<?xml version="1.0"?>
<div class="add">

<b>reconsider </b><font color="Maroon" title="c1">Zero1</font> =  <a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> , <font color="Maroon" title="c2">One</font> = 1, <font color="Maroon" title="c3">Two</font> = 2, <font color="Maroon" title="c4">Three</font> = 3 as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span> <b>by </b><i><a class="ref" href="enumset1.html#D2" title="ENUMSET1:def.2">ENUMSET1:def 2</a></i>;<br/>
<a NAME="E2:29_1_1"/>
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c1">Zero1</font>,<font color="Maroon" title="c2">One</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> where <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span> : <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">b</font> </span>}</span> 
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c5">Li</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> where <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span> : <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">b</font> </span>}</span>  as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<a NAME="E4:29_1_1"/>
<span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c1">Zero1</font>,<font color="Maroon" title="c2">One</font>,<font color="Maroon" title="c3">Two</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> where <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span> : ( <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">b</font> &amp; <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">c</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">c</font> ) </span>}</span> 
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c6">Pl</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> where <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span> : ( <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">b</font> &amp; <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">c</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">c</font> ) </span>}</span>  as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<a NAME="E6:29_1_1"/>
<span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">l</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>, <font color="Olive" title="b2">l</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">Li</font> : <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">l</font> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</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="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">l</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>, <font color="Olive" title="b2">l</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">Li</font> : <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">l</font> </span>}</span>  or <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )</span><br/>

<b>assume </b><a NAME="E1:29_1_1_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">l</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>, <font color="Olive" title="b2">l</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">Li</font> : <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">l</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="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span><br/>

<a NAME="E2:29_1_1_1"/><b>then </b>
 ex <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span> ex <font color="Olive" title="b2">l</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">Li</font> st <br/>( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">l</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">l</font> )
 
;<br/>
<b>hence </b><a NAME="E3:29_1_1_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 ; <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>then reconsider </b><font color="Maroon" title="c7">i1</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">l</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>, <font color="Olive" title="b2">l</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">Li</font> : <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">l</font> </span>}</span>  as    <a href="relset_1.html#M2" title="RELSET_1:mode.2">Relation</a> of <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font> <b>by </b><i><a class="ref" href="relset_1.html#D1" title="RELSET_1:def.1">RELSET_1:def 1</a></i>;<br/>
<a NAME="E8:29_1_1"/>
<span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">p</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>, <font color="Olive" title="b2">p</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">Pl</font> : <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">p</font> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c6">Pl</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</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="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">p</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>, <font color="Olive" title="b2">p</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">Pl</font> : <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">p</font> </span>}</span>  or <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c6">Pl</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )</span><br/>

<b>assume </b><a NAME="E1:29_1_1_2"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">p</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>, <font color="Olive" title="b2">p</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">Pl</font> : <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">p</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="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c6">Pl</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span><br/>

<a NAME="E2:29_1_1_2"/><b>then </b>
 ex <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span> ex <font color="Olive" title="b2">p</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">Pl</font> st <br/>( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">p</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">p</font> )
 
;<br/>
<b>hence </b><a NAME="E3:29_1_1_2"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c6">Pl</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 ; <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>then reconsider </b><font color="Maroon" title="c8">i2</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">p</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>, <font color="Olive" title="b2">p</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">Pl</font> : <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">p</font> </span>}</span>  as    <a href="relset_1.html#M2" title="RELSET_1:mode.2">Relation</a> of <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c6">Pl</font> <b>by </b><i><a class="ref" href="relset_1.html#D1" title="RELSET_1:def.1">RELSET_1:def 1</a></i>;<br/>
<a NAME="E10:29_1_1"/>
<span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">l</font>,<font color="Olive" title="b2">p</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> where <font color="Olive" title="b1">l</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">Li</font>, <font color="Olive" title="b2">p</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">Pl</font> : <font color="Olive" title="b1">l</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">p</font> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</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="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">l</font>,<font color="Olive" title="b2">p</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> where <font color="Olive" title="b1">l</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">Li</font>, <font color="Olive" title="b2">p</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">Pl</font> : <font color="Olive" title="b1">l</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">p</font> </span>}</span>  or <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )</span><br/>

<b>assume </b><a NAME="E1:29_1_1_3"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">l</font>,<font color="Olive" title="b2">p</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> where <font color="Olive" title="b1">l</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">Li</font>, <font color="Olive" title="b2">p</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">Pl</font> : <font color="Olive" title="b1">l</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">p</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="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span><br/>

<a NAME="E2:29_1_1_3"/><b>then </b>
 ex <font color="Olive" title="b1">l</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">Li</font> ex <font color="Olive" title="b2">p</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">Pl</font> st <br/>( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">l</font>,<font color="Olive" title="b2">p</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; <font color="Olive" title="b1">l</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">p</font> )
 
;<br/>
<b>hence </b><a NAME="E3:29_1_1_3"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 ; <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>then reconsider </b><font color="Maroon" title="c9">i3</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">l</font>,<font color="Olive" title="b2">p</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> where <font color="Olive" title="b1">l</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">Li</font>, <font color="Olive" title="b2">p</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">Pl</font> : <font color="Olive" title="b1">l</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">p</font> </span>}</span>  as    <a href="relset_1.html#M2" title="RELSET_1:mode.2">Relation</a> of <font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font> <b>by </b><i><a class="ref" href="relset_1.html#D1" title="RELSET_1:def.1">RELSET_1:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_4"><b>now </b></a><div class="add"><b>set </b><font color="Maroon" title="c10">S</font> =  <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #);<br/><a NAME="E1:29_1_1_4"/><i><font color="Green" title="E30">A1</font></i>: 
for <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span><br/> for <font color="Olive" title="b2">l</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">Li</font>  st <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">l</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">i1</font> holds <br/><font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">l</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">a</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></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">l</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">Li</font>  st <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">a</font>,<font color="Olive" title="b1">l</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">i1</font> holds <br/><font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">l</font></span><br/><b>let </b><font color="Maroon" title="c12">l</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">Li</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">a</font>,<font color="Maroon" title="c12">l</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">i1</font> implies <font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">l</font> )</span><br/>



<b>assume </b><a NAME="E1:29_1_1_4_1"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">a</font>,<font color="Maroon" title="c12">l</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">i1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">l</font></span><br/>

<b>then consider </b><font color="Maroon" title="c13">b</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>, <font color="Maroon" title="c14">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">Li</font><b> such that </b><br/><a NAME="E3:29_1_1_4_1"/><i><font color="Green" title="E31">A2</font></i>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">a</font>,<font color="Maroon" title="c12">l</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c13">b</font>,<font color="Maroon" title="c14">k</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <b>and </b><br/><a NAME="E4:29_1_1_4_1"/><i><font color="Green" title="E32">A3</font></i>: 
<font color="Maroon" title="c13">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">k</font>
 ;<br/>
<a NAME="E5:29_1_1_4_1"/>
( <font color="Maroon" title="c11">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">b</font> &amp; <font color="Maroon" title="c12">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">k</font> )
 
<b>by </b><i><a class="txt" href="#E3:29_1_1_4_1"><i><font color="Green" title="E31">A2</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><a NAME="E6:29_1_1_4_1"/>
<font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">l</font>
 <b>by </b><i><a class="txt" href="#E4:29_1_1_4_1"><i><font color="Green" title="E32">A3</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="E2:29_1_1_4"/><i><font color="Green" title="E31">A4</font></i>: 
for <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span><br/> for <font color="Olive" title="b2">p</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">Pl</font>  st <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">p</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">i2</font> holds <br/><font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">p</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_4_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">a</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></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">p</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">Pl</font>  st <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">a</font>,<font color="Olive" title="b1">p</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">i2</font> holds <br/><font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">p</font></span><br/><b>let </b><font color="Maroon" title="c12">p</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">Pl</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">a</font>,<font color="Maroon" title="c12">p</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">i2</font> implies <font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">p</font> )</span><br/>



<b>assume </b><a NAME="E1:29_1_1_4_2"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">a</font>,<font color="Maroon" title="c12">p</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">i2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">p</font></span><br/>

<b>then consider </b><font color="Maroon" title="c13">b</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>, <font color="Maroon" title="c14">q</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">Pl</font><b> such that </b><br/><a NAME="E3:29_1_1_4_2"/><i><font color="Green" title="E32">A5</font></i>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">a</font>,<font color="Maroon" title="c12">p</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c13">b</font>,<font color="Maroon" title="c14">q</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <b>and </b><br/><a NAME="E4:29_1_1_4_2"/><i><font color="Green" title="E33">A6</font></i>: 
<font color="Maroon" title="c13">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">q</font>
 ;<br/>
<a NAME="E5:29_1_1_4_2"/>
( <font color="Maroon" title="c11">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">b</font> &amp; <font color="Maroon" title="c12">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">q</font> )
 
<b>by </b><i><a class="txt" href="#E3:29_1_1_4_2"><i><font color="Green" title="E32">A5</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><a NAME="E6:29_1_1_4_2"/>
<font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">p</font>
 <b>by </b><i><a class="txt" href="#E4:29_1_1_4_2"><i><font color="Green" title="E33">A6</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="E3:29_1_1_4"/><i><font color="Green" title="E32">A7</font></i>: 
for <font color="Olive" title="b1">p</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">Pl</font><br/> for <font color="Olive" title="b2">l</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">Li</font>  st <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b2">l</font>,<font color="Olive" title="b1">p</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">i3</font> holds <br/><font color="Olive" title="b2">l</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">p</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_4_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">p</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">Pl</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">l</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">Li</font>  st <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">l</font>,<font color="Maroon" title="c11">p</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">i3</font> holds <br/><font color="Olive" title="b1">l</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">p</font></span><br/><b>let </b><font color="Maroon" title="c12">l</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">Li</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c12">l</font>,<font color="Maroon" title="c11">p</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">i3</font> implies <font color="Maroon" title="c12">l</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">p</font> )</span><br/>



<b>assume </b><a NAME="E1:29_1_1_4_3"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c12">l</font>,<font color="Maroon" title="c11">p</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">i3</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c12">l</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">p</font></span><br/>

<b>then consider </b><font color="Maroon" title="c13">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">Li</font>, <font color="Maroon" title="c14">q</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">Pl</font><b> such that </b><br/><a NAME="E3:29_1_1_4_3"/><i><font color="Green" title="E33">A8</font></i>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c12">l</font>,<font color="Maroon" title="c11">p</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c13">k</font>,<font color="Maroon" title="c14">q</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <b>and </b><br/><a NAME="E4:29_1_1_4_3"/><i><font color="Green" title="E34">A9</font></i>: 
<font color="Maroon" title="c13">k</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c14">q</font>
 ;<br/>
<a NAME="E5:29_1_1_4_3"/>
( <font color="Maroon" title="c12">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">k</font> &amp; <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">q</font> )
 
<b>by </b><i><a class="txt" href="#E3:29_1_1_4_3"><i><font color="Green" title="E33">A8</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><a NAME="E6:29_1_1_4_3"/>
<font color="Maroon" title="c12">l</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">p</font>
 <b>by </b><i><a class="txt" href="#E4:29_1_1_4_3"><i><font color="Green" title="E34">A9</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>thus </b><a NAME="E4:29_1_1_4"/>
 <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #) is <a href="incsp_1.html#V5" title="INCSP_1:attr.5">with_non-trivial_lines</a>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #) is <a href="incsp_1.html#V6" title="INCSP_1:attr.6">linear</a> &amp;  <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #) is <a href="incsp_1.html#V7" title="INCSP_1:attr.7">up-2-rank</a> &amp;  <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #) is <a href="incsp_1.html#V8" title="INCSP_1:attr.8">with_non-empty_planes</a> &amp;  <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #) is <a href="incsp_1.html#V9" title="INCSP_1:attr.9">planar</a> &amp;  <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #) is <a href="incsp_1.html#V10" title="INCSP_1:attr.10">with_&lt;=1_plane_per_3_pts</a> &amp;  <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #) is <a href="incsp_1.html#V11" title="INCSP_1:attr.11">with_lines_inside_planes</a> &amp;  <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #) is <a href="incsp_1.html#V12" title="INCSP_1:attr.12">with_planes_intersecting_in_2_pts</a> &amp;  <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #) is <a href="incsp_1.html#V13" title="INCSP_1:attr.13">up-3-dimensional</a> &amp;  <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #) is <a href="incsp_1.html#V14" title="INCSP_1:attr.14">inc-compatible</a> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_4_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">L</font> be   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #); <i><font color="Red">:: according to </font></i><a class="ref" href="incsp_1.html#D8" target="_self" title="INCSP_1:def.8">INCSP_1:def 8</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>, <font color="Olive" title="b2">B</font> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #) st <br/>( <font color="Olive" title="b1">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">B</font> &amp; <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c11">L</font> )</span><br/>

<b>reconsider </b><font color="Maroon" title="c12">l</font> = <font color="Maroon" title="c11">L</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">Li</font> ;<br/>
<a NAME="E2:29_1_1_4_4"/>
<font color="Maroon" title="c12">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Li</font>
 
;<br/>
<b>then consider </b><font color="Maroon" title="c13">a</font>, <font color="Maroon" title="c14">b</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span><b> such that </b><br/><a NAME="E4:29_1_1_4_4"/><i><font color="Green" title="E33">A10</font></i>: 
<font color="Maroon" title="c12">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c13">a</font>,<font color="Maroon" title="c14">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 <b>and </b><br/><a NAME="E5:29_1_1_4_4"/><i><font color="Green" title="E34">A11</font></i>: 
<font color="Maroon" title="c13">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">b</font>
 ;<br/>
<b>reconsider </b><font color="Maroon" title="c15">A</font> = <font color="Maroon" title="c13">a</font>, <font color="Maroon" title="c16">B</font> = <font color="Maroon" title="c14">b</font> as   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #) ;<br/>
<b>take </b>
<font color="Maroon" title="c15">A</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">B</font> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #) st <br/>( <font color="Maroon" title="c15">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">B</font> &amp; <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c15">A</font>,<font color="Olive" title="b1">B</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c11">L</font> )</span><br/>

<b>take </b>
<font color="Maroon" title="c16">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="c15">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c16">B</font> &amp; <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c15">A</font>,<font color="Maroon" title="c16">B</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c11">L</font> )</span><br/>

<b>thus </b><a NAME="E7:29_1_1_4_4"/>
<font color="Maroon" title="c15">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c16">B</font>
 <b>by </b><i><a class="txt" href="#E5:29_1_1_4_4"><i><font color="Green" title="E34">A11</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c15">A</font>,<font color="Maroon" title="c16">B</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c11">L</font></span><br/>

<a NAME="E8:29_1_1_4_4"/>
( <font color="Maroon" title="c13">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">l</font> &amp; <font color="Maroon" title="c14">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">l</font> )
 
