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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> 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="b3">A</font>, <font color="Olive" title="b4">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="Olive" title="b1">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b3">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="Olive" title="b4">K</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="b3">A</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="b4">K</font> &amp; not <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> holds <br/><font color="Olive" title="b3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">K</font></span><br/><b>let </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</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">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="c2">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="Maroon" title="c2">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b2">K</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">A</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="b2">K</font> &amp; not <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> holds <br/><font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">K</font></span><br/><b>let </b><font color="Maroon" title="c4">A</font>, <font color="Maroon" title="c5">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="c2">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c4">A</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="c5">K</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="c4">A</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="c5">K</font> &amp; not <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> implies <font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">K</font> )</span><br/>









<b>assume </b><b>that </b><br/><a NAME="E1:50"/><i><font color="Green" title="E39">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="c4">A</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="c5">K</font> )
 <b>and </b><br/><a NAME="E2:50"/><i><font color="Green" title="E40">A2</font></i>: 
( <font color="Maroon" title="c3">b</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c4">A</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="c5">K</font> )
 ; <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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> or <font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">K</font> )</span><br/>

<b>assume </b><a NAME="E3:50"/><i><font color="Green" title="E41">A3</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="c3">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">K</font></span><br/>

<b>consider </b><font color="Maroon" title="c6">X</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font><b> such that </b><br/><a NAME="E5:50"/><i><font color="Green" title="E42">A4</font></i>: 
( ( <font color="Maroon" title="c4">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="c6">X</font>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c6">X</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> ) or ( <font color="Maroon" title="c4">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="c6">X</font></span>)</span>,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c6">X</font> is <a href="aff_4.html#V1" title="AFF_4:attr.1">being_plane</a> ) )
 <b>by </b><i><a class="ref" href="afproj.html#T23" target="_self" title="AFPROJ:th.23">Th23</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">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="E7:50"/><i><font color="Green" title="E43">A5</font></i>: 
( ( <font color="Maroon" title="c5">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="c7">X'</font>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c7">X'</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> ) or ( <font color="Maroon" title="c5">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="c7">X'</font></span>)</span>,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c7">X'</font> is <a href="aff_4.html#V1" title="AFF_4:attr.1">being_plane</a> ) )
 <b>by </b><i><a class="ref" href="afproj.html#T23" target="_self" title="AFPROJ:th.23">Th23</a></i>;<br/>
<div><i><font color="Green" title="E44">A6</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:50_1"/>
<font color="Maroon" title="c2">a</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">K</font></span><br/><b>then reconsider </b><font color="Maroon" title="c8">x</font> = <font color="Maroon" title="c2">a</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font> ;<br/><a NAME="E3:50_1"/><i><font color="Green" title="E45">A7</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font>
