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

<b>reconsider </b><font color="Maroon">Zero1</font> = 0, <font color="Maroon">One</font> = 1, <font color="Maroon">Two</font> = 2, <font color="Maroon">Three</font> = 3 as    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span> <b>by </b><i><a class="ref" href="enumset1.html#D2">ENUMSET1:def 2</a></i>;<br/>
<a NAME="E2:28_1_1"/>
<span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">Zero1</font>,<font color="Maroon">One</font></span><a href="domain_1.html#K7">}</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K7">{</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="domain_1.html#K7">}</a></span> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>, <font color="Olive">b</font> is    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span> : <font color="Olive">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b</font> </span>}</span> 
 
;<br/>
<b>then reconsider </b><font color="Maroon">Li</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K7">{</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="domain_1.html#K7">}</a></span> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>, <font color="Olive">b</font> is    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span> : <font color="Olive">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b</font> </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<a NAME="E4:28_1_1"/>
<span class="p1"><a href="domain_1.html#K8">{</a><span class="default"><font color="Maroon">Zero1</font>,<font color="Maroon">One</font>,<font color="Maroon">Two</font></span><a href="domain_1.html#K8">}</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K8">{</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">c</font></span><a href="domain_1.html#K8">}</a></span> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>, <font color="Olive">b</font> is    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>, <font color="Olive">c</font> is    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span> : ( <font color="Olive">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b</font> &amp; <font color="Olive">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">c</font> &amp; <font color="Olive">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">c</font> ) </span>}</span> 
 
;<br/>
<b>then reconsider </b><font color="Maroon">Pl</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K8">{</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">c</font></span><a href="domain_1.html#K8">}</a></span> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>, <font color="Olive">b</font> is    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>, <font color="Olive">c</font> is    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span> : ( <font color="Olive">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b</font> &amp; <font color="Olive">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">c</font> &amp; <font color="Olive">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">c</font> ) </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<a NAME="E6:28_1_1"/>
<span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">a</font>,<font color="Olive">l</font></span><a href="domain_1.html#K1">]</a></span> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>, <font color="Olive">l</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Li</font> : <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Olive">l</font> </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>,<font color="Maroon">Li</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:28_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">a</font>,<font color="Olive">l</font></span><a href="domain_1.html#K1">]</a></span> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>, <font color="Olive">l</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Li</font> : <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Olive">l</font> </span>}</span> 
 ;<br/>

<a NAME="E2:28_1_1_1"/><b>then </b>
 ex <font color="Olive">a</font> being   <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span> ex <font color="Olive">l</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Li</font> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">a</font>,<font color="Olive">l</font></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Olive">l</font> )
 
;<br/>
<b>hence </b><a NAME="E3:28_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>,<font color="Maroon">Li</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">i1</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">a</font>,<font color="Olive">l</font></span><a href="domain_1.html#K1">]</a></span> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>, <font color="Olive">l</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Li</font> : <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Olive">l</font> </span>}</span>  as    <a href="relset_1.html#M2">Relation</a> of <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>,<font color="Maroon">Li</font> <b>by </b><i><a class="ref" href="relset_1.html#D1">RELSET_1:def 1</a></i>;<br/>
<a NAME="E8:28_1_1"/>
<span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">a</font>,<font color="Olive">p</font></span><a href="domain_1.html#K1">]</a></span> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>, <font color="Olive">p</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Pl</font> : <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Olive">p</font> </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>,<font color="Maroon">Pl</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:28_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">a</font>,<font color="Olive">p</font></span><a href="domain_1.html#K1">]</a></span> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>, <font color="Olive">p</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Pl</font> : <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Olive">p</font> </span>}</span> 
 ;<br/>

<a NAME="E2:28_1_1_2"/><b>then </b>
 ex <font color="Olive">a</font> being   <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span> ex <font color="Olive">p</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Pl</font> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">a</font>,<font color="Olive">p</font></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Olive">p</font> )
 