<b>by </b><i><a class="txt" href="#E4:29_1_1_4_4"><i><font color="Green" title="E33">A10</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>
<a NAME="E9:29_1_1_4_4"/><b>then </b>
( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c13">a</font>,<font color="Maroon" title="c12">l</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">i1</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c14">b</font>,<font color="Maroon" title="c12">l</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">i1</font> )
 
;<br/>
<a NAME="E10:29_1_1_4_4"/><b>then </b>
( <font color="Maroon" title="c15">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c11">L</font> &amp; <font color="Maroon" title="c16">B</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c11">L</font> )
 
<b>by </b><i><a class="ref" href="incsp_1.html#D1" target="_self" title="INCSP_1:def.1">Def1</a></i>;<br/>
<b>hence </b><a NAME="E11:29_1_1_4_4"/>
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c15">A</font>,<font color="Maroon" title="c16">B</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c11">L</font>
 <b>by </b><i><a class="ref" href="incsp_1.html#T11" target="_self" title="INCSP_1: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>thus </b><a NAME="E5:29_1_1_4"/><i><font color="Green" title="E33">A12</font></i>: 
 <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #) is <a href="incsp_1.html#V6" title="INCSP_1:attr.6">linear</a>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #) is <a href="incsp_1.html#V7" title="INCSP_1:attr.7">up-2-rank</a> &amp;  <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #) is <a href="incsp_1.html#V8" title="INCSP_1:attr.8">with_non-empty_planes</a> &amp;  <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #) is <a href="incsp_1.html#V9" title="INCSP_1:attr.9">planar</a> &amp;  <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #) is <a href="incsp_1.html#V10" title="INCSP_1:attr.10">with_&lt;=1_plane_per_3_pts</a> &amp;  <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #) is <a href="incsp_1.html#V11" title="INCSP_1:attr.11">with_lines_inside_planes</a> &amp;  <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #) is <a href="incsp_1.html#V12" title="INCSP_1:attr.12">with_planes_intersecting_in_2_pts</a> &amp;  <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #) is <a href="incsp_1.html#V13" title="INCSP_1:attr.13">up-3-dimensional</a> &amp;  <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #) is <a href="incsp_1.html#V14" title="INCSP_1:attr.14">inc-compatible</a> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_4_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">A</font>, <font color="Maroon" title="c12">B</font> be   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #); <i><font color="Red">:: according to </font></i><a class="ref" href="incsp_1.html#D9" target="_self" title="INCSP_1:def.9">INCSP_1:def 9</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">L</font> being  <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #) st <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c11">A</font>,<font color="Maroon" title="c12">B</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Olive" title="b1">L</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c13">a</font> = <font color="Maroon" title="c11">A</font>, <font color="Maroon" title="c14">b</font> = <font color="Maroon" title="c12">B</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span> ;<br/>
<div><i><font color="Green" title="E34">A13</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_4_5_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:29_1_1_4_5_1"/>
<font color="Maroon" title="c13">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">b</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">L</font> being  <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #) st <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c11">A</font>,<font color="Maroon" title="c12">B</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Olive" title="b1">L</font></span><br/><a NAME="E2:29_1_1_4_5_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c13">a</font>,<font color="Maroon" title="c14">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Li</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c15">l</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">Li</font><b> such that </b><br/><a NAME="E4:29_1_1_4_5_1"/><i><font color="Green" title="E35">A14</font></i>: 
<font color="Maroon" title="c15">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c13">a</font>,<font color="Maroon" title="c14">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 ;<br/><b>reconsider </b><font color="Maroon" title="c16">L</font> = <font color="Maroon" title="c15">l</font> as   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #) ;<br/><a NAME="E6:29_1_1_4_5_1"/>
( <font color="Maroon" title="c13">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">l</font> &amp; <font color="Maroon" title="c14">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">l</font> )
 <b>by </b><i><a class="txt" href="#E4:29_1_1_4_5_1"><i><font color="Green" title="E35">A14</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><a NAME="E7:29_1_1_4_5_1"/><b>then </b>
( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c13">a</font>,<font color="Maroon" title="c15">l</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">i1</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c14">b</font>,<font color="Maroon" title="c15">l</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">i1</font> )
 ;<br/><a NAME="E8:29_1_1_4_5_1"/><b>then </b>
( <font color="Maroon" title="c11">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c16">L</font> &amp; <font color="Maroon" title="c12">B</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c16">L</font> )
 <b>by </b><i><a class="ref" href="incsp_1.html#D1" target="_self" title="INCSP_1:def.1">Def1</a></i>;<br/><a NAME="E9:29_1_1_4_5_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c11">A</font>,<font color="Maroon" title="c12">B</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c16">L</font>
 <b>by </b><i><a class="ref" href="incsp_1.html#T11" target="_self" title="INCSP_1:th.11">Th11</a></i>;<br/><b>hence </b><a NAME="E10:29_1_1_4_5_1"/>
 ex <font color="Olive" title="b1">L</font> being  <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #) st <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c11">A</font>,<font color="Maroon" title="c12">B</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Olive" title="b1">L</font>
 ; <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="29_1_1_4_5_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:29_1_1_4_5_2"/><i><font color="Green" title="E35">A15</font></i>: 
<font color="Maroon" title="c13">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">b</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">L</font> being  <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #) st <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c11">A</font>,<font color="Maroon" title="c12">B</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Olive" title="b1">L</font></span><br/><a NAME="E2:29_1_1_4_5_2"/>
for <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>  ex <font color="Olive" title="b2">c</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span> st <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">c</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_4_5_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">a</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>; <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">c</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span> st <font color="Maroon" title="c15">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">c</font></span><br/>

<div><i><font color="Green" title="E36">A16</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_4_5_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:29_1_1_4_5_2_1_1"/>
<font color="Maroon" title="c15">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#NK2" title="NUMBERS:NK.2">0</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">c</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span> st <font color="Maroon" title="c15">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">c</font></span><br/><a NAME="E2:29_1_1_4_5_2_1_1"/><b>then </b>
<font color="Maroon" title="c15">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">One</font>
 ;<br/><b>hence </b><a NAME="E3:29_1_1_4_5_2_1_1"/>
 ex <font color="Olive" title="b1">c</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span> st <font color="Maroon" title="c15">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">c</font>
 ; <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="29_1_1_4_5_2_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:29_1_1_4_5_2_1_2"/>
( <font color="Maroon" title="c15">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c15">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 or <font color="Maroon" title="c15">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 )
 ; <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">c</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span> st <font color="Maroon" title="c15">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">c</font></span><br/><a NAME="E2:29_1_1_4_5_2_1_2"/><b>then </b>
<font color="Maroon" title="c15">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">Zero1</font>
 ;<br/><b>hence </b><a NAME="E3:29_1_1_4_5_2_1_2"/>
 ex <font color="Olive" title="b1">c</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span> st <font color="Maroon" title="c15">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">c</font>
 ; <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="E3:29_1_1_4_5_2_1"/>
 ex <font color="Olive" title="b1">c</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span> st <font color="Maroon" title="c15">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">c</font>
 <b>by </b><i><a class="txt" href="#E1:29_1_1_4_5_2_1"><i><font color="Green" title="E36">A16</font></i></a>, <a class="ref" href="enumset1.html#D2" title="ENUMSET1:def.2">ENUMSET1:def 2</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>then consider </b><font color="Maroon" title="c15">c</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span><b> such that </b><br/><a NAME="E4:29_1_1_4_5_2"/><i><font color="Green" title="E36">A17</font></i>: 
<font color="Maroon" title="c13">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">c</font>
 ;<br/><a NAME="E5:29_1_1_4_5_2"/>
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c13">a</font>,<font color="Maroon" title="c15">c</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Li</font>
 <b>by </b><i><a class="txt" href="#E4:29_1_1_4_5_2"><i><font color="Green" title="E36">A17</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c16">l</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">Li</font><b> such that </b><br/><a NAME="E7:29_1_1_4_5_2"/><i><font color="Green" title="E37">A18</font></i>: 
<font color="Maroon" title="c16">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c13">a</font>,<font color="Maroon" title="c15">c</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 ;<br/><b>reconsider </b><font color="Maroon" title="c17">L</font> = <font color="Maroon" title="c16">l</font> as   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #) ;<br/><a NAME="E9:29_1_1_4_5_2"/>
<font color="Maroon" title="c13">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">l</font>
 <b>by </b><i><a class="txt" href="#E7:29_1_1_4_5_2"><i><font color="Green" title="E37">A18</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><a NAME="E10:29_1_1_4_5_2"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c13">a</font>,<font color="Maroon" title="c16">l</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">i1</font>
 ;<br/><a NAME="E11:29_1_1_4_5_2"/><b>then </b>
( <font color="Maroon" title="c11">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c17">L</font> &amp; <font color="Maroon" title="c12">B</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c17">L</font> )
 <b>by </b><i><a class="txt" href="#E1:29_1_1_4_5_2"><i><font color="Green" title="E35">A15</font></i></a>, <a class="ref" href="incsp_1.html#D1" target="_self" title="INCSP_1:def.1">Def1</a></i>;<br/><a NAME="E12:29_1_1_4_5_2"/><b>then </b>
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c11">A</font>,<font color="Maroon" title="c12">B</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c17">L</font>
 <b>by </b><i><a class="ref" href="incsp_1.html#T11" target="_self" title="INCSP_1:th.11">Th11</a></i>;<br/><b>hence </b><a NAME="E13:29_1_1_4_5_2"/>
 ex <font color="Olive" title="b1">L</font> being  <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #) st <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c11">A</font>,<font color="Maroon" title="c12">B</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Olive" title="b1">L</font>
 ; <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="E4:29_1_1_4_5"/>
 ex <font color="Olive" title="b1">L</font> being  <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #) st <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c11">A</font>,<font color="Maroon" title="c12">B</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Olive" title="b1">L</font>
 <b>by </b><i><a class="txt" href="#E2:29_1_1_4_5"><i><font color="Green" title="E34">A13</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>thus </b><a NAME="E6:29_1_1_4"/>
 <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #) is <a href="incsp_1.html#V7" title="INCSP_1:attr.7">up-2-rank</a>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #) is <a href="incsp_1.html#V8" title="INCSP_1:attr.8">with_non-empty_planes</a> &amp;  <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #) is <a href="incsp_1.html#V9" title="INCSP_1:attr.9">planar</a> &amp;  <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #) is <a href="incsp_1.html#V10" title="INCSP_1:attr.10">with_&lt;=1_plane_per_3_pts</a> &amp;  <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #) is <a href="incsp_1.html#V11" title="INCSP_1:attr.11">with_lines_inside_planes</a> &amp;  <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #) is <a href="incsp_1.html#V12" title="INCSP_1:attr.12">with_planes_intersecting_in_2_pts</a> &amp;  <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #) is <a href="incsp_1.html#V13" title="INCSP_1:attr.13">up-3-dimensional</a> &amp;  <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #) is <a href="incsp_1.html#V14" title="INCSP_1:attr.14">inc-compatible</a> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_4_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">A</font>, <font color="Maroon" title="c12">B</font> be   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #); <i><font color="Red">:: according to </font></i><a class="ref" href="incsp_1.html#D10" target="_self" title="INCSP_1:def.10">INCSP_1:def 10</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">K</font>, <font color="Olive" title="b2">L</font> being  <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #)  st <font color="Maroon" title="c11">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">B</font> &amp; <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c11">A</font>,<font color="Maroon" title="c12">B</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Olive" title="b1">K</font> &amp; <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c11">A</font>,<font color="Maroon" title="c12">B</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Olive" title="b2">L</font> holds <br/><font color="Olive" title="b1">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">L</font></span><br/><b>let </b><font color="Maroon" title="c13">K</font>, <font color="Maroon" title="c14">L</font> be   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #); <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c11">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">B</font> &amp; <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c11">A</font>,<font color="Maroon" title="c12">B</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c13">K</font> &amp; <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c11">A</font>,<font color="Maroon" title="c12">B</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c14">L</font> implies <font color="Maroon" title="c13">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">L</font> )</span><br/>



<b>assume </b><b>that </b><br/><a NAME="E1:29_1_1_4_6"/><i><font color="Green" title="E34">A19</font></i>: 
<font color="Maroon" title="c11">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">B</font>
 <b>and </b><br/><a NAME="E2:29_1_1_4_6"/><i><font color="Green" title="E35">A20</font></i>: 
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c11">A</font>,<font color="Maroon" title="c12">B</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c13">K</font>
 <b>and </b><br/><a NAME="E3:29_1_1_4_6"/><i><font color="Green" title="E36">A21</font></i>: 
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c11">A</font>,<font color="Maroon" title="c12">B</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c14">L</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c13">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">L</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c15">k</font> = <font color="Maroon" title="c13">K</font>, <font color="Maroon" title="c16">l</font> = <font color="Maroon" title="c14">L</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">Li</font> ;<br/>
<a NAME="E5:29_1_1_4_6"/><i><font color="Green" title="E37">A22</font></i>: 
( <font color="Maroon" title="c11">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c13">K</font> &amp; <font color="Maroon" title="c12">B</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c13">K</font> )
 
<b>by </b><i><a class="txt" href="#E2:29_1_1_4_6"><i><font color="Green" title="E35">A20</font></i></a>, <a class="ref" href="incsp_1.html#T11" target="_self" title="INCSP_1:th.11">Th11</a></i>;<br/>
<a NAME="E6:29_1_1_4_6"/><i><font color="Green" title="E38">A23</font></i>: 
( <font color="Maroon" title="c11">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c14">L</font> &amp; <font color="Maroon" title="c12">B</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c14">L</font> )
 
<b>by </b><i><a class="txt" href="#E3:29_1_1_4_6"><i><font color="Green" title="E36">A21</font></i></a>, <a class="ref" href="incsp_1.html#T11" target="_self" title="INCSP_1:th.11">Th11</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c17">a</font> = <font color="Maroon" title="c11">A</font>, <font color="Maroon" title="c18">b</font> = <font color="Maroon" title="c12">B</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span> ;<br/>
<a NAME="E8:29_1_1_4_6"/>
( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c17">a</font>,<font color="Maroon" title="c15">k</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">i1</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c18">b</font>,<font color="Maroon" title="c15">k</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">i1</font> )
 
<b>by </b><i><a class="txt" href="#E5:29_1_1_4_6"><i><font color="Green" title="E37">A22</font></i></a>, <a class="ref" href="incsp_1.html#D1" target="_self" title="INCSP_1:def.1">Def1</a></i>;<br/>
<a NAME="E9:29_1_1_4_6"/><b>then </b><i><font color="Green" title="E39">A24</font></i>: 
( <font color="Maroon" title="c17">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">k</font> &amp; <font color="Maroon" title="c18">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">k</font> )
 