 ;<br/><a NAME="E4:50_1"/><i><font color="Green" title="E46">A8</font></i>: 
( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X</font> &amp; <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">X'</font> )
 <b>by </b><i><a class="txt" href="#E1:50"><i><font color="Green" title="E39">A1</font></i></a>, <a class="txt" href="#E5:50"><i><font color="Green" title="E42">A4</font></i></a>, <a class="txt" href="#E7:50"><i><font color="Green" title="E43">A5</font></i></a>, <a class="ref" href="afproj.html#T26" target="_self" title="AFPROJ:th.26">Th26</a>, <a class="ref" href="afproj.html#T27" target="_self" title="AFPROJ:th.27">Th27</a></i>;<br/><div><i><font color="Green" title="E47">A9</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:50_1_1"/>
<font color="Maroon" title="c3">b</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">K</font></span><br/><b>then reconsider </b><font color="Maroon" title="c9">y</font> = <font color="Maroon" title="c3">b</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font> ;<br/><a NAME="E3:50_1_1"/>
( <font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X</font> &amp; <font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">X'</font> )
 <b>by </b><i><a class="txt" href="#E2:50"><i><font color="Green" title="E40">A2</font></i></a>, <a class="txt" href="#E5:50"><i><font color="Green" title="E42">A4</font></i></a>, <a class="txt" href="#E7:50"><i><font color="Green" title="E43">A5</font></i></a>, <a class="ref" href="afproj.html#T26" target="_self" title="AFPROJ:th.26">Th26</a>, <a class="ref" href="afproj.html#T27" target="_self" title="AFPROJ:th.27">Th27</a></i>;<br/><b>hence </b><a NAME="E4:50_1_1"/>
<font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">K</font>
 <b>by </b><i><a class="txt" href="#E1:50"><i><font color="Green" title="E39">A1</font></i></a>, <a class="txt" href="#E3:50"><i><font color="Green" title="E41">A3</font></i></a>, <a class="txt" href="#E5:50"><i><font color="Green" title="E42">A4</font></i></a>, <a class="txt" href="#E7:50"><i><font color="Green" title="E43">A5</font></i></a>, <a class="txt" href="#E4:50_1"><i><font color="Green" title="E46">A8</font></i></a>, <a class="ref" href="afproj.html#T27" target="_self" title="AFPROJ:th.27">Th27</a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</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="50_1_2"><b>now </b></a><div class="add"><b>given </b><font color="Maroon" title="c9">Y</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font><b> such that </b><a NAME="E2:50_1_2"/><i><font color="Green" title="E48">A10</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="c9">Y</font> &amp; <font color="Maroon" title="c9">Y</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">K</font></span><br/><a NAME="E3:50_1_2"/>
( <font color="Maroon" title="c9">Y</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c6">X</font> &amp; <font color="Maroon" title="c9">Y</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c7">X'</font> )
 <b>by </b><i><a class="txt" href="#E1:50"><i><font color="Green" title="E39">A1</font></i></a>, <a class="txt" href="#E2:50"><i><font color="Green" title="E40">A2</font></i></a>, <a class="txt" href="#E5:50"><i><font color="Green" title="E42">A4</font></i></a>, <a class="txt" href="#E7:50"><i><font color="Green" title="E43">A5</font></i></a>, <a class="txt" href="#E3:50_1"><i><font color="Green" title="E45">A7</font></i></a>, <a class="txt" href="#E2:50_1_2"><i><font color="Green" title="E48">A10</font></i></a>, <a class="ref" href="afproj.html#T27" target="_self" title="AFPROJ:th.27">Th27</a>, <a class="ref" href="afproj.html#T28" target="_self" title="AFPROJ:th.28">Th28</a></i>;<br/><a NAME="E4:50_1_2"/><b>then </b>
( <font color="Maroon" title="c9">Y</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c6">X</font> &amp; <font color="Maroon" title="c9">Y</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c7">X'</font> )
 <b>by </b><i><a class="txt" href="#E1:50"><i><font color="Green" title="E39">A1</font></i></a>, <a class="txt" href="#E5:50"><i><font color="Green" title="E42">A4</font></i></a>, <a class="txt" href="#E7:50"><i><font color="Green" title="E43">A5</font></i></a>, <a class="txt" href="#E3:50_1"><i><font color="Green" title="E45">A7</font></i></a>, <a class="txt" href="#E2:50_1_2"><i><font color="Green" title="E48">A10</font></i></a>, <a class="ref" href="afproj.html#T27" target="_self" title="AFPROJ:th.27">Th27</a>, <a class="ref" href="aff_4.html#T40" title="AFF_4:th.40">AFF_4:40</a></i>;<br/><a NAME="E5:50_1_2"/><b>then </b>
