="E48">A7</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="53_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:53_2"/><i><font color="Green" title="E46">A13</font></i>: 
( <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">X</font>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c4">X</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_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">a</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">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b1">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c3">K</font> )</span><br/><div><i><font color="Green" title="E47">A14</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="53_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:53_2_1"/><i><font color="Green" title="E48">A15</font></i>: 
( <font color="Maroon" title="c3">K</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="c5">X'</font></span>)</span>,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c5">X'</font> <a href="aff_4.html#NR1" title="AFF_4:NR.1">is_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">a</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">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b1">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c3">K</font> )</span><br/><a NAME="E2:53_2_1"/><b>then </b>
<font color="Maroon" title="c5">X'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">AS</font>
 <b>by </b><i><a class="txt" href="afproj.html#E1:53"><i><font color="Green" title="E42">A1</font></i></a>, <a class="ref" href="afproj.html#T2" target="_self" title="AFPROJ:th.2">Th2</a></i>;<br/><a NAME="E3:53_2_1"/><b>then </b><i><font color="Green" title="E49">A16</font></i>: 
<font color="Maroon" title="c4">X</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c5">X'</font>
 <b>by </b><i><a class="txt" href="afproj.html#E1:53_2"><i><font color="Green" title="E46">A13</font></i></a>, <a class="txt" href="afproj.html#E1:53_2_1"><i><font color="Green" title="E48">A15</font></i></a>, <a class="ref" href="aff_4.html#T42" title="AFF_4:th.42">AFF_4:42</a></i>;<br/><a NAME="E4:53_2_1"/>
<font color="Maroon" title="c4">X</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c4">X</font>
 <b>by </b><i><a class="txt" href="afproj.html#E1:53_2"><i><font color="Green" title="E46">A13</font></i></a>, <a class="ref" href="aff_1.html#T55" title="AFF_1:th.55">AFF_1:55</a></i>;<br/><a NAME="E5:53_2_1"/><b>then </b><i><font color="Green" title="E50">A17</font></i>: 
<font color="Maroon" title="c4">X</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c4">X</font>
 <b>by </b><i><a class="txt" href="afproj.html#E1:53_2"><i><font color="Green" title="E46">A13</font></i></a>, <a class="ref" href="aff_4.html#T40" title="AFF_4:th.40">AFF_4:40</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c6">a</font> =  <a href="afproj.html#K5" title="AFPROJ:func.5">LDir</a> <font color="Maroon" title="c4">X</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <span class="p1">(<span class="default"><a href="afproj.html#K13" title="AFPROJ:func.13">IncProjSp_of</a> <font color="Maroon" title="c1">AS</font></span>)</span> <b>by </b><i><a class="txt" href="afproj.html#E1:53_2"><i><font color="Green" title="E46">A13</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="c7">a</font> = <font color="Maroon" title="c6">a</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c7">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c3">K</font> )</span><br/><b>thus </b><a NAME="E7:53_2_1"/>
( <font color="Maroon" title="c7">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c7">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c3">K</font> )
 <b>by </b><i><a class="txt" href="afproj.html#E1:53_2"><i><font color="Green" title="E46">A13</font></i></a>, <a class="txt" href="afproj.html#E1:53_2_1"><i><font color="Green" title="E48">A15</font></i></a>, <a class="txt" href="afproj.html#E3:53_2_1"><i><font color="Green" title="E49">A16</font></i></a>, <a class="txt" href="afproj.html#E5:53_2_1"><i><font color="Green" title="E50">A17</font></i></a>, <a class="ref" href="afproj.html#T28" target="_self" title="AFPROJ:th.28">Th28</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><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:53_2_2"/><i><font color="Green" title="E48">A18</font></i>: 
( <font color="Maroon" title="c3">K</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="c5">X'</font>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c5">X'</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_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">a</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">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b1">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c3">K</font> )</span><br/><div><i><font color="Green" title="E49">A19</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="53_2_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:53_2_2_1"/><i><font color="Green" title="E50">A20</font></i>: 
<font color="Maroon" title="c4">X</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c5">X'</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">a</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">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b1">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c3">K</font> )</span><br/><b>reconsider </b><font color="Maroon" title="c6">a</font> =  <a href="afproj.html#K5" title="AFPROJ:func.5">LDir</a> <font color="Maroon" title="c4">X</font>, <font color="Maroon" title="c7">b</font> =  <a href="afproj.html#K5" title="AFPROJ:func.5">LDir</a> <font color="Maroon" title="c5">X'</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="incsp_1.html#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="afproj.html#E1:53_2"><i><font color="Green" title="E46">A13</font></i></a>, <a class="txt" href="afproj.html#E1:53_2_2"><i><font color="Green" title="E48">A18</font></i></a>, <a class="ref" href="afproj.html#T20" target="_self" title="AFPROJ:th.20">Th20</a></i>;<br/><b>take </b><font color="Maroon" title="c8">a</font> = <font color="Maroon" title="c6">a</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c8">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c8">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c3">K</font> )</span><br/><a NAME="E3:53_2_2_1"/><i><font color="Green" title="E51">A21</font></i>: 