<b>by </b><i><a class="txt" href="#E1:29_1_1_4"><i><font color="Green" title="E30">A1</font></i></a></i>;<br/>
<a NAME="E10:29_1_1_4_6"/>
( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c17">a</font>,<font color="Maroon" title="c16">l</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">i1</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c18">b</font>,<font color="Maroon" title="c16">l</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">i1</font> )
 
<b>by </b><i><a class="txt" href="#E6:29_1_1_4_6"><i><font color="Green" title="E38">A23</font></i></a>, <a class="ref" href="incsp_1.html#D1" target="_self" title="INCSP_1:def.1">Def1</a></i>;<br/>
<a NAME="E11:29_1_1_4_6"/><b>then </b><i><font color="Green" title="E40">A25</font></i>: 
( <font color="Maroon" title="c17">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">l</font> &amp; <font color="Maroon" title="c18">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">l</font> )
 
<b>by </b><i><a class="txt" href="#E1:29_1_1_4"><i><font color="Green" title="E30">A1</font></i></a></i>;<br/>
<a NAME="E12:29_1_1_4_6"/>
<font color="Maroon" title="c15">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Li</font>
 
;<br/>
<b>then consider </b><font color="Maroon" title="c19">x1</font>, <font color="Maroon" title="c20">x2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span><b> such that </b><br/><a NAME="E14:29_1_1_4_6"/><i><font color="Green" title="E41">A26</font></i>: 
<font color="Maroon" title="c15">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c19">x1</font>,<font color="Maroon" title="c20">x2</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 <b>and </b><br/><a NAME="E15:29_1_1_4_6"/>
<font color="Maroon" title="c19">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c20">x2</font>
 ;<br/>
<a NAME="E16:29_1_1_4_6"/><i><font color="Green" title="E42">A27</font></i>: 
( ( <font color="Maroon" title="c17">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">x1</font> or <font color="Maroon" title="c17">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">x2</font> ) &amp; ( <font color="Maroon" title="c18">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">x1</font> or <font color="Maroon" title="c18">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">x2</font> ) )
 
<b>by </b><i><a class="txt" href="#E9:29_1_1_4_6"><i><font color="Green" title="E39">A24</font></i></a>, <a class="txt" href="#E14:29_1_1_4_6"><i><font color="Green" title="E41">A26</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>
<a NAME="E17:29_1_1_4_6"/>
<font color="Maroon" title="c16">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Li</font>
 
;<br/>
<b>then consider </b><font color="Maroon" title="c21">x3</font>, <font color="Maroon" title="c22">x4</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span><b> such that </b><br/><a NAME="E19:29_1_1_4_6"/><i><font color="Green" title="E43">A28</font></i>: 
<font color="Maroon" title="c16">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c21">x3</font>,<font color="Maroon" title="c22">x4</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 <b>and </b><br/><a NAME="E20:29_1_1_4_6"/>
<font color="Maroon" title="c21">x3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c22">x4</font>
 ;<br/>
<a NAME="E21:29_1_1_4_6"/>
( ( <font color="Maroon" title="c17">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">x3</font> or <font color="Maroon" title="c17">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">x4</font> ) &amp; ( <font color="Maroon" title="c18">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">x3</font> or <font color="Maroon" title="c18">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">x4</font> ) )
 
<b>by </b><i><a class="txt" href="#E11:29_1_1_4_6"><i><font color="Green" title="E40">A25</font></i></a>, <a class="txt" href="#E19:29_1_1_4_6"><i><font color="Green" title="E43">A28</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>
<b>hence </b><a NAME="E22:29_1_1_4_6"/>
<font color="Maroon" title="c13">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">L</font>
 <b>by </b><i><a class="txt" href="#E1:29_1_1_4_6"><i><font color="Green" title="E34">A19</font></i></a>, <a class="txt" href="#E14:29_1_1_4_6"><i><font color="Green" title="E41">A26</font></i></a>, <a class="txt" href="#E16:29_1_1_4_6"><i><font color="Green" title="E42">A27</font></i></a>, <a class="txt" href="#E19:29_1_1_4_6"><i><font color="Green" title="E43">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>thus </b><a NAME="E7:29_1_1_4"/>
 <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #) is <a href="incsp_1.html#V8" title="INCSP_1:attr.8">with_non-empty_planes</a>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #) is <a href="incsp_1.html#V9" title="INCSP_1:attr.9">planar</a> &amp;  <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #) is <a href="incsp_1.html#V10" title="INCSP_1:attr.10">with_&lt;=1_plane_per_3_pts</a> &amp;  <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #) is <a href="incsp_1.html#V11" title="INCSP_1:attr.11">with_lines_inside_planes</a> &amp;  <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #) is <a href="incsp_1.html#V12" title="INCSP_1:attr.12">with_planes_intersecting_in_2_pts</a> &amp;  <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #) is <a href="incsp_1.html#V13" title="INCSP_1:attr.13">up-3-dimensional</a> &amp;  <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #) is <a href="incsp_1.html#V14" title="INCSP_1:attr.14">inc-compatible</a> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_4_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">P</font> be   <a href="incsp_1.html#NM3" title="INCSP_1:NM.3">PLANE</a> of <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #); <i><font color="Red">:: according to </font></i><a class="ref" href="incsp_1.html#D11" target="_self" title="INCSP_1:def.11">INCSP_1:def 11</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 <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #) st <font color="Olive" title="b1">A</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c11">P</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c12">p</font> = <font color="Maroon" title="c11">P</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">Pl</font> ;<br/>
<a NAME="E2:29_1_1_4_7"/>
<font color="Maroon" title="c12">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Pl</font>
 
;<br/>
<b>then consider </b><font color="Maroon" title="c13">a</font>, <font color="Maroon" title="c14">b</font>, <font color="Maroon" title="c15">c</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span><b> such that </b><br/><a NAME="E4:29_1_1_4_7"/><i><font color="Green" title="E34">A29</font></i>: 
<font color="Maroon" title="c12">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c13">a</font>,<font color="Maroon" title="c14">b</font>,<font color="Maroon" title="c15">c</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span>
 <b>and </b><br/><a NAME="E5:29_1_1_4_7"/>
( <font color="Maroon" title="c13">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">b</font> &amp; <font color="Maroon" title="c13">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">c</font> &amp; <font color="Maroon" title="c14">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">c</font> )
 ;<br/>
<b>reconsider </b><font color="Maroon" title="c16">A</font> = <font color="Maroon" title="c13">a</font> as   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #) ;<br/>
<b>take </b>
<font color="Maroon" title="c16">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="c16">A</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c11">P</font></span><br/>

<a NAME="E7:29_1_1_4_7"/>
<font color="Maroon" title="c13">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">p</font>
 
<b>by </b><i><a class="txt" href="#E4:29_1_1_4_7"><i><font color="Green" title="E34">A29</font></i></a>, <a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></i>;<br/>
<a NAME="E8:29_1_1_4_7"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c13">a</font>,<font color="Maroon" title="c12">p</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">i2</font>
 
;<br/>
<b>hence </b><a NAME="E9:29_1_1_4_7"/>
<font color="Maroon" title="c16">A</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c11">P</font>
 <b>by </b><i><a class="ref" href="incsp_1.html#D2" target="_self" title="INCSP_1:def.2">Def2</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>thus </b><a NAME="E8:29_1_1_4"/>
 <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #) is <a href="incsp_1.html#V9" title="INCSP_1:attr.9">planar</a>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #) is <a href="incsp_1.html#V10" title="INCSP_1:attr.10">with_&lt;=1_plane_per_3_pts</a> &amp;  <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #) is <a href="incsp_1.html#V11" title="INCSP_1:attr.11">with_lines_inside_planes</a> &amp;  <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #) is <a href="incsp_1.html#V12" title="INCSP_1:attr.12">with_planes_intersecting_in_2_pts</a> &amp;  <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #) is <a href="incsp_1.html#V13" title="INCSP_1:attr.13">up-3-dimensional</a> &amp;  <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #) is <a href="incsp_1.html#V14" title="INCSP_1:attr.14">inc-compatible</a> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_4_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">A</font>, <font color="Maroon" title="c12">B</font>, <font color="Maroon" title="c13">C</font> be   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #); <i><font color="Red">:: according to </font></i><a class="ref" href="incsp_1.html#D12" target="_self" title="INCSP_1:def.12">INCSP_1:def 12</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">P</font> being  <a href="incsp_1.html#NM3" title="INCSP_1:NM.3">PLANE</a> of <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #) st <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c11">A</font>,<font color="Maroon" title="c12">B</font>,<font color="Maroon" title="c13">C</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R5" title="INCSP_1:pred.5">on</a> <font color="Olive" title="b1">P</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c14">a</font> = <font color="Maroon" title="c11">A</font>, <font color="Maroon" title="c15">b</font> = <font color="Maroon" title="c12">B</font>, <font color="Maroon" title="c16">c</font> = <font color="Maroon" title="c13">C</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span> ;<br/>
<div><i><font color="Green" title="E34">A30</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_4_8_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:29_1_1_4_8_1"/>
( <font color="Maroon" title="c14">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">b</font> &amp; <font color="Maroon" title="c14">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c16">c</font> &amp; <font color="Maroon" title="c15">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c16">c</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">P</font> being  <a href="incsp_1.html#NM3" title="INCSP_1:NM.3">PLANE</a> of <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #) st <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c11">A</font>,<font color="Maroon" title="c12">B</font>,<font color="Maroon" title="c13">C</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R5" title="INCSP_1:pred.5">on</a> <font color="Olive" title="b1">P</font></span><br/><a NAME="E2:29_1_1_4_8_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c14">a</font>,<font color="Maroon" title="c15">b</font>,<font color="Maroon" title="c16">c</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Pl</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c17">p</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">Pl</font><b> such that </b><br/><a NAME="E4:29_1_1_4_8_1"/><i><font color="Green" title="E35">A31</font></i>: 
<font color="Maroon" title="c17">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c14">a</font>,<font color="Maroon" title="c15">b</font>,<font color="Maroon" title="c16">c</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span>
 ;<br/><b>reconsider </b><font color="Maroon" title="c18">P</font> = <font color="Maroon" title="c17">p</font> as   <a href="incsp_1.html#NM3" title="INCSP_1:NM.3">PLANE</a> of <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #) ;<br/><a NAME="E6:29_1_1_4_8_1"/>
( <font color="Maroon" title="c14">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">p</font> &amp; <font color="Maroon" title="c15">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">p</font> &amp; <font color="Maroon" title="c16">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">p</font> )
 <b>by </b><i><a class="txt" href="#E4:29_1_1_4_8_1"><i><font color="Green" title="E35">A31</font></i></a>, <a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></i>;<br/><a NAME="E7:29_1_1_4_8_1"/><b>then </b>
( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c14">a</font>,<font color="Maroon" title="c17">p</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">i2</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c15">b</font>,<font color="Maroon" title="c17">p</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">i2</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c16">c</font>,<font color="Maroon" title="c17">p</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">i2</font> )
 ;<br/><a NAME="E8:29_1_1_4_8_1"/><b>then </b>
( <font color="Maroon" title="c11">A</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c18">P</font> &amp; <font color="Maroon" title="c12">B</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c18">P</font> &amp; <font color="Maroon" title="c13">C</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c18">P</font> )
 <b>by </b><i><a class="ref" href="incsp_1.html#D2" target="_self" title="INCSP_1:def.2">Def2</a></i>;<br/><a NAME="E9:29_1_1_4_8_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c11">A</font>,<font color="Maroon" title="c12">B</font>,<font color="Maroon" title="c13">C</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R5" title="INCSP_1:pred.5">on</a> <font color="Maroon" title="c18">P</font>
 <b>by </b><i><a class="ref" href="incsp_1.html#T14" target="_self" title="INCSP_1:th.14">Th14</a></i>;<br/><b>hence </b><a NAME="E10:29_1_1_4_8_1"/>
 ex <font color="Olive" title="b1">P</font> being  <a href="incsp_1.html#NM3" title="INCSP_1:NM.3">PLANE</a> of <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #) st <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c11">A</font>,<font color="Maroon" title="c12">B</font>,<font color="Maroon" title="c13">C</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R5" title="INCSP_1:pred.5">on</a> <font color="Olive" title="b1">P</font>
 ; <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><i><font color="Green" title="E35">A32</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_4_8_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:29_1_1_4_8_2"/><i><font color="Green" title="E36">A33</font></i>: 
( <font color="Maroon" title="c14">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">b</font> &amp; <font color="Maroon" title="c14">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">c</font> &amp; <font color="Maroon" title="c15">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">c</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">P</font> being  <a href="incsp_1.html#NM3" title="INCSP_1:NM.3">PLANE</a> of <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #) st <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c11">A</font>,<font color="Maroon" title="c12">B</font>,<font color="Maroon" title="c13">C</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R5" title="INCSP_1:pred.5">on</a> <font color="Olive" title="b1">P</font></span><br/><a NAME="E2:29_1_1_4_8_2"/>
for <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>  ex <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span> st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">b</font> &amp; <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">c</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">c</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_4_8_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c17">a</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>; <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">b</font>, <font color="Olive" title="b2">c</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span> st <br/>( <font color="Maroon" title="c17">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">b</font> &amp; <font color="Maroon" title="c17">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">c</font> &amp; <font color="Olive" title="b1">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">c</font> )</span><br/>