<font color="Maroon" title="c6">X</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c7">X'</font>
 <b>by </b><i><a class="ref" href="aff_1.html#T58" title="AFF_1:th.58">AFF_1:58</a></i>;<br/><b>hence </b><a NAME="E6:50_1_2"/>
<font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">K</font>
 <b>by </b><i><a class="txt" href="#E1:50"><i><font color="Green" title="E39">A1</font></i></a>, <a class="txt" href="#E5:50"><i><font color="Green" title="E42">A4</font></i></a>, <a class="txt" href="#E7:50"><i><font color="Green" title="E43">A5</font></i></a>, <a class="txt" href="#E4:50_1"><i><font color="Green" title="E46">A8</font></i></a>, <a class="ref" href="afproj.html#T27" target="_self" title="AFPROJ:th.27">Th27</a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</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="E7:50_1"/>
<font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">K</font>
 <b>by </b><i><a class="txt" href="#E5:50_1"><i><font color="Green" title="E47">A9</font></i></a>, <a class="ref" href="afproj.html#T20" target="_self" title="AFPROJ:th.20">Th20</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="50_2"><b>now </b></a><div class="add"><b>given </b><font color="Maroon" title="c8">Y</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font><b> such that </b><a NAME="E2:50_2"/><i><font color="Green" title="E45">A11</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="c8">Y</font> &amp; <font color="Maroon" title="c8">Y</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">K</font></span><br/><div><i><font color="Green" title="E46">A12</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="50_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:50_2_1"/>
<font color="Maroon" title="c3">b</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">K</font></span><br/><b>then reconsider </b><font color="Maroon" title="c9">y</font> = <font color="Maroon" title="c3">b</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font> ;<br/><a NAME="E3:50_2_1"/><i><font color="Green" title="E47">A13</font></i>: 
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font>
 ;<br/><a NAME="E4:50_2_1"/><i><font color="Green" title="E48">A14</font></i>: 
( <font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X</font> &amp; <font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">X'</font> )
 <b>by </b><i><a class="txt" href="#E2:50"><i><font color="Green" title="E40">A2</font></i></a>, <a class="txt" href="#E5:50"><i><font color="Green" title="E42">A4</font></i></a>, <a class="txt" href="#E7:50"><i><font color="Green" title="E43">A5</font></i></a>, <a class="ref" href="afproj.html#T26" target="_self" title="AFPROJ:th.26">Th26</a>, <a class="ref" href="afproj.html#T27" target="_self" title="AFPROJ:th.27">Th27</a></i>;<br/><a NAME="E5:50_2_1"/>
( <font color="Maroon" title="c8">Y</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c6">X</font> &amp; <font color="Maroon" title="c8">Y</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c7">X'</font> )
 <b>by </b><i><a class="txt" href="#E1:50"><i><font color="Green" title="E39">A1</font></i></a>, <a class="txt" href="#E2:50"><i><font color="Green" title="E40">A2</font></i></a>, <a class="txt" href="#E5:50"><i><font color="Green" title="E42">A4</font></i></a>, <a class="txt" href="#E7:50"><i><font color="Green" title="E43">A5</font></i></a>, <a class="txt" href="#E2:50_2"><i><font color="Green" title="E45">A11</font></i></a>, <a class="txt" href="#E3:50_2_1"><i><font color="Green" title="E47">A13</font></i></a>, <a class="ref" href="afproj.html#T27" target="_self" title="AFPROJ:th.27">Th27</a>, <a class="ref" href="afproj.html#T28" target="_self" title="AFPROJ:th.28">Th28</a></i>;<br/><a NAME="E6:50_2_1"/><b>then </b>
( <font color="Maroon" title="c8">Y</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c6">X</font> &amp; <font color="Maroon" title="c8">Y</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c7">X'</font> )
 <b>by </b><i><a class="txt" href="#E2:50"><i><font color="Green" title="E40">A2</font></i></a>, <a class="txt" href="#E5:50"><i><font color="Green" title="E42">A4</font></i></a>, <a class="txt" href="#E7:50"><i><font color="Green" title="E43">A5</font></i></a>, <a class="txt" href="#E2:50_2"><i><font color="Green" title="E45">A11</font></i></a>, <a class="txt" href="#E3:50_2_1"><i><font color="Green" title="E47">A13</font></i></a>, <a class="ref" href="afproj.html#T27" target="_self" title="AFPROJ:th.27">Th27</a>, <a class="ref" href="aff_4.html#T40" title="AFF_4:th.40">AFF_4:40</a></i>;<br/><a NAME="E7:50_2_1"/><b>then </b>
<font color="Maroon" title="c6">X</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c7">X'</font>