;<br/>
<b>hence </b><a NAME="E3:28_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>,<font color="Maroon">Pl</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">i2</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">a</font>,<font color="Olive">p</font></span><a href="domain_1.html#K1">]</a></span> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>, <font color="Olive">p</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Pl</font> : <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Olive">p</font> </span>}</span>  as    <a href="relset_1.html#M2">Relation</a> of <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>,<font color="Maroon">Pl</font> <b>by </b><i><a class="ref" href="relset_1.html#D1">RELSET_1:def 1</a></i>;<br/>
<a NAME="E10:28_1_1"/>
<span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">l</font>,<font color="Olive">p</font></span><a href="domain_1.html#K1">]</a></span> where <font color="Olive">l</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Li</font>, <font color="Olive">p</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Pl</font> : <font color="Olive">l</font> <a href="tarski.html#R1">c=</a> <font color="Olive">p</font> </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">Li</font>,<font color="Maroon">Pl</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:28_1_1_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">l</font>,<font color="Olive">p</font></span><a href="domain_1.html#K1">]</a></span> where <font color="Olive">l</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Li</font>, <font color="Olive">p</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Pl</font> : <font color="Olive">l</font> <a href="tarski.html#R1">c=</a> <font color="Olive">p</font> </span>}</span> 
 ;<br/>

<a NAME="E2:28_1_1_3"/><b>then </b>
 ex <font color="Olive">l</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Li</font> ex <font color="Olive">p</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Pl</font> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">l</font>,<font color="Olive">p</font></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Olive">l</font> <a href="tarski.html#R1">c=</a> <font color="Olive">p</font> )
 
;<br/>
<b>hence </b><a NAME="E3:28_1_1_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">Li</font>,<font color="Maroon">Pl</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">i3</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">l</font>,<font color="Olive">p</font></span><a href="domain_1.html#K1">]</a></span> where <font color="Olive">l</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Li</font>, <font color="Olive">p</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Pl</font> : <font color="Olive">l</font> <a href="tarski.html#R1">c=</a> <font color="Olive">p</font> </span>}</span>  as    <a href="relset_1.html#M2">Relation</a> of <font color="Maroon">Li</font>,<font color="Maroon">Pl</font> <b>by </b><i><a class="ref" href="relset_1.html#D1">RELSET_1:def 1</a></i>;<br/>
<div><i><font color="Green">E23</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_4"><b>now </b></a><div class="add"><b>set </b><font color="Maroon">S</font> =  <a href="incsp_1.html#G2" target="_self">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>,<font color="Maroon">Li</font>,<font color="Maroon">Pl</font>,<font color="Maroon">i1</font>,<font color="Maroon">i2</font>,<font color="Maroon">i3</font> #);<br/><a NAME="E1:28_1_1_4"/><i><font color="Green">E24</font></i>: 
for <font color="Olive">a</font> being   <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span><br/> for <font color="Olive">l</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Li</font>  st <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">a</font>,<font color="Olive">l</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">i1</font> holds <br/><font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Olive">l</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>;<br/><b>let </b><font color="Maroon">l</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Li</font>;<br/>



<b>assume </b><a NAME="E1:28_1_1_4_1"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">l</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">i1</font>
 ;<br/>

<b>then consider </b><font color="Maroon">b</font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>, <font color="Maroon">k</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Li</font><b> such that </b><br/><a NAME="E3:28_1_1_4_1"/><i><font color="Green">E36</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">l</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">k</font></span><a href="domain_1.html#K1">]</a></span>
 <b>and </b><br/><a NAME="E4:28_1_1_4_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">k</font>
 ;<br/>
<a NAME="E5:28_1_1_4_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">l</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> )
 
<b>by </b><i><a class="ref" href="incsp_1.html#T11" target="_self">Th11</a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><a NAME="E6:28_1_1_4_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">l</font>
 <b>by </b><i><a class="txt" href="incsp_1.html#E12:28_1_1"><i><font color="Green">E23</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E2:28_1_1_4"/><i><font color="Green">E58</font></i>: 
for <font color="Olive">a</font> being   <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span><br/> for <font color="Olive">p</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Pl</font>  st <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">a</font>,<font color="Olive">p</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">i2</font> holds <br/><font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Olive">p</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_4_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>;<br/><b>let </b><font color="Maroon">p</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Pl</font>;<br/>