<div><i><font color="Green" title="E37">A34</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_4_8_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:29_1_1_4_8_2_1_1"/>
( <font color="Maroon" title="c17">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a>  or <font color="Maroon" title="c17">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 ; <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">b</font>, <font color="Olive" title="b2">c</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span> st <br/>( <font color="Maroon" title="c17">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">b</font> &amp; <font color="Maroon" title="c17">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">c</font> &amp; <font color="Olive" title="b1">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">c</font> )</span><br/><a NAME="E2:29_1_1_4_8_2_1_1"/><b>then </b>
( <font color="Maroon" title="c17">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">Two</font> &amp; <font color="Maroon" title="c17">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">Three</font> &amp; <font color="Maroon" title="c3">Two</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">Three</font> )
 ;<br/><b>hence </b><a NAME="E3:29_1_1_4_8_2_1_1"/>
 ex <font color="Olive" title="b1">b</font>, <font color="Olive" title="b2">c</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span> st <br/>( <font color="Maroon" title="c17">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">b</font> &amp; <font color="Maroon" title="c17">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">c</font> &amp; <font color="Olive" title="b1">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">c</font> )
 ; <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="29_1_1_4_8_2_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:29_1_1_4_8_2_1_2"/>
( <font color="Maroon" title="c17">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 or <font color="Maroon" title="c17">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 )
 ; <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">b</font>, <font color="Olive" title="b2">c</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span> st <br/>( <font color="Maroon" title="c17">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">b</font> &amp; <font color="Maroon" title="c17">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">c</font> &amp; <font color="Olive" title="b1">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">c</font> )</span><br/><a NAME="E2:29_1_1_4_8_2_1_2"/><b>then </b>
( <font color="Maroon" title="c17">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">Zero1</font> &amp; <font color="Maroon" title="c17">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">One</font> &amp; <font color="Maroon" title="c1">Zero1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">One</font> )
 ;<br/><b>hence </b><a NAME="E3:29_1_1_4_8_2_1_2"/>
 ex <font color="Olive" title="b1">b</font>, <font color="Olive" title="b2">c</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span> st <br/>( <font color="Maroon" title="c17">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">b</font> &amp; <font color="Maroon" title="c17">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">c</font> &amp; <font color="Olive" title="b1">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">c</font> )
 ; <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="E3:29_1_1_4_8_2_1"/>
 ex <font color="Olive" title="b1">b</font>, <font color="Olive" title="b2">c</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span> st <br/>( <font color="Maroon" title="c17">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">b</font> &amp; <font color="Maroon" title="c17">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">c</font> &amp; <font color="Olive" title="b1">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">c</font> )
 <b>by </b><i><a class="txt" href="#E1:29_1_1_4_8_2_1"><i><font color="Green" title="E37">A34</font></i></a>, <a class="ref" href="enumset1.html#D2" title="ENUMSET1:def.2">ENUMSET1:def 2</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>then consider </b><font color="Maroon" title="c17">x</font>, <font color="Maroon" title="c18">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span><b> such that </b><br/><a NAME="E4:29_1_1_4_8_2"/><i><font color="Green" title="E37">A35</font></i>: 
( <font color="Maroon" title="c14">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c17">x</font> &amp; <font color="Maroon" title="c14">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c18">y</font> &amp; <font color="Maroon" title="c17">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c18">y</font> )
 ;<br/><a NAME="E5:29_1_1_4_8_2"/>
<span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c14">a</font>,<font color="Maroon" title="c17">x</font>,<font color="Maroon" title="c18">y</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Pl</font>
 <b>by </b><i><a class="txt" href="#E4:29_1_1_4_8_2"><i><font color="Green" title="E37">A35</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c19">p</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">Pl</font><b> such that </b><br/><a NAME="E7:29_1_1_4_8_2"/><i><font color="Green" title="E38">A36</font></i>: 
<font color="Maroon" title="c19">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c14">a</font>,<font color="Maroon" title="c17">x</font>,<font color="Maroon" title="c18">y</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span>
 ;<br/><b>reconsider </b><font color="Maroon" title="c20">P</font> = <font color="Maroon" title="c19">p</font> as   <a href="incsp_1.html#NM3" title="INCSP_1:NM.3">PLANE</a> of <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #) ;<br/><a NAME="E9:29_1_1_4_8_2"/>
<font color="Maroon" title="c14">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">p</font>
 <b>by </b><i><a class="txt" href="#E7:29_1_1_4_8_2"><i><font color="Green" title="E38">A36</font></i></a>, <a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></i>;<br/><a NAME="E10:29_1_1_4_8_2"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c14">a</font>,<font color="Maroon" title="c19">p</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">i2</font>
 ;<br/><a NAME="E11:29_1_1_4_8_2"/><b>then </b>
<font color="Maroon" title="c11">A</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c20">P</font>
 <b>by </b><i><a class="ref" href="incsp_1.html#D2" target="_self" title="INCSP_1:def.2">Def2</a></i>;<br/><a NAME="E12:29_1_1_4_8_2"/><b>then </b>
<span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c11">A</font>,<font color="Maroon" title="c12">B</font>,<font color="Maroon" title="c13">C</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R5" title="INCSP_1:pred.5">on</a> <font color="Maroon" title="c20">P</font>
 <b>by </b><i><a class="txt" href="#E1:29_1_1_4_8_2"><i><font color="Green" title="E36">A33</font></i></a>, <a class="ref" href="incsp_1.html#T14" target="_self" title="INCSP_1:th.14">Th14</a></i>;<br/><b>hence </b><a NAME="E13:29_1_1_4_8_2"/>
 ex <font color="Olive" title="b1">P</font> being  <a href="incsp_1.html#NM3" title="INCSP_1:NM.3">PLANE</a> of <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #) st <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c11">A</font>,<font color="Maroon" title="c12">B</font>,<font color="Maroon" title="c13">C</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R5" title="INCSP_1:pred.5">on</a> <font color="Olive" title="b1">P</font>
 ; <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="29_1_1_4_8_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:29_1_1_4_8_3"/><i><font color="Green" title="E36">A37</font></i>: 
( ( <font color="Maroon" title="c14">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">b</font> &amp; <font color="Maroon" title="c14">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c16">c</font> ) or ( <font color="Maroon" title="c14">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">c</font> &amp; <font color="Maroon" title="c14">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">b</font> ) or ( <font color="Maroon" title="c15">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">c</font> &amp; <font color="Maroon" title="c14">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">b</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">P</font> being  <a href="incsp_1.html#NM3" title="INCSP_1:NM.3">PLANE</a> of <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #) st <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c11">A</font>,<font color="Maroon" title="c12">B</font>,<font color="Maroon" title="c13">C</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R5" title="INCSP_1:pred.5">on</a> <font color="Olive" title="b1">P</font></span><br/><b>then consider </b><font color="Maroon" title="c17">x</font>, <font color="Maroon" title="c18">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span><b> such that </b><br/><a NAME="E3:29_1_1_4_8_3"/><i><font color="Green" title="E37">A38</font></i>: 
( ( <font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">a</font> or <font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">b</font> or <font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">c</font> ) &amp; ( <font color="Maroon" title="c18">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">a</font> or <font color="Maroon" title="c18">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">b</font> or <font color="Maroon" title="c18">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">c</font> ) )
 <b>and </b><br/><a NAME="E4:29_1_1_4_8_3"/><i><font color="Green" title="E38">A39</font></i>: 
<font color="Maroon" title="c17">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c18">y</font>
 ;<br/><a NAME="E5:29_1_1_4_8_3"/>
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>  ex <font color="Olive" title="b3">c</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span> st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">c</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">c</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_4_8_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c19">a</font>, <font color="Maroon" title="c20">b</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>; <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">c</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span> st <br/>( <font color="Maroon" title="c19">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">c</font> &amp; <font color="Maroon" title="c20">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">c</font> )</span><br/>



<div><i><font color="Green" title="E39">A40</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_4_8_3_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:29_1_1_4_8_3_1_1"/>
( <font color="Maroon" title="c19">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a>  &amp; ( <font color="Maroon" title="c20">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a>  or <font color="Maroon" title="c20">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c20">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 ) )
 ; <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">c</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span> st <br/>( <font color="Maroon" title="c19">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">c</font> &amp; <font color="Maroon" title="c20">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">c</font> )</span><br/><a NAME="E2:29_1_1_4_8_3_1_1"/><b>then </b>
( <font color="Maroon" title="c19">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">Three</font> &amp; <font color="Maroon" title="c20">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">Three</font> )
 ;<br/><b>hence </b><a NAME="E3:29_1_1_4_8_3_1_1"/>
 ex <font color="Olive" title="b1">c</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span> st <br/>( <font color="Maroon" title="c19">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">c</font> &amp; <font color="Maroon" title="c20">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">c</font> )
 ; <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><i><font color="Green" title="E40">A41</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_4_8_3_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:29_1_1_4_8_3_1_2"/>
( <font color="Maroon" title="c19">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a>  &amp; <font color="Maroon" title="c20">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 )
 ; <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">c</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span> st <br/>( <font color="Maroon" title="c19">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">c</font> &amp; <font color="Maroon" title="c20">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">c</font> )</span><br/><a NAME="E2:29_1_1_4_8_3_1_2"/><b>then </b>
( <font color="Maroon" title="c19">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">One</font> &amp; <font color="Maroon" title="c20">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">One</font> )
 ;<br/><b>hence </b><a NAME="E3:29_1_1_4_8_3_1_2"/>
 ex <font color="Olive" title="b1">c</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span> st <br/>( <font color="Maroon" title="c19">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">c</font> &amp; <font color="Maroon" title="c20">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">c</font> )
 ; <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><i><font color="Green" title="E41">A42</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_4_8_3_1_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:29_1_1_4_8_3_1_3"/>
( ( <font color="Maroon" title="c19">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c19">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 or <font color="Maroon" title="c19">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 ) &amp; ( <font color="Maroon" title="c20">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c20">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 or <font color="Maroon" title="c20">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 ) )
 ; <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">c</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span> st <br/>( <font color="Maroon" title="c19">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">c</font> &amp; <font color="Maroon" title="c20">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">c</font> )</span><br/><a NAME="E2:29_1_1_4_8_3_1_3"/><b>then </b>
( <font color="Maroon" title="c19">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">Zero1</font> &amp; <font color="Maroon" title="c20">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">Zero1</font> )
 ;<br/><b>hence </b><a NAME="E3:29_1_1_4_8_3_1_3"/>
 ex <font color="Olive" title="b1">c</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span> st <br/>( <font color="Maroon" title="c19">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">c</font> &amp; <font color="Maroon" title="c20">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">c</font> )
 ; <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><i><font color="Green" title="E42">A43</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_4_8_3_1_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:29_1_1_4_8_3_1_4"/>
( ( <font color="Maroon" title="c19">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c19">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 ) &amp; <font color="Maroon" title="c20">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#NK2" title="NUMBERS:NK.2">0</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">c</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span> st <br/>( <font color="Maroon" title="c19">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">c</font> &amp; <font color="Maroon" title="c20">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">c</font> )</span><br/><a NAME="E2:29_1_1_4_8_3_1_4"/><b>then </b>
( <font color="Maroon" title="c19">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">Three</font> &amp; <font color="Maroon" title="c20">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">Three</font> )
 ;<br/><b>hence </b><a NAME="E3:29_1_1_4_8_3_1_4"/>
 ex <font color="Olive" title="b1">c</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span> st <br/>( <font color="Maroon" title="c19">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">c</font> &amp; <font color="Maroon" title="c20">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">c</font> )
 ; <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="29_1_1_4_8_3_1_5"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:29_1_1_4_8_3_1_5"/>
( <font color="Maroon" title="c19">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 &amp; <font color="Maroon" title="c20">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#NK2" title="NUMBERS:NK.2">0</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">c</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span> st <br/>( <font color="Maroon" title="c19">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">c</font> &amp; <font color="Maroon" title="c20">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">c</font> )</span><br/><a NAME="E2:29_1_1_4_8_3_1_5"/><b>then </b>
( <font color="Maroon" title="c19">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">Two</font> &amp; <font color="Maroon" title="c20">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">Two</font> )
 ;<br/><b>hence </b><a NAME="E3:29_1_1_4_8_3_1_5"/>
 ex <font color="Olive" title="b1">c</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span> st <br/>( <font color="Maroon" title="c19">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">c</font> &amp; <font color="Maroon" title="c20">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">c</font> )
 ; <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="E6:29_1_1_4_8_3_1"/>
 ex <font color="Olive" title="b1">c</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span> st <br/>( <font color="Maroon" title="c19">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">c</font> &amp; <font color="Maroon" title="c20">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">c</font> )
 <b>by </b><i><a class="txt" href="#E1:29_1_1_4_8_3_1"><i><font color="Green" title="E39">A40</font></i></a>, <a class="txt" href="#E2:29_1_1_4_8_3_1"><i><font color="Green" title="E40">A41</font></i></a>, <a class="txt" href="#E3:29_1_1_4_8_3_1"><i><font color="Green" title="E41">A42</font></i></a>, <a class="txt" href="#E4:29_1_1_4_8_3_1"><i><font color="Green" title="E42">A43</font></i></a>, <a class="ref" href="enumset1.html#D2" title="ENUMSET1:def.2">ENUMSET1:def 2</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>then consider </b><font color="Maroon" title="c19">z</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span><b> such that </b><br/><a NAME="E7:29_1_1_4_8_3"/><i><font color="Green" title="E39">A44</font></i>: 
( <font color="Maroon" title="c17">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c19">z</font> &amp; <font color="Maroon" title="c18">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c19">z</font> )
 ;<br/><a NAME="E8:29_1_1_4_8_3"/>
<span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c17">x</font>,<font color="Maroon" title="c18">y</font>,<font color="Maroon" title="c19">z</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Pl</font>
 <b>by </b><i><a class="txt" href="#E4:29_1_1_4_8_3"><i><font color="Green" title="E38">A39</font></i></a>, <a class="txt" href="#E7:29_1_1_4_8_3"><i><font color="Green" title="E39">A44</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c20">p</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">Pl</font><b> such that </b><br/><a NAME="E10:29_1_1_4_8_3"/><i><font color="Green" title="E40">A45</font></i>: 
<font color="Maroon" title="c20">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c17">x</font>,<font color="Maroon" title="c18">y</font>,<font color="Maroon" title="c19">z</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span>
 ;<br/><b>reconsider </b><font color="Maroon" title="c21">P</font> = <font color="Maroon" title="c20">p</font> as   <a href="incsp_1.html#NM3" title="INCSP_1:NM.3">PLANE</a> of <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #) ;<br/><a NAME="E12:29_1_1_4_8_3"/>
( <font color="Maroon" title="c14">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">p</font> &amp; <font color="Maroon" title="c15">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">p</font> &amp; <font color="Maroon" title="c16">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">p</font> )
 <b>by </b><i><a class="txt" href="#E1:29_1_1_4_8_3"><i><font color="Green" title="E36">A37</font></i></a>, <a class="txt" href="#E3:29_1_1_4_8_3"><i><font color="Green" title="E37">A38</font></i></a>, <a class="txt" href="#E4:29_1_1_4_8_3"><i><font color="Green" title="E38">A39</font></i></a>, <a class="txt" href="#E10:29_1_1_4_8_3"><i><font color="Green" title="E40">A45</font></i></a>, <a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></i>;<br/><a NAME="E13:29_1_1_4_8_3"/><b>then </b>
( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c14">a</font>,<font color="Maroon" title="c20">p</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">i2</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c15">b</font>,<font color="Maroon" title="c20">p</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">i2</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c16">c</font>,<font color="Maroon" title="c20">p</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">i2</font> )
 ;<br/><a NAME="E14:29_1_1_4_8_3"/><b>then </b>
( <font color="Maroon" title="c11">A</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c21">P</font> &amp; <font color="Maroon" title="c12">B</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c21">P</font> &amp; <font color="Maroon" title="c13">C</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c21">P</font> )
 <b>by </b><i><a class="ref" href="incsp_1.html#D2" target="_self" title="INCSP_1:def.2">Def2</a></i>;<br/><a NAME="E15:29_1_1_4_8_3"/><b>then </b>
<span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c11">A</font>,<font color="Maroon" title="c12">B</font>,<font color="Maroon" title="c13">C</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R5" title="INCSP_1:pred.5">on</a> <font color="Maroon" title="c21">P</font>
 <b>by </b><i><a class="ref" href="incsp_1.html#T14" target="_self" title="INCSP_1:th.14">Th14</a></i>;<br/><b>hence </b><a NAME="E16:29_1_1_4_8_3"/>
 ex <font color="Olive" title="b1">P</font> being  <a href="incsp_1.html#NM3" title="INCSP_1:NM.3">PLANE</a> of <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #) st <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c11">A</font>,<font color="Maroon" title="c12">B</font>,<font color="Maroon" title="c13">C</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R5" title="INCSP_1:pred.5">on</a> <font color="Olive" title="b1">P</font>
 ; <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="E5:29_1_1_4_8"/>
 ex <font color="Olive" title="b1">P</font> being  <a href="incsp_1.html#NM3" title="INCSP_1:NM.3">PLANE</a> of <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #) st <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c11">A</font>,<font color="Maroon" title="c12">B</font>,<font color="Maroon" title="c13">C</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R5" title="INCSP_1:pred.5">on</a> <font color="Olive" title="b1">P</font>
 <b>by </b><i><a class="txt" href="#E2:29_1_1_4_8"><i><font color="Green" title="E34">A30</font></i></a>, <a class="txt" href="#E3:29_1_1_4_8"><i><font color="Green" title="E35">A32</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>thus </b><a NAME="E9:29_1_1_4"/>
 <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #) is <a href="incsp_1.html#V10" title="INCSP_1:attr.10">with_&lt;=1_plane_per_3_pts</a>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #) is <a href="incsp_1.html#V11" title="INCSP_1:attr.11">with_lines_inside_planes</a> &amp;  <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #) is <a href="incsp_1.html#V12" title="INCSP_1:attr.12">with_planes_intersecting_in_2_pts</a> &amp;  <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #) is <a href="incsp_1.html#V13" title="INCSP_1:attr.13">up-3-dimensional</a> &amp;  <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #) is <a href="incsp_1.html#V14" title="INCSP_1:attr.14">inc-compatible</a> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_4_9"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">A</font>, <font color="Maroon" title="c12">B</font>, <font color="Maroon" title="c13">C</font> be   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #); <i><font color="Red">:: according to </font></i><a class="ref" href="incsp_1.html#D13" target="_self" title="INCSP_1:def.13">INCSP_1:def 13</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">P</font>, <font color="Olive" title="b2">Q</font> being  <a href="incsp_1.html#NM3" title="INCSP_1:NM.3">PLANE</a> of <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #)  st not <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c11">A</font>,<font color="Maroon" title="c12">B</font>,<font color="Maroon" title="c13">C</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> is <a href="incsp_1.html#V3" title="INCSP_1:attr.3">linear</a> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c11">A</font>,<font color="Maroon" title="c12">B</font>,<font color="Maroon" title="c13">C</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R5" title="INCSP_1:pred.5">on</a> <font color="Olive" title="b1">P</font> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c11">A</font>,<font color="Maroon" title="c12">B</font>,<font color="Maroon" title="c13">C</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R5" title="INCSP_1:pred.5">on</a> <font color="Olive" title="b2">Q</font> holds <br/><font color="Olive" title="b1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">Q</font></span><br/><b>let </b><font color="Maroon" title="c14">P</font>, <font color="Maroon" title="c15">Q</font> be   <a href="incsp_1.html#NM3" title="INCSP_1:NM.3">PLANE</a> of <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #); <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( not <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c11">A</font>,<font color="Maroon" title="c12">B</font>,<font color="Maroon" title="c13">C</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> is <a href="incsp_1.html#V3" title="INCSP_1:attr.3">linear</a> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c11">A</font>,<font color="Maroon" title="c12">B</font>,<font color="Maroon" title="c13">C</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R5" title="INCSP_1:pred.5">on</a> <font color="Maroon" title="c14">P</font> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c11">A</font>,<font color="Maroon" title="c12">B</font>,<font color="Maroon" title="c13">C</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R5" title="INCSP_1:pred.5">on</a> <font color="Maroon" title="c15">Q</font> implies <font color="Maroon" title="c14">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">Q</font> )</span><br/>