 <b>by </b><i><a class="ref" href="aff_1.html#T58" title="AFF_1:th.58">AFF_1:58</a></i>;<br/><b>hence </b><a NAME="E8:50_2_1"/>
<font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">K</font>
 <b>by </b><i><a class="txt" href="#E2:50"><i><font color="Green" title="E40">A2</font></i></a>, <a class="txt" href="#E5:50"><i><font color="Green" title="E42">A4</font></i></a>, <a class="txt" href="#E7:50"><i><font color="Green" title="E43">A5</font></i></a>, <a class="txt" href="#E4:50_2_1"><i><font color="Green" title="E48">A14</font></i></a>, <a class="ref" href="afproj.html#T27" target="_self" title="AFPROJ:th.27">Th27</a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</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="50_2_2"><b>now </b></a><div class="add"><b>given </b><font color="Maroon" title="c9">Y'</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font><b> such that </b><a NAME="E2:50_2_2"/><i><font color="Green" title="E47">A15</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="c9">Y'</font> &amp; <font color="Maroon" title="c9">Y'</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">K</font></span><br/><a NAME="E3:50_2_2"/><i><font color="Green" title="E48">A16</font></i>: 
for <font color="Olive" title="b1">Z</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font><br/> for <font color="Olive" title="b2">M</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="b2">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">Z</font>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b1">Z</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &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="b2">M</font> holds <br/> not <font color="Maroon" title="c3">b</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b2">M</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_2_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">Z</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</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"> for <font color="Olive" title="b1">M</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">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">Z</font>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c10">Z</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &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="b1">M</font> holds <br/> not <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></span><br/><b>let </b><font color="Maroon" title="c11">M</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="c11">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">Z</font>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c10">Z</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &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="c11">M</font> implies  not <font color="Maroon" title="c3">b</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c11">M</font> )</span><br/>



<b>assume </b><a NAME="E1:50_2_2_1"/><i><font color="Green" title="E49">A17</font></i>: 
( <font color="Maroon" title="c11">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">Z</font>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c10">Z</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c2">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c11">M</font> or  not <font color="Maroon" title="c3">b</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c11">M</font> )</span><br/>

<b>assume </b><a NAME="E2:50_2_2_1"/>
( <font color="Maroon" title="c2">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c11">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="c11">M</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="E3:50_2_2_1"/><b>then </b>
( <font color="Maroon" title="c8">Y</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c10">Z</font> &amp; <font color="Maroon" title="c9">Y'</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c10">Z</font> )
 
<b>by </b><i><a class="txt" href="#E2:50_2"><i><font color="Green" title="E45">A11</font></i></a>, <a class="txt" href="#E2:50_2_2"><i><font color="Green" title="E47">A15</font></i></a>, <a class="txt" href="#E1:50_2_2_1"><i><font color="Green" title="E49">A17</font></i></a>, <a class="ref" href="afproj.html#T28" target="_self" title="AFPROJ:th.28">Th28</a></i>;<br/>
<a NAME="E4:50_2_2_1"/><b>then </b>
( <font color="Maroon" title="c8">Y</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c10">Z</font> &amp; <font color="Maroon" title="c9">Y'</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c10">Z</font> )
 