<b>assume </b><a NAME="E1:28_1_1_4_2"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">p</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">i2</font>
 ;<br/>

<b>then consider </b><font color="Maroon">b</font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>, <font color="Maroon">q</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Pl</font><b> such that </b><br/><a NAME="E3:28_1_1_4_2"/><i><font color="Green">E60</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">p</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">q</font></span><a href="domain_1.html#K1">]</a></span>
 <b>and </b><br/><a NAME="E4:28_1_1_4_2"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">q</font>
 ;<br/>
<a NAME="E5:28_1_1_4_2"/>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> )
 
<b>by </b><i><a class="txt" href="incsp_1.html#E12:28_1_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><a NAME="E6:28_1_1_4_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="txt" href="incsp_1.html#E1:28_1_1_4"><i><font color="Green">E24</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:28_1_1_4"/><i><font color="Green">E62</font></i>: 
for <font color="Olive">p</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Pl</font><br/> for <font color="Olive">l</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Li</font>  st <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">l</font>,<font color="Olive">p</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">i3</font> holds <br/><font color="Olive">l</font> <a href="tarski.html#R1">c=</a> <font color="Olive">p</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_4_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Pl</font>;<br/><b>let </b><font color="Maroon">l</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Li</font>;<br/>



<b>assume </b><a NAME="E1:28_1_1_4_3"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">l</font>,<font color="Maroon">p</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">i3</font>
 ;<br/>

<b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Li</font>, <font color="Maroon">q</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Pl</font><b> such that </b><br/><a NAME="E3:28_1_1_4_3"/><i><font color="Green">E63</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">l</font>,<font color="Maroon">p</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">k</font>,<font color="Maroon">q</font></span><a href="domain_1.html#K1">]</a></span>
 <b>and </b><br/><a NAME="E4:28_1_1_4_3"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">k</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">q</font>
 ;<br/>
<a NAME="E5:28_1_1_4_3"/>
( <font color="Maroon">l</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> )
 
<b>by </b><i><a class="txt" href="incsp_1.html#E1:28_1_1_4"><i><font color="Green">E24</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><a NAME="E6:28_1_1_4_3"/>
<font color="Maroon">l</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="incsp_1.html#T12" target="_self">Th12</a></i>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E4:28_1_1_4"/>
for <font color="Olive">L</font> being  <a href="incsp_1.html#NM2">LINE</a> of <a href="incsp_1.html#G2" target="_self">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>,<font color="Maroon">Li</font>,<font color="Maroon">Pl</font>,<font color="Maroon">i1</font>,<font color="Maroon">i2</font>,<font color="Maroon">i3</font> #)  ex <font color="Olive">A</font>, <font color="Olive">B</font> being  <a href="incsp_1.html#NM1">POINT</a> of <a href="incsp_1.html#G2" target="_self">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>,<font color="Maroon">Li</font>,<font color="Maroon">Pl</font>,<font color="Maroon">i1</font>,<font color="Maroon">i2</font>,<font color="Maroon">i3</font> #) st <br/>( <font color="Olive">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">B</font> &amp; <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Olive">A</font>,<font color="Olive">B</font></span><a href="domain_1.html#K7">}</a></span> <a href="incsp_1.html#R4" target="_self">on</a> <font color="Olive">L</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_4_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">L</font> be   <a href="incsp_1.html#NM2">LINE</a> of <a href="incsp_1.html#G2" target="_self">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>,<font color="Maroon">Li</font>,<font color="Maroon">Pl</font>,<font color="Maroon">i1</font>,<font color="Maroon">i2</font>,<font color="Maroon">i3</font> #);<br/>

<b>reconsider </b><font color="Maroon">l</font> = <font color="Maroon">L</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Li</font> ;<br/>
<a NAME="E2:28_1_1_4_4"/>
<font color="Maroon">l</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Li</font>
 