<b>assume </b><b>that </b><br/><a NAME="E1:29_1_1_4_9"/><i><font color="Green" title="E34">A46</font></i>: 
not <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c11">A</font>,<font color="Maroon" title="c12">B</font>,<font color="Maroon" title="c13">C</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> is <a href="incsp_1.html#V3" title="INCSP_1:attr.3">linear</a>
 <b>and </b><br/><a NAME="E2:29_1_1_4_9"/><i><font color="Green" title="E35">A47</font></i>: 
<span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c11">A</font>,<font color="Maroon" title="c12">B</font>,<font color="Maroon" title="c13">C</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R5" title="INCSP_1:pred.5">on</a> <font color="Maroon" title="c14">P</font>
 <b>and </b><br/><a NAME="E3:29_1_1_4_9"/><i><font color="Green" title="E36">A48</font></i>: 
<span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c11">A</font>,<font color="Maroon" title="c12">B</font>,<font color="Maroon" title="c13">C</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R5" title="INCSP_1:pred.5">on</a> <font color="Maroon" title="c15">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="c14">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">Q</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c16">p</font> = <font color="Maroon" title="c14">P</font>, <font color="Maroon" title="c17">q</font> = <font color="Maroon" title="c15">Q</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">Pl</font> ;<br/>
<div><i><font color="Green" title="E37">A49</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_4_9_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:29_1_1_4_9_1"/><i><font color="Green" title="E38">A50</font></i>: 
( <font color="Maroon" title="c11">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">B</font> or <font color="Maroon" title="c11">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">C</font> or <font color="Maroon" title="c12">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">C</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><b>consider </b><font color="Maroon" title="c18">K</font> being   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #)<b> such that </b><br/><a NAME="E3:29_1_1_4_9_1"/><i><font color="Green" title="E39">A51</font></i>: 
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c11">A</font>,<font color="Maroon" title="c12">B</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c18">K</font>
 <b>by </b><i><a class="txt" href="#E5:29_1_1_4"><i><font color="Green" title="E33">A12</font></i></a>, <a class="ref" href="incsp_1.html#D9" target="_self" title="INCSP_1:def.9">Def9</a></i>;<br/><b>consider </b><font color="Maroon" title="c19">L</font> being   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #)<b> such that </b><br/><a NAME="E5:29_1_1_4_9_1"/><i><font color="Green" title="E40">A52</font></i>: 
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c11">A</font>,<font color="Maroon" title="c13">C</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c19">L</font>
 <b>by </b><i><a class="txt" href="#E5:29_1_1_4"><i><font color="Green" title="E33">A12</font></i></a>, <a class="ref" href="incsp_1.html#D9" target="_self" title="INCSP_1:def.9">Def9</a></i>;<br/><a NAME="E6:29_1_1_4_9_1"/><i><font color="Green" title="E41">A53</font></i>: 
( <font color="Maroon" title="c11">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c18">K</font> &amp; <font color="Maroon" title="c12">B</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c18">K</font> &amp; <font color="Maroon" title="c11">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c19">L</font> &amp; <font color="Maroon" title="c13">C</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c19">L</font> )
 <b>by </b><i><a class="txt" href="#E3:29_1_1_4_9_1"><i><font color="Green" title="E39">A51</font></i></a>, <a class="txt" href="#E5:29_1_1_4_9_1"><i><font color="Green" title="E40">A52</font></i></a>, <a class="ref" href="incsp_1.html#T11" target="_self" title="INCSP_1:th.11">Th11</a></i>;<br/><a NAME="E7:29_1_1_4_9_1"/>
( not <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c11">A</font>,<font color="Maroon" title="c12">B</font>,<font color="Maroon" title="c13">C</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c18">K</font> &amp; not <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c11">A</font>,<font color="Maroon" title="c12">B</font>,<font color="Maroon" title="c13">C</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c19">L</font> )
 <b>by </b><i><a class="txt" href="#E1:29_1_1_4_9"><i><font color="Green" title="E34">A46</font></i></a>, <a class="ref" href="incsp_1.html#D6" target="_self" title="INCSP_1:def.6">Def6</a></i>;<br/><b>hence </b><a NAME="E8:29_1_1_4_9_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:29_1_1_4_9_1"><i><font color="Green" title="E38">A50</font></i></a>, <a class="txt" href="#E6:29_1_1_4_9_1"><i><font color="Green" title="E41">A53</font></i></a>, <a class="ref" href="incsp_1.html#T12" target="_self" title="INCSP_1:th.12">Th12</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="E6:29_1_1_4_9"/><i><font color="Green" title="E38">A54</font></i>: 
( <font color="Maroon" title="c11">A</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c14">P</font> &amp; <font color="Maroon" title="c12">B</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c14">P</font> &amp; <font color="Maroon" title="c13">C</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c14">P</font> )
 
<b>by </b><i><a class="txt" href="#E2:29_1_1_4_9"><i><font color="Green" title="E35">A47</font></i></a>, <a class="ref" href="incsp_1.html#T14" target="_self" title="INCSP_1:th.14">Th14</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c18">a</font> = <font color="Maroon" title="c11">A</font>, <font color="Maroon" title="c19">b</font> = <font color="Maroon" title="c12">B</font>, <font color="Maroon" title="c20">c</font> = <font color="Maroon" title="c13">C</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span> ;<br/>
<a NAME="E8:29_1_1_4_9"/>
( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c18">a</font>,<font color="Maroon" title="c16">p</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">i2</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c19">b</font>,<font color="Maroon" title="c16">p</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">i2</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c20">c</font>,<font color="Maroon" title="c16">p</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">i2</font> )
 
<b>by </b><i><a class="txt" href="#E6:29_1_1_4_9"><i><font color="Green" title="E38">A54</font></i></a>, <a class="ref" href="incsp_1.html#D2" target="_self" title="INCSP_1:def.2">Def2</a></i>;<br/>
<a NAME="E9:29_1_1_4_9"/><b>then </b><i><font color="Green" title="E39">A55</font></i>: 
( <font color="Maroon" title="c18">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">p</font> &amp; <font color="Maroon" title="c19">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">p</font> &amp; <font color="Maroon" title="c20">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">p</font> )
 
<b>by </b><i><a class="txt" href="#E2:29_1_1_4"><i><font color="Green" title="E31">A4</font></i></a></i>;<br/>
<a NAME="E10:29_1_1_4_9"/>
<font color="Maroon" title="c16">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Pl</font>
 
;<br/>
<b>then consider </b><font color="Maroon" title="c21">x1</font>, <font color="Maroon" title="c22">x2</font>, <font color="Maroon" title="c23">x3</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span><b> such that </b><br/><a NAME="E12:29_1_1_4_9"/><i><font color="Green" title="E40">A56</font></i>: 
<font color="Maroon" title="c16">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c21">x1</font>,<font color="Maroon" title="c22">x2</font>,<font color="Maroon" title="c23">x3</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span>
 <b>and </b><br/><a NAME="E13:29_1_1_4_9"/>
( <font color="Maroon" title="c21">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c22">x2</font> &amp; <font color="Maroon" title="c21">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c23">x3</font> &amp; <font color="Maroon" title="c22">x2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c23">x3</font> )
 ;<br/>
<a NAME="E14:29_1_1_4_9"/><i><font color="Green" title="E41">A57</font></i>: 
( ( <font color="Maroon" title="c18">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">x1</font> or <font color="Maroon" title="c18">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">x2</font> or <font color="Maroon" title="c18">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">x3</font> ) &amp; ( <font color="Maroon" title="c19">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">x1</font> or <font color="Maroon" title="c19">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">x2</font> or <font color="Maroon" title="c19">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">x3</font> ) &amp; ( <font color="Maroon" title="c20">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">x1</font> or <font color="Maroon" title="c20">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">x2</font> or <font color="Maroon" title="c20">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">x3</font> ) )
 
<b>by </b><i><a class="txt" href="#E9:29_1_1_4_9"><i><font color="Green" title="E39">A55</font></i></a>, <a class="txt" href="#E12:29_1_1_4_9"><i><font color="Green" title="E40">A56</font></i></a>, <a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></i>;<br/>
<a NAME="E15:29_1_1_4_9"/>
( <font color="Maroon" title="c11">A</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c15">Q</font> &amp; <font color="Maroon" title="c12">B</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c15">Q</font> &amp; <font color="Maroon" title="c13">C</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c15">Q</font> )
 
<b>by </b><i><a class="txt" href="#E3:29_1_1_4_9"><i><font color="Green" title="E36">A48</font></i></a>, <a class="ref" href="incsp_1.html#T14" target="_self" title="INCSP_1:th.14">Th14</a></i>;<br/>
<a NAME="E16:29_1_1_4_9"/><b>then </b>
( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c18">a</font>,<font color="Maroon" title="c17">q</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">i2</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c19">b</font>,<font color="Maroon" title="c17">q</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">i2</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c20">c</font>,<font color="Maroon" title="c17">q</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">i2</font> )
 
<b>by </b><i><a class="ref" href="incsp_1.html#D2" target="_self" title="INCSP_1:def.2">Def2</a></i>;<br/>
<a NAME="E17:29_1_1_4_9"/><b>then </b><i><font color="Green" title="E42">A58</font></i>: 
( <font color="Maroon" title="c18">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">q</font> &amp; <font color="Maroon" title="c19">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">q</font> &amp; <font color="Maroon" title="c20">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">q</font> )
 
<b>by </b><i><a class="txt" href="#E2:29_1_1_4"><i><font color="Green" title="E31">A4</font></i></a></i>;<br/>
<a NAME="E18:29_1_1_4_9"/>
<font color="Maroon" title="c17">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Pl</font>
 
;<br/>
<b>then consider </b><font color="Maroon" title="c24">x1</font>, <font color="Maroon" title="c25">x2</font>, <font color="Maroon" title="c26">x3</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span><b> such that </b><br/><a NAME="E20:29_1_1_4_9"/><i><font color="Green" title="E43">A59</font></i>: 
<font color="Maroon" title="c17">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c24">x1</font>,<font color="Maroon" title="c25">x2</font>,<font color="Maroon" title="c26">x3</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span>
 <b>and </b><br/><a NAME="E21:29_1_1_4_9"/>
( <font color="Maroon" title="c24">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c25">x2</font> &amp; <font color="Maroon" title="c24">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c26">x3</font> &amp; <font color="Maroon" title="c25">x2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c26">x3</font> )
 ;<br/>
<a NAME="E22:29_1_1_4_9"/>
( ( <font color="Maroon" title="c18">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">x1</font> or <font color="Maroon" title="c18">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">x2</font> or <font color="Maroon" title="c18">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">x3</font> ) &amp; ( <font color="Maroon" title="c19">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">x1</font> or <font color="Maroon" title="c19">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">x2</font> or <font color="Maroon" title="c19">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">x3</font> ) &amp; ( <font color="Maroon" title="c20">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">x1</font> or <font color="Maroon" title="c20">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">x2</font> or <font color="Maroon" title="c20">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">x3</font> ) )
 