<font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b</font>
 <b>by </b><i><a class="txt" href="afproj.html#E1:53_2"><i><font color="Green" title="E46">A13</font></i></a>, <a class="txt" href="afproj.html#E1:53_2_2"><i><font color="Green" title="E48">A18</font></i></a>, <a class="txt" href="afproj.html#E1:53_2_2_1"><i><font color="Green" title="E50">A20</font></i></a>, <a class="ref" href="afproj.html#T11" target="_self" title="AFPROJ:th.11">Th11</a></i>;<br/><a NAME="E4:53_2_2_1"/>
( <font color="Maroon" title="c4">X</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c4">X</font> &amp; <font color="Maroon" title="c5">X'</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c5">X'</font> )
 <b>by </b><i><a class="txt" href="afproj.html#E1:53_2"><i><font color="Green" title="E46">A13</font></i></a>, <a class="txt" href="afproj.html#E1:53_2_2"><i><font color="Green" title="E48">A18</font></i></a>, <a class="ref" href="aff_1.html#T55" title="AFF_1:th.55">AFF_1:55</a></i>;<br/><a NAME="E5:53_2_2_1"/><b>then </b>
( <font color="Maroon" title="c4">X</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c4">X</font> &amp; <font color="Maroon" title="c5">X'</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c5">X'</font> )
 <b>by </b><i><a class="txt" href="afproj.html#E1:53_2"><i><font color="Green" title="E46">A13</font></i></a>, <a class="txt" href="afproj.html#E1:53_2_2"><i><font color="Green" title="E48">A18</font></i></a>, <a class="ref" href="aff_4.html#T40" title="AFF_4:th.40">AFF_4:40</a></i>;<br/><b>hence </b><a NAME="E6:53_2_2_1"/>
( <font color="Maroon" title="c8">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c8">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c3">K</font> )
 <b>by </b><i><a class="txt" href="afproj.html#E1:53_2"><i><font color="Green" title="E46">A13</font></i></a>, <a class="txt" href="afproj.html#E1:53_2_2"><i><font color="Green" title="E48">A18</font></i></a>, <a class="txt" href="afproj.html#E3:53_2_2_1"><i><font color="Green" title="E51">A21</font></i></a>, <a class="ref" href="afproj.html#T28" target="_self" title="AFPROJ:th.28">Th28</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="53_2_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:53_2_2_2"/>
not <font color="Maroon" title="c4">X</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c5">X'</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">a</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">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b1">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c3">K</font> )</span><br/><b>then consider </b><font color="Maroon" title="c6">x</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="E3:53_2_2_2"/><i><font color="Green" title="E50">A22</font></i>: 
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> &amp; <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X'</font> )
 <b>by </b><i><a class="txt" href="afproj.html#E1:53"><i><font color="Green" title="E42">A1</font></i></a>, <a class="txt" href="afproj.html#E1:53_2"><i><font color="Green" title="E46">A13</font></i></a>, <a class="txt" href="afproj.html#E1:53_2_2"><i><font color="Green" title="E48">A18</font></i></a>, <a class="ref" href="aff_1.html#T72" title="AFF_1:th.72">AFF_1:72</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c7">a</font> = <font color="Maroon" title="c6">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <span class="p1">(<span class="default"><a href="afproj.html#K13" title="AFPROJ:func.13">IncProjSp_of</a> <font color="Maroon" title="c1">AS</font></span>)</span> <b>by </b><i><a class="ref" href="afproj.html#T20" target="_self" title="AFPROJ:th.20">Th20</a></i>;<br/><b>take </b><font color="Maroon" title="c8">a</font> = <font color="Maroon" title="c7">a</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c8">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c8">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c3">K</font> )</span><br/><b>thus </b><a NAME="E5:53_2_2_2"/>
( <font color="Maroon" title="c8">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c8">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c3">K</font> )
 <b>by </b><i><a class="txt" href="afproj.html#E1:53_2"><i><font color="Green" title="E46">A13</font></i></a>, <a class="txt" href="afproj.html#E1:53_2_2"><i><font color="Green" title="E48">A18</font></i></a>, <a class="txt" href="afproj.html#E3:53_2_2_2"><i><font color="Green" title="E50">A22</font></i></a>, <a class="ref" href="afproj.html#T26" target="_self" title="AFPROJ:th.26">Th26</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="E4:53_2_2"/>
 ex <font color="Olive" title="b1">a</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">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b1">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c3">K</font> )
 <b>by </b><i><a class="txt" href="afproj.html#E2:53_2_2"><i><font color="Green" title="E49">A19</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="E4:53_2"/>
 ex <font color="Olive" title="b1">a</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">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b1">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c3">K</font> )
 <b>by </b><i><a class="txt" href="afproj.html#E5:53"><i><font color="Green" title="E44">A3</font></i></a>, <a class="txt" href="afproj.html#E2:53_2"><i><font color="Green" title="E47">A14</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="E8:53"/>
 ex <font color="Olive" title="b1">a</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">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b1">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c3">K</font> )
 <b>by </b><i><a class="txt" href="afproj.html#E3:53"><i><font color="Green" title="E43">A2</font></i></a>, <a class="txt" href="afproj.html#E6:53"><i><font color="Green" title="E45">A4</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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