;<br/>
<b>then consider </b><font color="Maroon">a</font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>, <font color="Maroon">b</font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span><b> such that </b><br/><a NAME="E4:28_1_1_4_4"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">l</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K7">}</a></span>
 <b>and </b><br/><a NAME="E5:28_1_1_4_4"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font>
 ;<br/>
<b>reconsider </b><font color="Maroon">A</font> = <font color="Maroon">a</font>, <font color="Maroon">B</font> = <font color="Maroon">b</font> as   <a href="incsp_1.html#NM1">POINT</a> of <a href="incsp_1.html#G2" target="_self">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>,<font color="Maroon">Li</font>,<font color="Maroon">Pl</font>,<font color="Maroon">i1</font>,<font color="Maroon">i2</font>,<font color="Maroon">i3</font> #) ;<br/>
<b>take </b>
<font color="Maroon">A</font>
;<br/>

<b>take </b>
<font color="Maroon">B</font>
;<br/>

<b>thus </b><a NAME="E7:28_1_1_4_4"/>
<font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="incsp_1.html#T12" target="_self">Th12</a></i>;<br/>

<a NAME="E8:28_1_1_4_4"/>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">l</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">l</font> )
 
<b>by </b><i><a class="txt" href="incsp_1.html#E1:28_1_1_4"><i><font color="Green">E24</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E9:28_1_1_4_4"/><b>then </b>
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">l</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">i1</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">l</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">i1</font> )
 
;<br/>
<a NAME="E10:28_1_1_4_4"/><b>then </b>
( <font color="Maroon">A</font> <a href="incsp_1.html#R1" target="_self">on</a> <font color="Maroon">L</font> &amp; <font color="Maroon">B</font> <a href="incsp_1.html#R1" target="_self">on</a> <font color="Maroon">L</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E11:28_1_1_4_4"/>
<span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="domain_1.html#K7">}</a></span> <a href="incsp_1.html#R4" target="_self">on</a> <font color="Maroon">L</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E5:28_1_1_4"/><i><font color="Green">E67</font></i>: 
for <font color="Olive">A</font>, <font color="Olive">B</font> being  <a href="incsp_1.html#NM1">POINT</a> of <a href="incsp_1.html#G2" target="_self">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>,<font color="Maroon">Li</font>,<font color="Maroon">Pl</font>,<font color="Maroon">i1</font>,<font color="Maroon">i2</font>,<font color="Maroon">i3</font> #)  ex <font color="Olive">L</font> being  <a href="incsp_1.html#NM2">LINE</a> of <a href="incsp_1.html#G2" target="_self">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>,<font color="Maroon">Li</font>,<font color="Maroon">Pl</font>,<font color="Maroon">i1</font>,<font color="Maroon">i2</font>,<font color="Maroon">i3</font> #) st <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Olive">A</font>,<font color="Olive">B</font></span><a href="domain_1.html#K7">}</a></span> <a href="incsp_1.html#R4" target="_self">on</a> <font color="Olive">L</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_4_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">A</font> be   <a href="incsp_1.html#NM1">POINT</a> of <a href="incsp_1.html#G2" target="_self">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>,<font color="Maroon">Li</font>,<font color="Maroon">Pl</font>,<font color="Maroon">i1</font>,<font color="Maroon">i2</font>,<font color="Maroon">i3</font> #), <font color="Maroon">B</font> be   <a href="incsp_1.html#NM1">POINT</a> of <a href="incsp_1.html#G2" target="_self">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>,<font color="Maroon">Li</font>,<font color="Maroon">Pl</font>,<font color="Maroon">i1</font>,<font color="Maroon">i2</font>,<font color="Maroon">i3</font> #);<br/>