<b>by </b><i><a class="txt" href="#E17:29_1_1_4_9"><i><font color="Green" title="E42">A58</font></i></a>, <a class="txt" href="#E20:29_1_1_4_9"><i><font color="Green" title="E43">A59</font></i></a>, <a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E23:29_1_1_4_9"/>
<font color="Maroon" title="c14">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">Q</font>
 <b>by </b><i><a class="txt" href="#E5:29_1_1_4_9"><i><font color="Green" title="E37">A49</font></i></a>, <a class="txt" href="#E12:29_1_1_4_9"><i><font color="Green" title="E40">A56</font></i></a>, <a class="txt" href="#E14:29_1_1_4_9"><i><font color="Green" title="E41">A57</font></i></a>, <a class="txt" href="#E20:29_1_1_4_9"><i><font color="Green" title="E43">A59</font></i></a>, <a class="ref" href="enumset1.html#T98" title="ENUMSET1:th.98">ENUMSET1:98</a>, <a class="ref" href="enumset1.html#T99" title="ENUMSET1:th.99">ENUMSET1:99</a>, <a class="ref" href="enumset1.html#T100" title="ENUMSET1:th.100">ENUMSET1:100</a>, <a class="ref" href="enumset1.html#T102" title="ENUMSET1:th.102">ENUMSET1:102</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>thus </b><a NAME="E10:29_1_1_4"/>
 <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #) is <a href="incsp_1.html#V11" title="INCSP_1:attr.11">with_lines_inside_planes</a>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #) is <a href="incsp_1.html#V12" title="INCSP_1:attr.12">with_planes_intersecting_in_2_pts</a> &amp;  <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #) is <a href="incsp_1.html#V13" title="INCSP_1:attr.13">up-3-dimensional</a> &amp;  <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #) is <a href="incsp_1.html#V14" title="INCSP_1:attr.14">inc-compatible</a> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_4_10"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">L</font> be   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #); <i><font color="Red">:: according to </font></i><a class="ref" href="incsp_1.html#D14" target="_self" title="INCSP_1:def.14">INCSP_1:def 14</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">P</font> being  <a href="incsp_1.html#NM3" title="INCSP_1:NM.3">PLANE</a> of <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #)  st  ex <font color="Olive" title="b2">A</font>, <font color="Olive" title="b3">B</font> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #) st <br/>( <font color="Olive" title="b2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">B</font> &amp; <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b2">A</font>,<font color="Olive" title="b3">B</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c11">L</font> &amp; <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b2">A</font>,<font color="Olive" title="b3">B</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R5" title="INCSP_1:pred.5">on</a> <font color="Olive" title="b1">P</font> ) holds <br/><font color="Maroon" title="c11">L</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Olive" title="b1">P</font></span><br/><b>let </b><font color="Maroon" title="c12">P</font> be   <a href="incsp_1.html#NM3" title="INCSP_1:NM.3">PLANE</a> of <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</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>, <font color="Olive" title="b2">B</font> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #) st <br/>( <font color="Olive" title="b1">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">B</font> &amp; <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c11">L</font> &amp; <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R5" title="INCSP_1:pred.5">on</a> <font color="Maroon" title="c12">P</font> ) implies <font color="Maroon" title="c11">L</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Maroon" title="c12">P</font> )</span><br/>



<b>given </b><font color="Maroon" title="c13">A</font>, <font color="Maroon" title="c14">B</font> being   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #)<b> such that </b><a NAME="E2:29_1_1_4_10"/><i><font color="Green" title="E34">A60</font></i>: 
<font color="Maroon" title="c13">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">B</font>
 <b>and </b><br/><a NAME="E3:29_1_1_4_10"/><i><font color="Green" title="E35">A61</font></i>: 
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c13">A</font>,<font color="Maroon" title="c14">B</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c11">L</font>
 <b>and </b><br/><a NAME="E4:29_1_1_4_10"/><i><font color="Green" title="E36">A62</font></i>: 
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c13">A</font>,<font color="Maroon" title="c14">B</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R5" title="INCSP_1:pred.5">on</a> <font color="Maroon" title="c12">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c11">L</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Maroon" title="c12">P</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c15">l</font> = <font color="Maroon" title="c11">L</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">Li</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c16">p</font> = <font color="Maroon" title="c12">P</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">Pl</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c17">a</font> = <font color="Maroon" title="c13">A</font>, <font color="Maroon" title="c18">b</font> = <font color="Maroon" title="c14">B</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span> ;<br/>
<a NAME="E8:29_1_1_4_10"/>
( <font color="Maroon" title="c13">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c11">L</font> &amp; <font color="Maroon" title="c14">B</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c11">L</font> )
 
<b>by </b><i><a class="txt" href="#E3:29_1_1_4_10"><i><font color="Green" title="E35">A61</font></i></a>, <a class="ref" href="incsp_1.html#T11" target="_self" title="INCSP_1:th.11">Th11</a></i>;<br/>
<a NAME="E9:29_1_1_4_10"/><b>then </b>
( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c17">a</font>,<font color="Maroon" title="c15">l</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">i1</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c18">b</font>,<font color="Maroon" title="c15">l</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">i1</font> )
 
<b>by </b><i><a class="ref" href="incsp_1.html#D1" target="_self" title="INCSP_1:def.1">Def1</a></i>;<br/>
<a NAME="E10:29_1_1_4_10"/><b>then </b><i><font color="Green" title="E37">A63</font></i>: 
( <font color="Maroon" title="c17">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">l</font> &amp; <font color="Maroon" title="c18">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">l</font> )
 
<b>by </b><i><a class="txt" href="#E1:29_1_1_4"><i><font color="Green" title="E30">A1</font></i></a></i>;<br/>
<a NAME="E11:29_1_1_4_10"/>
( <font color="Maroon" title="c13">A</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c12">P</font> &amp; <font color="Maroon" title="c14">B</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c12">P</font> )
 
<b>by </b><i><a class="txt" href="#E4:29_1_1_4_10"><i><font color="Green" title="E36">A62</font></i></a>, <a class="ref" href="incsp_1.html#T13" target="_self" title="INCSP_1:th.13">Th13</a></i>;<br/>
<a NAME="E12:29_1_1_4_10"/><b>then </b>
( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c17">a</font>,<font color="Maroon" title="c16">p</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">i2</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c18">b</font>,<font color="Maroon" title="c16">p</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">i2</font> )
 
<b>by </b><i><a class="ref" href="incsp_1.html#D2" target="_self" title="INCSP_1:def.2">Def2</a></i>;<br/>
<a NAME="E13:29_1_1_4_10"/><b>then </b><i><font color="Green" title="E38">A64</font></i>: 
( <font color="Maroon" title="c17">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">p</font> &amp; <font color="Maroon" title="c18">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">p</font> )
 
<b>by </b><i><a class="txt" href="#E2:29_1_1_4"><i><font color="Green" title="E31">A4</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_4_10_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c19">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">l</font> implies <font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">p</font> )</span><br/><b>assume </b><a NAME="E1:29_1_1_4_10_1"/><i><font color="Green" title="E39">A65</font></i>: 
<font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">l</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">p</font></span><br/><a NAME="E2:29_1_1_4_10_1"/>
<font color="Maroon" title="c15">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Li</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c20">x1</font>, <font color="Maroon" title="c21">x2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span><b> such that </b><br/><a NAME="E4:29_1_1_4_10_1"/><i><font color="Green" title="E40">A66</font></i>: 
<font color="Maroon" title="c15">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c20">x1</font>,<font color="Maroon" title="c21">x2</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 <b>and </b><br/><a NAME="E5:29_1_1_4_10_1"/>
<font color="Maroon" title="c20">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c21">x2</font>
 ;<br/><a NAME="E6:29_1_1_4_10_1"/>
( ( <font color="Maroon" title="c17">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">x1</font> or <font color="Maroon" title="c17">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">x2</font> ) &amp; ( <font color="Maroon" title="c18">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">x1</font> or <font color="Maroon" title="c18">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">x2</font> ) )
 <b>by </b><i><a class="txt" href="#E10:29_1_1_4_10"><i><font color="Green" title="E37">A63</font></i></a>, <a class="txt" href="#E4:29_1_1_4_10_1"><i><font color="Green" title="E40">A66</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><b>hence </b><a NAME="E7:29_1_1_4_10_1"/>
<font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">p</font>
 <b>by </b><i><a class="txt" href="#E2:29_1_1_4_10"><i><font color="Green" title="E34">A60</font></i></a>, <a class="txt" href="#E13:29_1_1_4_10"><i><font color="Green" title="E38">A64</font></i></a>, <a class="txt" href="#E1:29_1_1_4_10_1"><i><font color="Green" title="E39">A65</font></i></a>, <a class="txt" href="#E4:29_1_1_4_10_1"><i><font color="Green" title="E40">A66</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</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="E15:29_1_1_4_10"/><b>then </b>
<font color="Maroon" title="c15">l</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c16">p</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E16:29_1_1_4_10"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c15">l</font>,<font color="Maroon" title="c16">p</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">i3</font>
 
;<br/>
<b>hence </b><a NAME="E17:29_1_1_4_10"/>
<font color="Maroon" title="c11">L</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Maroon" title="c12">P</font>
 <b>by </b><i><a class="ref" href="incsp_1.html#D3" target="_self" title="INCSP_1:def.3">Def3</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>thus </b><a NAME="E11:29_1_1_4"/>
 <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #) is <a href="incsp_1.html#V12" title="INCSP_1:attr.12">with_planes_intersecting_in_2_pts</a>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #) is <a href="incsp_1.html#V13" title="INCSP_1:attr.13">up-3-dimensional</a> &amp;  <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #) is <a href="incsp_1.html#V14" title="INCSP_1:attr.14">inc-compatible</a> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_4_11"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">A</font> be   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #); <i><font color="Red">:: according to </font></i><a class="ref" href="incsp_1.html#D15" target="_self" title="INCSP_1:def.15">INCSP_1:def 15</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">P</font>, <font color="Olive" title="b2">Q</font> being  <a href="incsp_1.html#NM3" title="INCSP_1:NM.3">PLANE</a> of <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #)  st <font color="Maroon" title="c11">A</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Olive" title="b1">P</font> &amp; <font color="Maroon" title="c11">A</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Olive" title="b2">Q</font> holds <br/> ex <font color="Olive" title="b3">B</font> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #) st <br/>( <font color="Maroon" title="c11">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">B</font> &amp; <font color="Olive" title="b3">B</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Olive" title="b1">P</font> &amp; <font color="Olive" title="b3">B</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Olive" title="b2">Q</font> )</span><br/><b>let </b><font color="Maroon" title="c12">P</font>, <font color="Maroon" title="c13">Q</font> be   <a href="incsp_1.html#NM3" title="INCSP_1:NM.3">PLANE</a> of <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #); <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c11">A</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c12">P</font> &amp; <font color="Maroon" title="c11">A</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c13">Q</font> implies  ex <font color="Olive" title="b1">B</font> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #) st <br/>( <font color="Maroon" title="c11">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">B</font> &amp; <font color="Olive" title="b1">B</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c12">P</font> &amp; <font color="Olive" title="b1">B</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c13">Q</font> ) )</span><br/>



<b>assume </b><b>that </b><br/><a NAME="E1:29_1_1_4_11"/><i><font color="Green" title="E34">A67</font></i>: 
<font color="Maroon" title="c11">A</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c12">P</font>
 <b>and </b><br/><a NAME="E2:29_1_1_4_11"/><i><font color="Green" title="E35">A68</font></i>: 
<font color="Maroon" title="c11">A</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c13">Q</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">B</font> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #) st <br/>( <font color="Maroon" title="c11">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">B</font> &amp; <font color="Olive" title="b1">B</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c12">P</font> &amp; <font color="Olive" title="b1">B</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c13">Q</font> )</span><br/>

<b>reconsider </b><font color="Maroon" title="c14">a</font> = <font color="Maroon" title="c11">A</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span> ;<br/>
<b>reconsider </b><font color="Maroon" title="c15">p</font> = <font color="Maroon" title="c12">P</font>, <font color="Maroon" title="c16">q</font> = <font color="Maroon" title="c13">Q</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">Pl</font> ;<br/>
<a NAME="E5:29_1_1_4_11"/>
<font color="Maroon" title="c15">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Pl</font>
 
;<br/>
<b>then consider </b><font color="Maroon" title="c17">x1</font>, <font color="Maroon" title="c18">x2</font>, <font color="Maroon" title="c19">x3</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span><b> such that </b><br/><a NAME="E7:29_1_1_4_11"/><i><font color="Green" title="E36">A69</font></i>: 
<font color="Maroon" title="c15">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c17">x1</font>,<font color="Maroon" title="c18">x2</font>,<font color="Maroon" title="c19">x3</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span>
 <b>and </b><br/><a NAME="E8:29_1_1_4_11"/><i><font color="Green" title="E37">A70</font></i>: 
( <font color="Maroon" title="c17">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c18">x2</font> &amp; <font color="Maroon" title="c17">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c19">x3</font> &amp; <font color="Maroon" title="c18">x2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c19">x3</font> )
 ;<br/>
<a NAME="E9:29_1_1_4_11"/>
<font color="Maroon" title="c16">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Pl</font>
 
;<br/>
<b>then consider </b><font color="Maroon" title="c20">y1</font>, <font color="Maroon" title="c21">y2</font>, <font color="Maroon" title="c22">y3</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span><b> such that </b><br/><a NAME="E11:29_1_1_4_11"/><i><font color="Green" title="E38">A71</font></i>: 
<font color="Maroon" title="c16">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c20">y1</font>,<font color="Maroon" title="c21">y2</font>,<font color="Maroon" title="c22">y3</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span>
 <b>and </b><br/><a NAME="E12:29_1_1_4_11"/><i><font color="Green" title="E39">A72</font></i>: 
( <font color="Maroon" title="c20">y1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c21">y2</font> &amp; <font color="Maroon" title="c20">y1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c22">y3</font> &amp; <font color="Maroon" title="c21">y2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c22">y3</font> )
 ;<br/>
<a NAME="E13:29_1_1_4_11"/><i><font color="Green" title="E40">A73</font></i>: 
( <font color="Maroon" title="c17">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">p</font> &amp; <font color="Maroon" title="c18">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">p</font> &amp; <font color="Maroon" title="c19">x3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">p</font> )
 