<b>by </b><i><a class="txt" href="#E2:50_2"><i><font color="Green" title="E45">A11</font></i></a>, <a class="txt" href="#E2:50_2_2"><i><font color="Green" title="E47">A15</font></i></a>, <a class="txt" href="#E1:50_2_2_1"><i><font color="Green" title="E49">A17</font></i></a>, <a class="ref" href="aff_4.html#T40" title="AFF_4:th.40">AFF_4:40</a></i>;<br/>
<a NAME="E5:50_2_2_1"/><b>then </b>
<font color="Maroon" title="c8">Y</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c9">Y'</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T58" title="AFF_1:th.58">AFF_1:58</a></i>;<br/>
<b>hence </b><a NAME="E6:50_2_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E3:50"><i><font color="Green" title="E41">A3</font></i></a>, <a class="txt" href="#E2:50_2"><i><font color="Green" title="E45">A11</font></i></a>, <a class="txt" href="#E2:50_2_2"><i><font color="Green" title="E47">A15</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="E4:50_2_2"/><i><font color="Green" title="E49">A18</font></i>: 
not <font color="Maroon" title="c8">Y</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c9">Y'</font>
 <b>by </b><i><a class="txt" href="#E3:50"><i><font color="Green" title="E41">A3</font></i></a>, <a class="txt" href="#E2:50_2"><i><font color="Green" title="E45">A11</font></i></a>, <a class="txt" href="#E2:50_2_2"><i><font color="Green" title="E47">A15</font></i></a>, <a class="ref" href="afproj.html#T11" target="_self" title="AFPROJ:th.11">Th11</a></i>;<br/><a NAME="E5:50_2_2"/>
( <font color="Maroon" title="c8">Y</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c6">X</font> &amp; <font color="Maroon" title="c8">Y</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c7">X'</font> &amp; <font color="Maroon" title="c9">Y'</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c6">X</font> &amp; <font color="Maroon" title="c9">Y'</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c7">X'</font> )
 <b>by </b><i><a class="txt" href="#E1:50"><i><font color="Green" title="E39">A1</font></i></a>, <a class="txt" href="#E2:50"><i><font color="Green" title="E40">A2</font></i></a>, <a class="txt" href="#E5:50"><i><font color="Green" title="E42">A4</font></i></a>, <a class="txt" href="#E7:50"><i><font color="Green" title="E43">A5</font></i></a>, <a class="txt" href="#E2:50_2"><i><font color="Green" title="E45">A11</font></i></a>, <a class="txt" href="#E2:50_2_2"><i><font color="Green" title="E47">A15</font></i></a>, <a class="txt" href="#E3:50_2_2"><i><font color="Green" title="E48">A16</font></i></a>, <a class="ref" href="afproj.html#T29" target="_self" title="AFPROJ:th.29">Th29</a></i>;<br/><a NAME="E6:50_2_2"/><b>then </b>
<font color="Maroon" title="c6">X</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c7">X'</font>
 <b>by </b><i><a class="txt" href="#E1:50"><i><font color="Green" title="E39">A1</font></i></a>, <a class="txt" href="#E2:50"><i><font color="Green" title="E40">A2</font></i></a>, <a class="txt" href="#E5:50"><i><font color="Green" title="E42">A4</font></i></a>, <a class="txt" href="#E7:50"><i><font color="Green" title="E43">A5</font></i></a>, <a class="txt" href="#E2:50_2"><i><font color="Green" title="E45">A11</font></i></a>, <a class="txt" href="#E2:50_2_2"><i><font color="Green" title="E47">A15</font></i></a>, <a class="txt" href="#E3:50_2_2"><i><font color="Green" title="E48">A16</font></i></a>, <a class="txt" href="#E4:50_2_2"><i><font color="Green" title="E49">A18</font></i></a>, <a class="ref" href="afproj.html#T5" target="_self" title="AFPROJ:th.5">Th5</a></i>;<br/><b>hence </b><a NAME="E7:50_2_2"/>
<font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">K</font>
 <b>by </b><i><a class="txt" href="#E1:50"><i><font color="Green" title="E39">A1</font></i></a>, <a class="txt" href="#E2:50"><i><font color="Green" title="E40">A2</font></i></a>, <a class="txt" href="#E5:50"><i><font color="Green" title="E42">A4</font></i></a>, <a class="txt" href="#E7:50"><i><font color="Green" title="E43">A5</font></i></a>, <a class="txt" href="#E3:50_2_2"><i><font color="Green" title="E48">A16</font></i></a>, <a class="ref" href="afproj.html#T13" target="_self" title="AFPROJ:th.13">Th13</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>hence </b><a NAME="E5:50_2"/>
<font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">K</font>
 <b>by </b><i><a class="txt" href="#E3:50_2"><i><font color="Green" title="E46">A12</font></i></a>, <a class="ref" href="afproj.html#T20" target="_self" title="AFPROJ:th.20">Th20</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="E10:50"/>
<font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">K</font>
 <b>by </b><i><a class="txt" href="#E8:50"><i><font color="Green" title="E44">A6</font></i></a>, <a class="ref" href="afproj.html#T20" target="_self" title="AFPROJ:th.20">Th20</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>