<b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">A</font>, <font color="Maroon">b</font> = <font color="Maroon">B</font> as    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span> ;<br/>
<div><i><font color="Green">E68</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_4_5_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:28_1_1_4_5_1"/>
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font>
 ;<br/><a NAME="E2:28_1_1_4_5_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K7">}</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">Li</font>
 ;<br/><b>then consider </b><font color="Maroon">l</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Li</font><b> such that </b><br/><a NAME="E4:28_1_1_4_5_1"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">l</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K7">}</a></span>
 ;<br/><b>reconsider </b><font color="Maroon">L</font> = <font color="Maroon">l</font> as   <a href="incsp_1.html#NM2">LINE</a> of <a href="incsp_1.html#G2" target="_self">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>,<font color="Maroon">Li</font>,<font color="Maroon">Pl</font>,<font color="Maroon">i1</font>,<font color="Maroon">i2</font>,<font color="Maroon">i3</font> #) ;<br/><a NAME="E6:28_1_1_4_5_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">l</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">l</font> )
 <b>by </b><i><a class="ref" href="incsp_1.html#T13" target="_self">Th13</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E7:28_1_1_4_5_1"/><b>then </b>
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">l</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">i1</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">l</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">i1</font> )
 ;<br/><a NAME="E8:28_1_1_4_5_1"/><b>then </b>
( <font color="Maroon">A</font> <a href="incsp_1.html#R1" target="_self">on</a> <font color="Maroon">L</font> &amp; <font color="Maroon">B</font> <a href="incsp_1.html#R1" target="_self">on</a> <font color="Maroon">L</font> )
 <b>by </b><i/>;<br/><a NAME="E9:28_1_1_4_5_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="domain_1.html#K7">}</a></span> <a href="incsp_1.html#R4" target="_self">on</a> <font color="Maroon">L</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E10:28_1_1_4_5_1"/>
 ex <font color="Olive">L</font> being  <a href="incsp_1.html#NM2">LINE</a> of <a href="incsp_1.html#G2" target="_self">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>,<font color="Maroon">Li</font>,<font color="Maroon">Pl</font>,<font color="Maroon">i1</font>,<font color="Maroon">i2</font>,<font color="Maroon">i3</font> #) st <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="domain_1.html#K7">}</a></span> <a href="incsp_1.html#R4" target="_self">on</a> <font color="Olive">L</font>
 ;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_4_5_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:28_1_1_4_5_2"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 ;<br/><a NAME="E2:28_1_1_4_5_2"/>
for <font color="Olive">a</font> being   <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>  ex <font color="Olive">c</font> being   <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span> st <font color="Olive">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">c</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_4_5_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>;<br/>