<b>by </b><i><a class="txt" href="#E7:29_1_1_4_11"><i><font color="Green" title="E36">A69</font></i></a>, <a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></i>;<br/>
<a NAME="E14:29_1_1_4_11"/><i><font color="Green" title="E41">A74</font></i>: 
( <font color="Maroon" title="c20">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">q</font> &amp; <font color="Maroon" title="c21">y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">q</font> &amp; <font color="Maroon" title="c22">y3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">q</font> )
 
<b>by </b><i><a class="txt" href="#E11:29_1_1_4_11"><i><font color="Green" title="E38">A71</font></i></a>, <a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></i>;<br/>
<a NAME="E15:29_1_1_4_11"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c14">a</font>,<font color="Maroon" title="c15">p</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">i2</font>
 
<b>by </b><i><a class="txt" href="#E1:29_1_1_4_11"><i><font color="Green" title="E34">A67</font></i></a>, <a class="ref" href="incsp_1.html#D2" target="_self" title="INCSP_1:def.2">Def2</a></i>;<br/>
<a NAME="E16:29_1_1_4_11"/><b>then </b>
<font color="Maroon" title="c14">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">p</font>
 
<b>by </b><i><a class="txt" href="#E2:29_1_1_4"><i><font color="Green" title="E31">A4</font></i></a></i>;<br/>
<a NAME="E17:29_1_1_4_11"/><b>then </b>
( <font color="Maroon" title="c14">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">x1</font> or <font color="Maroon" title="c14">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">x2</font> or <font color="Maroon" title="c14">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">x3</font> )
 
<b>by </b><i><a class="txt" href="#E7:29_1_1_4_11"><i><font color="Green" title="E36">A69</font></i></a>, <a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c23">z1</font>, <font color="Maroon" title="c24">z2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span><b> such that </b><br/><a NAME="E19:29_1_1_4_11"/><i><font color="Green" title="E42">A75</font></i>: 
( <font color="Maroon" title="c23">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">p</font> &amp; <font color="Maroon" title="c24">z2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">p</font> )
 <b>and </b><br/><a NAME="E20:29_1_1_4_11"/><i><font color="Green" title="E43">A76</font></i>: 
( <font color="Maroon" title="c23">z1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">a</font> &amp; <font color="Maroon" title="c24">z2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">a</font> )
 <b>and </b><br/><a NAME="E21:29_1_1_4_11"/><i><font color="Green" title="E44">A77</font></i>: 
<font color="Maroon" title="c23">z1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c24">z2</font>
 <b>by </b><i><a class="txt" href="#E8:29_1_1_4_11"><i><font color="Green" title="E37">A70</font></i></a>, <a class="txt" href="#E13:29_1_1_4_11"><i><font color="Green" title="E40">A73</font></i></a></i>;<br/>
<a NAME="E22:29_1_1_4_11"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c14">a</font>,<font color="Maroon" title="c16">q</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">i2</font>
 
<b>by </b><i><a class="txt" href="#E2:29_1_1_4_11"><i><font color="Green" title="E35">A68</font></i></a>, <a class="ref" href="incsp_1.html#D2" target="_self" title="INCSP_1:def.2">Def2</a></i>;<br/>
<a NAME="E23:29_1_1_4_11"/><b>then </b>
<font color="Maroon" title="c14">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">q</font>
 
<b>by </b><i><a class="txt" href="#E2:29_1_1_4"><i><font color="Green" title="E31">A4</font></i></a></i>;<br/>
<a NAME="E24:29_1_1_4_11"/><b>then </b>
( <font color="Maroon" title="c14">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">y1</font> or <font color="Maroon" title="c14">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">y2</font> or <font color="Maroon" title="c14">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">y3</font> )
 
<b>by </b><i><a class="txt" href="#E11:29_1_1_4_11"><i><font color="Green" title="E38">A71</font></i></a>, <a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c25">z3</font>, <font color="Maroon" title="c26">z4</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span><b> such that </b><br/><a NAME="E26:29_1_1_4_11"/><i><font color="Green" title="E45">A78</font></i>: 
( <font color="Maroon" title="c25">z3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">q</font> &amp; <font color="Maroon" title="c26">z4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">q</font> )
 <b>and </b><br/><a NAME="E27:29_1_1_4_11"/><i><font color="Green" title="E46">A79</font></i>: 
( <font color="Maroon" title="c25">z3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">a</font> &amp; <font color="Maroon" title="c26">z4</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">a</font> )
 <b>and </b><br/><a NAME="E28:29_1_1_4_11"/><i><font color="Green" title="E47">A80</font></i>: 
<font color="Maroon" title="c25">z3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c26">z4</font>
 <b>by </b><i><a class="txt" href="#E12:29_1_1_4_11"><i><font color="Green" title="E39">A72</font></i></a>, <a class="txt" href="#E14:29_1_1_4_11"><i><font color="Green" title="E41">A74</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_4_11_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:29_1_1_4_11_1"/><i><font color="Green" title="E48">A81</font></i>: 
( <font color="Maroon" title="c23">z1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c25">z3</font> &amp; <font color="Maroon" title="c23">z1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c26">z4</font> &amp; <font color="Maroon" title="c24">z2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c25">z3</font> &amp; <font color="Maroon" title="c24">z2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c26">z4</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_4_11_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:29_1_1_4_11_1_1"/>
( <font color="Maroon" title="c14">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a>  or <font color="Maroon" title="c14">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c14">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 or <font color="Maroon" title="c14">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 )
 </span><b>by </b><i><a class="ref" href="enumset1.html#D2" title="ENUMSET1:def.2">ENUMSET1:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_4_11_1_1_1"><b>suppose </b></a><a NAME="E1:29_1_1_4_11_1_1_1"/><i><font color="Green" title="E49">A82</font></i>: 
<font color="Maroon" title="c14">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:29_1_1_4_11_1_1_1"/><b>then </b><i><font color="Green" title="E50">A83</font></i>: 
( ( <font color="Maroon" title="c23">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c23">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 or <font color="Maroon" title="c23">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 ) &amp; ( <font color="Maroon" title="c24">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c24">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 or <font color="Maroon" title="c24">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 ) )
 <b>by </b><i><a class="txt" href="#E20:29_1_1_4_11"><i><font color="Green" title="E43">A76</font></i></a>, <a class="ref" href="enumset1.html#D2" title="ENUMSET1:def.2">ENUMSET1:def 2</a></i>;<br/><a NAME="E3:29_1_1_4_11_1_1_1"/>
( ( <font color="Maroon" title="c25">z3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c25">z3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 or <font color="Maroon" title="c25">z3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 ) &amp; ( <font color="Maroon" title="c26">z4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c26">z4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 or <font color="Maroon" title="c26">z4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 ) )
 <b>by </b><i><a class="txt" href="#E27:29_1_1_4_11"><i><font color="Green" title="E46">A79</font></i></a>, <a class="txt" href="#E1:29_1_1_4_11_1_1_1"><i><font color="Green" title="E49">A82</font></i></a>, <a class="ref" href="enumset1.html#D2" title="ENUMSET1:def.2">ENUMSET1:def 2</a></i>;<br/><b>hence </b><a NAME="E4:29_1_1_4_11_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E21:29_1_1_4_11"><i><font color="Green" title="E44">A77</font></i></a>, <a class="txt" href="#E28:29_1_1_4_11"><i><font color="Green" title="E47">A80</font></i></a>, <a class="txt" href="#E1:29_1_1_4_11_1"><i><font color="Green" title="E48">A81</font></i></a>, <a class="txt" href="#E2:29_1_1_4_11_1_1_1"><i><font color="Green" title="E50">A83</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="hsNdiv(this)" href="javascript:()" title="29_1_1_4_11_1_1_2"><b>suppose </b></a><a NAME="E1:29_1_1_4_11_1_1_2"/><i><font color="Green" title="E49">A84</font></i>: 
<font color="Maroon" title="c14">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:29_1_1_4_11_1_1_2"/><b>then </b><i><font color="Green" title="E50">A85</font></i>: 
( ( <font color="Maroon" title="c23">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a>  or <font color="Maroon" title="c23">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 or <font color="Maroon" title="c23">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 ) &amp; ( <font color="Maroon" title="c24">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a>  or <font color="Maroon" title="c24">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 or <font color="Maroon" title="c24">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 ) )
 <b>by </b><i><a class="txt" href="#E20:29_1_1_4_11"><i><font color="Green" title="E43">A76</font></i></a>, <a class="ref" href="enumset1.html#D2" title="ENUMSET1:def.2">ENUMSET1:def 2</a></i>;<br/><a NAME="E3:29_1_1_4_11_1_1_2"/>
( ( <font color="Maroon" title="c25">z3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a>  or <font color="Maroon" title="c25">z3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 or <font color="Maroon" title="c25">z3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 ) &amp; ( <font color="Maroon" title="c26">z4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a>  or <font color="Maroon" title="c26">z4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 or <font color="Maroon" title="c26">z4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 ) )
 <b>by </b><i><a class="txt" href="#E27:29_1_1_4_11"><i><font color="Green" title="E46">A79</font></i></a>, <a class="txt" href="#E1:29_1_1_4_11_1_1_2"><i><font color="Green" title="E49">A84</font></i></a>, <a class="ref" href="enumset1.html#D2" title="ENUMSET1:def.2">ENUMSET1:def 2</a></i>;<br/><b>hence </b><a NAME="E4:29_1_1_4_11_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="#E21:29_1_1_4_11"><i><font color="Green" title="E44">A77</font></i></a>, <a class="txt" href="#E28:29_1_1_4_11"><i><font color="Green" title="E47">A80</font></i></a>, <a class="txt" href="#E1:29_1_1_4_11_1"><i><font color="Green" title="E48">A81</font></i></a>, <a class="txt" href="#E2:29_1_1_4_11_1_1_2"><i><font color="Green" title="E50">A85</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="hsNdiv(this)" href="javascript:()" title="29_1_1_4_11_1_1_3"><b>suppose </b></a><a NAME="E1:29_1_1_4_11_1_1_3"/><i><font color="Green" title="E49">A86</font></i>: 
<font color="Maroon" title="c14">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:29_1_1_4_11_1_1_3"/><b>then </b><i><font color="Green" title="E50">A87</font></i>: 
( ( <font color="Maroon" title="c23">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a>  or <font color="Maroon" title="c23">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c23">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 ) &amp; ( <font color="Maroon" title="c24">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a>  or <font color="Maroon" title="c24">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c24">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 ) )
 <b>by </b><i><a class="txt" href="#E20:29_1_1_4_11"><i><font color="Green" title="E43">A76</font></i></a>, <a class="ref" href="enumset1.html#D2" title="ENUMSET1:def.2">ENUMSET1:def 2</a></i>;<br/><a NAME="E3:29_1_1_4_11_1_1_3"/>
( ( <font color="Maroon" title="c25">z3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a>  or <font color="Maroon" title="c25">z3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c25">z3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 ) &amp; ( <font color="Maroon" title="c26">z4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a>  or <font color="Maroon" title="c26">z4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c26">z4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 ) )
 <b>by </b><i><a class="txt" href="#E27:29_1_1_4_11"><i><font color="Green" title="E46">A79</font></i></a>, <a class="txt" href="#E1:29_1_1_4_11_1_1_3"><i><font color="Green" title="E49">A86</font></i></a>, <a class="ref" href="enumset1.html#D2" title="ENUMSET1:def.2">ENUMSET1:def 2</a></i>;<br/><b>hence </b><a NAME="E4:29_1_1_4_11_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="#E21:29_1_1_4_11"><i><font color="Green" title="E44">A77</font></i></a>, <a class="txt" href="#E28:29_1_1_4_11"><i><font color="Green" title="E47">A80</font></i></a>, <a class="txt" href="#E1:29_1_1_4_11_1"><i><font color="Green" title="E48">A81</font></i></a>, <a class="txt" href="#E2:29_1_1_4_11_1_1_3"><i><font color="Green" title="E50">A87</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="hsNdiv(this)" href="javascript:()" title="29_1_1_4_11_1_1_4"><b>suppose </b></a><a NAME="E1:29_1_1_4_11_1_1_4"/><i><font color="Green" title="E49">A88</font></i>: 
<font color="Maroon" title="c14">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:29_1_1_4_11_1_1_4"/><b>then </b><i><font color="Green" title="E50">A89</font></i>: 
( ( <font color="Maroon" title="c23">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a>  or <font color="Maroon" title="c23">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c23">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 ) &amp; ( <font color="Maroon" title="c24">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a>  or <font color="Maroon" title="c24">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c24">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 ) )
 <b>by </b><i><a class="txt" href="#E20:29_1_1_4_11"><i><font color="Green" title="E43">A76</font></i></a>, <a class="ref" href="enumset1.html#D2" title="ENUMSET1:def.2">ENUMSET1:def 2</a></i>;<br/><a NAME="E3:29_1_1_4_11_1_1_4"/>
( ( <font color="Maroon" title="c25">z3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a>  or <font color="Maroon" title="c25">z3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c25">z3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 ) &amp; ( <font color="Maroon" title="c26">z4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a>  or <font color="Maroon" title="c26">z4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c26">z4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 ) )
 <b>by </b><i><a class="txt" href="#E27:29_1_1_4_11"><i><font color="Green" title="E46">A79</font></i></a>, <a class="txt" href="#E1:29_1_1_4_11_1_1_4"><i><font color="Green" title="E49">A88</font></i></a>, <a class="ref" href="enumset1.html#D2" title="ENUMSET1:def.2">ENUMSET1:def 2</a></i>;<br/><b>hence </b><a NAME="E4:29_1_1_4_11_1_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="#E21:29_1_1_4_11"><i><font color="Green" title="E44">A77</font></i></a>, <a class="txt" href="#E28:29_1_1_4_11"><i><font color="Green" title="E47">A80</font></i></a>, <a class="txt" href="#E1:29_1_1_4_11_1"><i><font color="Green" title="E48">A81</font></i></a>, <a class="txt" href="#E2:29_1_1_4_11_1_1_4"><i><font color="Green" title="E50">A89</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><b>end;</b></div></div><b>end;</b></div>
<b>then consider </b><font color="Maroon" title="c27">z</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span><b> such that </b><br/><a NAME="E31:29_1_1_4_11"/><i><font color="Green" title="E48">A90</font></i>: 
( <font color="Maroon" title="c27">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">p</font> &amp; <font color="Maroon" title="c27">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">q</font> )
 <b>and </b><br/><a NAME="E32:29_1_1_4_11"/><i><font color="Green" title="E49">A91</font></i>: 
<font color="Maroon" title="c14">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c27">z</font>
 <b>by </b><i><a class="txt" href="#E19:29_1_1_4_11"><i><font color="Green" title="E42">A75</font></i></a>, <a class="txt" href="#E20:29_1_1_4_11"><i><font color="Green" title="E43">A76</font></i></a>, <a class="txt" href="#E26:29_1_1_4_11"><i><font color="Green" title="E45">A78</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c28">B</font> = <font color="Maroon" title="c27">z</font> as   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #) ;<br/>
<b>take </b>
<font color="Maroon" title="c28">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="c11">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c28">B</font> &amp; <font color="Maroon" title="c28">B</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c12">P</font> &amp; <font color="Maroon" title="c28">B</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c13">Q</font> )</span><br/>

<b>thus </b><a NAME="E34:29_1_1_4_11"/>
<font color="Maroon" title="c11">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c28">B</font>
 <b>by </b><i><a class="txt" href="#E32:29_1_1_4_11"><i><font color="Green" title="E49">A91</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c28">B</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c12">P</font> &amp; <font color="Maroon" title="c28">B</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c13">Q</font> )</span><br/>

<a NAME="E35:29_1_1_4_11"/>
( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c27">z</font>,<font color="Maroon" title="c15">p</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">i2</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c27">z</font>,<font color="Maroon" title="c16">q</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">i2</font> )
 
<b>by </b><i><a class="txt" href="#E31:29_1_1_4_11"><i><font color="Green" title="E48">A90</font></i></a></i>;<br/>
<b>hence </b><a NAME="E36:29_1_1_4_11"/>
( <font color="Maroon" title="c28">B</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c12">P</font> &amp; <font color="Maroon" title="c28">B</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c13">Q</font> )
 <b>by </b><i><a class="ref" href="incsp_1.html#D2" target="_self" title="INCSP_1:def.2">Def2</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>thus </b><a NAME="E12:29_1_1_4"/>
 <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #) is <a href="incsp_1.html#V13" title="INCSP_1:attr.13">up-3-dimensional</a>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #) is <a href="incsp_1.html#V14" title="INCSP_1:attr.14">inc-compatible</a></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_4_12"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon" title="c11">Three</font> = 3 as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span> <b>by </b><i><a class="ref" href="enumset1.html#D2" title="ENUMSET1:def.2">ENUMSET1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">A</font> = <font color="Maroon" title="c1">Zero1</font>, <font color="Maroon" title="c13">B</font> = <font color="Maroon" title="c2">One</font>, <font color="Maroon" title="c14">C</font> = <font color="Maroon" title="c3">Two</font>, <font color="Maroon" title="c15">D</font> = <font color="Maroon" title="c11">Three</font> as   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #) ;<br/>
<b>take </b>
<font color="Maroon" title="c12">A</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="incsp_1.html#D16" target="_self" title="INCSP_1:def.16">INCSP_1:def 16</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> not for <font color="Olive" title="b1">B</font>, <font color="Olive" title="b2">C</font>, <font color="Olive" title="b3">D</font> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #) holds  <span class="p1"><a href="domain_1.html#K9" title="DOMAIN_1:func.9">{</a><span class="default"><font color="Maroon" title="c12">A</font>,<font color="Olive" title="b1">B</font>,<font color="Olive" title="b2">C</font>,<font color="Olive" title="b3">D</font></span><a href="domain_1.html#K9" title="DOMAIN_1:func.9">}</a></span> is <a href="incsp_1.html#V4" title="INCSP_1:attr.4">planar</a></span><br/>

<b>take </b>
<font color="Maroon" title="c13">B</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> not for <font color="Olive" title="b1">C</font>, <font color="Olive" title="b2">D</font> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #) holds  <span class="p1"><a href="domain_1.html#K9" title="DOMAIN_1:func.9">{</a><span class="default"><font color="Maroon" title="c12">A</font>,<font color="Maroon" title="c13">B</font>,<font color="Olive" title="b1">C</font>,<font color="Olive" title="b2">D</font></span><a href="domain_1.html#K9" title="DOMAIN_1:func.9">}</a></span> is <a href="incsp_1.html#V4" title="INCSP_1:attr.4">planar</a></span><br/>

<b>take </b>
<font color="Maroon" title="c14">C</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> not for <font color="Olive" title="b1">D</font> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #) holds  <span class="p1"><a href="domain_1.html#K9" title="DOMAIN_1:func.9">{</a><span class="default"><font color="Maroon" title="c12">A</font>,<font color="Maroon" title="c13">B</font>,<font color="Maroon" title="c14">C</font>,<font color="Olive" title="b1">D</font></span><a href="domain_1.html#K9" title="DOMAIN_1:func.9">}</a></span> is <a href="incsp_1.html#V4" title="INCSP_1:attr.4">planar</a></span><br/>

<b>take </b>
<font color="Maroon" title="c15">D</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> not <span class="p1"><a href="domain_1.html#K9" title="DOMAIN_1:func.9">{</a><span class="default"><font color="Maroon" title="c12">A</font>,<font color="Maroon" title="c13">B</font>,<font color="Maroon" title="c14">C</font>,<font color="Maroon" title="c15">D</font></span><a href="domain_1.html#K9" title="DOMAIN_1:func.9">}</a></span> is <a href="incsp_1.html#V4" title="INCSP_1:attr.4">planar</a></span><br/>

<b>assume </b><a NAME="E3:29_1_1_4_12"/>
<span class="p1"><a href="domain_1.html#K9" title="DOMAIN_1:func.9">{</a><span class="default"><font color="Maroon" title="c12">A</font>,<font color="Maroon" title="c13">B</font>,<font color="Maroon" title="c14">C</font>,<font color="Maroon" title="c15">D</font></span><a href="domain_1.html#K9" title="DOMAIN_1:func.9">}</a></span> is <a href="incsp_1.html#V4" title="INCSP_1:attr.4">planar</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<b>then consider </b><font color="Maroon" title="c16">P</font> being   <a href="incsp_1.html#NM3" title="INCSP_1:NM.3">PLANE</a> of <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #)<b> such that </b><br/><a NAME="E5:29_1_1_4_12"/><i><font color="Green" title="E34">A92</font></i>: 
<span class="p1"><a href="domain_1.html#K9" title="DOMAIN_1:func.9">{</a><span class="default"><font color="Maroon" title="c12">A</font>,<font color="Maroon" title="c13">B</font>,<font color="Maroon" title="c14">C</font>,<font color="Maroon" title="c15">D</font></span><a href="domain_1.html#K9" title="DOMAIN_1:func.9">}</a></span> <a href="incsp_1.html#R5" title="INCSP_1:pred.5">on</a> <font color="Maroon" title="c16">P</font>
 <b>by </b><i><a class="ref" href="incsp_1.html#D7" target="_self" title="INCSP_1:def.7">Def7</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c17">p</font> = <font color="Maroon" title="c16">P</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">Pl</font> ;<br/>
<a NAME="E7:29_1_1_4_12"/>
<font color="Maroon" title="c17">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Pl</font>
 
;<br/>
<b>then consider </b><font color="Maroon" title="c18">a</font>, <font color="Maroon" title="c19">b</font>, <font color="Maroon" title="c20">c</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span><b> such that </b><br/><a NAME="E9:29_1_1_4_12"/><i><font color="Green" title="E35">A93</font></i>: 
<font color="Maroon" title="c17">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c18">a</font>,<font color="Maroon" title="c19">b</font>,<font color="Maroon" title="c20">c</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span>
 <b>and </b><br/><a NAME="E10:29_1_1_4_12"/>
( <font color="Maroon" title="c18">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c19">b</font> &amp; <font color="Maroon" title="c18">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c20">c</font> &amp; <font color="Maroon" title="c19">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c20">c</font> )
 ;<br/>
<a NAME="E11:29_1_1_4_12"/>
( <font color="Maroon" title="c12">A</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c16">P</font> &amp; <font color="Maroon" title="c13">B</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c16">P</font> &amp; <font color="Maroon" title="c14">C</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c16">P</font> &amp; <font color="Maroon" title="c15">D</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c16">P</font> )
 
<b>by </b><i><a class="txt" href="#E5:29_1_1_4_12"><i><font color="Green" title="E34">A92</font></i></a>, <a class="ref" href="incsp_1.html#T15" target="_self" title="INCSP_1:th.15">Th15</a></i>;<br/>
<a NAME="E12:29_1_1_4_12"/><b>then </b>
( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c1">Zero1</font>,<font color="Maroon" title="c17">p</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">i2</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c2">One</font>,<font color="Maroon" title="c17">p</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">i2</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c3">Two</font>,<font color="Maroon" title="c17">p</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">i2</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">Three</font>,<font color="Maroon" title="c17">p</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">i2</font> )
 
<b>by </b><i><a class="ref" href="incsp_1.html#D2" target="_self" title="INCSP_1:def.2">Def2</a></i>;<br/>
<a NAME="E13:29_1_1_4_12"/><b>then </b>
( <font color="Maroon" title="c1">Zero1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">p</font> &amp; <font color="Maroon" title="c2">One</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">p</font> &amp; <font color="Maroon" title="c3">Two</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">p</font> &amp; <font color="Maroon" title="c11">Three</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">p</font> )
 
<b>by </b><i><a class="txt" href="#E2:29_1_1_4"><i><font color="Green" title="E31">A4</font></i></a></i>;<br/>
<a NAME="E14:29_1_1_4_12"/><b>then </b>
( ( <font color="Maroon" title="c1">Zero1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">a</font> or <font color="Maroon" title="c1">Zero1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">b</font> or <font color="Maroon" title="c1">Zero1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">c</font> ) &amp; ( <font color="Maroon" title="c2">One</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">a</font> or <font color="Maroon" title="c2">One</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">b</font> or <font color="Maroon" title="c2">One</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">c</font> ) &amp; ( <font color="Maroon" title="c3">Two</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">a</font> or <font color="Maroon" title="c3">Two</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">b</font> or <font color="Maroon" title="c3">Two</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">c</font> ) &amp; ( <font color="Maroon" title="c11">Three</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">a</font> or <font color="Maroon" title="c11">Three</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">b</font> or <font color="Maroon" title="c11">Three</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">c</font> ) )
 
<b>by </b><i><a class="txt" href="#E9:29_1_1_4_12"><i><font color="Green" title="E35">A93</font></i></a>, <a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E15:29_1_1_4_12"/>
contradiction
 ; <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>thus </b><a NAME="E13:29_1_1_4"/>
 <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #) is <a href="incsp_1.html#V14" title="INCSP_1:attr.14">inc-compatible</a>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_4_13"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">A</font> be   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #); <i><font color="Red">:: according to </font></i><a class="ref" href="incsp_1.html#D17" target="_self" title="INCSP_1:def.17">INCSP_1:def 17</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">L</font> being  <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #)<br/> for <font color="Olive" title="b2">P</font> being  <a href="incsp_1.html#NM3" title="INCSP_1:NM.3">PLANE</a> of <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #)  st <font color="Maroon" title="c11">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b1">L</font> &amp; <font color="Olive" title="b1">L</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Olive" title="b2">P</font> holds <br/><font color="Maroon" title="c11">A</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Olive" title="b2">P</font></span><br/><b>let </b><font color="Maroon" title="c12">L</font> be   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</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">P</font> being  <a href="incsp_1.html#NM3" title="INCSP_1:NM.3">PLANE</a> of <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #)  st <font color="Maroon" title="c11">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c12">L</font> &amp; <font color="Maroon" title="c12">L</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Olive" title="b1">P</font> holds <br/><font color="Maroon" title="c11">A</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Olive" title="b1">P</font></span><br/><b>let </b><font color="Maroon" title="c13">P</font> be   <a href="incsp_1.html#NM3" title="INCSP_1:NM.3">PLANE</a> of <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #); <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c11">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c12">L</font> &amp; <font color="Maroon" title="c12">L</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Maroon" title="c13">P</font> implies <font color="Maroon" title="c11">A</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c13">P</font> )</span><br/>





<b>reconsider </b><font color="Maroon" title="c14">a</font> = <font color="Maroon" title="c11">A</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span> ;<br/>
<b>reconsider </b><font color="Maroon" title="c15">l</font> = <font color="Maroon" title="c12">L</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">Li</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c16">p</font> = <font color="Maroon" title="c13">P</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">Pl</font> ;<br/>
<b>assume </b><a NAME="E4:29_1_1_4_13"/>
( <font color="Maroon" title="c11">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c12">L</font> &amp; <font color="Maroon" title="c12">L</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Maroon" title="c13">P</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c11">A</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c13">P</font></span><br/>

<a NAME="E5:29_1_1_4_13"/><b>then </b>
( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c14">a</font>,<font color="Maroon" title="c15">l</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">i1</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c15">l</font>,<font color="Maroon" title="c16">p</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">i3</font> )
 
<b>by </b><i><a class="ref" href="incsp_1.html#D1" target="_self" title="INCSP_1:def.1">Def1</a>, <a class="ref" href="incsp_1.html#D3" target="_self" title="INCSP_1:def.3">Def3</a></i>;<br/>
<a NAME="E6:29_1_1_4_13"/><b>then </b>
( <font color="Maroon" title="c14">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">l</font> &amp; <font color="Maroon" title="c15">l</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c16">p</font> )
 
<b>by </b><i><a class="txt" href="#E1:29_1_1_4"><i><font color="Green" title="E30">A1</font></i></a>, <a class="txt" href="#E3:29_1_1_4"><i><font color="Green" title="E32">A7</font></i></a></i>;<br/>
<a NAME="E7:29_1_1_4_13"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c14">a</font>,<font color="Maroon" title="c16">p</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">i2</font>
 
;<br/>
<b>hence </b><a NAME="E8:29_1_1_4_13"/>
<font color="Maroon" title="c11">A</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c13">P</font>
 <b>by </b><i><a class="ref" href="incsp_1.html#D2" target="_self" title="INCSP_1:def.2">Def2</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><b>end;</b></div>
<a NAME="E13:29_1_1"/><b>then </b>
 <a href="incsp_1.html#G2" title="INCSP_1:aggr.2">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#NK2" title="NUMBERS:NK.2">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>,<font color="Maroon" title="c5">Li</font>,<font color="Maroon" title="c6">Pl</font>,<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c9">i3</font> #) is <a href="incsp_1.html#V15" title="INCSP_1:attr.15">IncSpace-like</a>
 
<b>by </b><i><a class="ref" href="incsp_1.html#D18" target="_self" title="INCSP_1:def.18">Def18</a></i>;<br/>
<b>hence </b><a NAME="E14:29_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being   <a href="incsp_1.html#L2" title="INCSP_1:struct.2">IncStruct</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="incsp_1.html#V2" title="INCSP_1:attr.2">strict</a> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="incsp_1.html#V15" title="INCSP_1:attr.15">IncSpace-like</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