<div><i><font color="Green">E71</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_4_5_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:28_1_1_4_5_2_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:28_1_1_4_5_2_1_1"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">One</font>
 ;<br/><b>hence </b><a NAME="E3:28_1_1_4_5_2_1_1"/>
 ex <font color="Olive">c</font> being   <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span> st <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">c</font>
 ;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_4_5_2_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:28_1_1_4_5_2_1_2"/>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 2 or <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 3 )
 ;<br/><a NAME="E2:28_1_1_4_5_2_1_2"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">Zero1</font>
 ;<br/><b>hence </b><a NAME="E3:28_1_1_4_5_2_1_2"/>
 ex <font color="Olive">c</font> being   <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span> st <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">c</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:28_1_1_4_5_2_1"/>
 ex <font color="Olive">c</font> being   <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span> st <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">c</font>
 <b>by </b><i><a class="ref" href="incsp_1.html#T14" target="_self">Th14</a>, <a class="ref" href="enumset1.html#D2">ENUMSET1:def 2</a></i>;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon">c</font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span><b> such that </b><br/><a NAME="E4:28_1_1_4_5_2"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c</font>
 ;<br/><a NAME="E5:28_1_1_4_5_2"/>
<span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="domain_1.html#K7">}</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">Li</font>
 <b>by </b><i><a class="ref" href="incsp_1.html#T14" target="_self">Th14</a></i>;<br/><b>then consider </b><font color="Maroon">l</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Li</font><b> such that </b><br/><a NAME="E7:28_1_1_4_5_2"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">l</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="domain_1.html#K7">}</a></span>
 ;<br/><b>reconsider </b><font color="Maroon">L</font> = <font color="Maroon">l</font> as   <a href="incsp_1.html#NM2">LINE</a> of <a href="incsp_1.html#G2" target="_self">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>,<font color="Maroon">Li</font>,<font color="Maroon">Pl</font>,<font color="Maroon">i1</font>,<font color="Maroon">i2</font>,<font color="Maroon">i3</font> #) ;<br/><a NAME="E9:28_1_1_4_5_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">l</font>
 <b>by </b><i><a class="ref" href="incsp_1.html#T15" target="_self">Th15</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E10:28_1_1_4_5_2"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">l</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">i1</font>
 ;<br/><a NAME="E11:28_1_1_4_5_2"/><b>then </b>
( <font color="Maroon">A</font> <a href="incsp_1.html#R1" target="_self">on</a> <font color="Maroon">L</font> &amp; <font color="Maroon">B</font> <a href="incsp_1.html#R1" target="_self">on</a> <font color="Maroon">L</font> )
 <b>by </b><i><a class="ref" href="incsp_1.html#T13" target="_self">Th13</a>, </i>;<br/><a NAME="E12:28_1_1_4_5_2"/><b>then </b>
<span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="domain_1.html#K7">}</a></span> <a href="incsp_1.html#R4" target="_self">on</a> <font color="Maroon">L</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E13:28_1_1_4_5_2"/>
 ex <font color="Olive">L</font> being  <a href="incsp_1.html#NM2">LINE</a> of <a href="incsp_1.html#G2" target="_self">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>,<font color="Maroon">Li</font>,<font color="Maroon">Pl</font>,<font color="Maroon">i1</font>,<font color="Maroon">i2</font>,<font color="Maroon">i3</font> #) st <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="domain_1.html#K7">}</a></span> <a href="incsp_1.html#R4" target="_self">on</a> <font color="Olive">L</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:28_1_1_4_5"/>
 ex <font color="Olive">L</font> being  <a href="incsp_1.html#NM2">LINE</a> of <a href="incsp_1.html#G2" target="_self">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>,<font color="Maroon">Li</font>,<font color="Maroon">Pl</font>,<font color="Maroon">i1</font>,<font color="Maroon">i2</font>,<font color="Maroon">i3</font> #) st <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="domain_1.html#K7">}</a></span> <a href="incsp_1.html#R4" target="_self">on</a> <font color="Olive">L</font>
 <b>by </b><i><a class="ref" href="incsp_1.html#T12" target="_self">Th12</a></i>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E6:28_1_1_4"/>
for <font color="Olive">A</font>, <font color="Olive">B</font> being  <a href="incsp_1.html#NM1">POINT</a> of <a href="incsp_1.html#G2" target="_self">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>,<font color="Maroon">Li</font>,<font color="Maroon">Pl</font>,<font color="Maroon">i1</font>,<font color="Maroon">i2</font>,<font color="Maroon">i3</font> #)<br/> for <font color="Olive">K</font>, <font color="Olive">L</font> being  <a href="incsp_1.html#NM2">LINE</a> of <a href="incsp_1.html#G2" target="_self">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>,<font color="Maroon">Li</font>,<font color="Maroon">Pl</font>,<font color="Maroon">i1</font>,<font color="Maroon">i2</font>,<font color="Maroon">i3</font> #)  st <font color="Olive">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">B</font> &amp; <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Olive">A</font>,<font color="Olive">B</font></span><a href="domain_1.html#K7">}</a></span> <a href="incsp_1.html#R4" target="_self">on</a> <font color="Olive">K</font> &amp; <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Olive">A</font>,<font color="Olive">B</font></span><a href="domain_1.html#K7">}</a></span> <a href="incsp_1.html#R4" target="_self">on</a> <font color="Olive">L</font> holds <br/><font color="Olive">K</font> <a href="hidden.html#R1">=</a> <font color="Olive">L</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_4_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">A</font> be   <a href="incsp_1.html#NM1">POINT</a> of <a href="incsp_1.html#G2" target="_self">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>,<font color="Maroon">Li</font>,<font color="Maroon">Pl</font>,<font color="Maroon">i1</font>,<font color="Maroon">i2</font>,<font color="Maroon">i3</font> #), <font color="Maroon">B</font> be   <a href="incsp_1.html#NM1">POINT</a> of <a href="incsp_1.html#G2" target="_self">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>,<font color="Maroon">Li</font>,<font color="Maroon">Pl</font>,<font color="Maroon">i1</font>,<font color="Maroon">i2</font>,<font color="Maroon">i3</font> #);<br/><b>let </b><font color="Maroon">K</font> be   <a href="incsp_1.html#NM2">LINE</a> of <a href="incsp_1.html#G2" target="_self">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>,<font color="Maroon">Li</font>,<font color="Maroon">Pl</font>,<font color="Maroon">i1</font>,<font color="Maroon">i2</font>,<font color="Maroon">i3</font> #), <font color="Maroon">L</font> be   <a href="incsp_1.html#NM2">LINE</a> of <a href="incsp_1.html#G2" target="_self">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>,<font color="Maroon">Li</font>,<font color="Maroon">Pl</font>,<font color="Maroon">i1</font>,<font color="Maroon">i2</font>,<font color="Maroon">i3</font> #);<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:28_1_1_4_6"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">B</font>
 <b>and </b><br/><a NAME="E2:28_1_1_4_6"/><i><font color="Green">E75</font></i>: 
<span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="domain_1.html#K7">}</a></span> <a href="incsp_1.html#R4" target="_self">on</a> <font color="Maroon">K</font>
 <b>and </b><br/><a NAME="E3:28_1_1_4_6"/><i><font color="Green">E76</font></i>: 
<span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="domain_1.html#K7">}</a></span> <a href="incsp_1.html#R4" target="_self">on</a> <font color="Maroon">L</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">k</font> = <font color="Maroon">K</font>, <font color="Maroon">l</font> = <font color="Maroon">L</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Li</font> ;<br/>
<a NAME="E5:28_1_1_4_6"/><i><font color="Green">E77</font></i>: 
( <font color="Maroon">A</font> <a href="incsp_1.html#R1" target="_self">on</a> <font color="Maroon">K</font> &amp; <font color="Maroon">B</font> <a href="incsp_1.html#R1" target="_self">on</a> <font color="Maroon">K</font> )
 
<b>by </b><i><a class="ref" href="incsp_1.html#T13" target="_self">Th13</a>, </i>;<br/>
<a NAME="E6:28_1_1_4_6"/><i><font color="Green">E78</font></i>: 
( <font color="Maroon">A</font> <a href="incsp_1.html#R1" target="_self">on</a> <font color="Maroon">L</font> &amp; <font color="Maroon">B</font> <a href="incsp_1.html#R1" target="_self">on</a> <font color="Maroon">L</font> )
 
<b>by </b><i><a class="ref" href="incsp_1.html#T14" target="_self">Th14</a>, </i>;<br/>
<b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">A</font>, <font color="Maroon">b</font> = <font color="Maroon">B</font> as    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span> ;<br/>
<a NAME="E8:28_1_1_4_6"/>
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">k</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">i1</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">k</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">i1</font> )
 
<b>by </b><i><a class="ref" href="incsp_1.html#T15" target="_self">Th15</a>, </i>;<br/>
<a NAME="E9:28_1_1_4_6"/><b>then </b><i><font color="Green">E79</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">k</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">k</font> )
 
<b>by </b><i><a class="ref" href="incsp_1.html#D7" target="_self">Def7</a></i>;<br/>
<a NAME="E10:28_1_1_4_6"/>
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">l</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">i1</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">l</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">i1</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E11:28_1_1_4_6"/><b>then </b><i><font color="Green">E80</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">l</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">l</font> )
 
<b>by </b><i><a class="ref" href="incsp_1.html#D7" target="_self">Def7</a></i>;<br/>
<a NAME="E12:28_1_1_4_6"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Li</font>
 
;<br/>
<b>then consider </b><font color="Maroon">x1</font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>, <font color="Maroon">x2</font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span><b> such that </b><br/><a NAME="E14:28_1_1_4_6"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="domain_1.html#K7">}</a></span>
 <b>and </b><br/><a NAME="E15:28_1_1_4_6"/>
<font color="Maroon">x1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x2</font>
 ;<br/>
<a NAME="E16:28_1_1_4_6"/><i><font color="Green">E84</font></i>: 
( ( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> or <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font> ) &amp; ( <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> or <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font> ) )
 
<b>by </b><i><a class="ref" href="incsp_1.html#T16" target="_self">Th16</a>, <a class="ref" href="incsp_1.html#T18" target="_self">Th18</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E17:28_1_1_4_6"/>
<font color="Maroon">l</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Li</font>
 
;<br/>
<b>then consider </b><font color="Maroon">x3</font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>, <font color="Maroon">x4</font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span><b> such that </b><br/><a NAME="E19:28_1_1_4_6"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">l</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">x3</font>,<font color="Maroon">x4</font></span><a href="domain_1.html#K7">}</a></span>
 <b>and </b><br/><a NAME="E20:28_1_1_4_6"/>
<font color="Maroon">x3</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x4</font>
 ;<br/>
<a NAME="E21:28_1_1_4_6"/>
( ( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x3</font> or <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x4</font> ) &amp; ( <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x3</font> or <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x4</font> ) )
 
<b>by </b><i><a class="ref" href="incsp_1.html#T17" target="_self">Th17</a>, <a class="ref" href="incsp_1.html#T20" target="_self">Th20</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<b>hence </b><a NAME="E22:28_1_1_4_6"/>
<font color="Maroon">K</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="incsp_1.html#T12" target="_self">Th12</a>, <a class="ref" href="incsp_1.html#T18" target="_self">Th18</a>, <a class="ref" href="incsp_1.html#T19" target="_self">Th19</a>, <a class="ref" href="incsp_1.html#T20" target="_self">Th20</a></i>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E7:28_1_1_4"/>
for <font color="Olive">P</font> being  <a href="incsp_1.html#NM3">PLANE</a> of <a href="incsp_1.html#G2" target="_self">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>,<font color="Maroon">Li</font>,<font color="Maroon">Pl</font>,<font color="Maroon">i1</font>,<font color="Maroon">i2</font>,<font color="Maroon">i3</font> #)  ex <font color="Olive">A</font> being  <a href="incsp_1.html#NM1">POINT</a> of <a href="incsp_1.html#G2" target="_self">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>,<font color="Maroon">Li</font>,<font color="Maroon">Pl</font>,<font color="Maroon">i1</font>,<font color="Maroon">i2</font>,<font color="Maroon">i3</font> #) st <font color="Olive">A</font> <a href="incsp_1.html#R2" target="_self">on</a> <font color="Olive">P</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_4_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">P</font> be   <a href="incsp_1.html#NM3">PLANE</a> of <a href="incsp_1.html#G2" target="_self">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>,<font color="Maroon">Li</font>,<font color="Maroon">Pl</font>,<font color="Maroon">i1</font>,<font color="Maroon">i2</font>,<font color="Maroon">i3</font> #);<br/>

<b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">P</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Pl</font> ;<br/>
<a NAME="E2:28_1_1_4_7"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Pl</font>
 
;<br/>
<b>then consider </b><font color="Maroon">a</font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>, <font color="Maroon">b</font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>, <font color="Maroon">c</font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span><b> such that </b><br/><a NAME="E4:28_1_1_4_7"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K8">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font></span><a href="domain_1.html#K8">}</a></span>
 <b>and </b><br/><a NAME="E5:28_1_1_4_7"/>
( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c</font> )
 ;<br/>
<b>reconsider </b><font color="Maroon">A</font> = <font color="Maroon">a</font> as   <a href="incsp_1.html#NM1">POINT</a> of <a href="incsp_1.html#G2" target="_self">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>,<font color="Maroon">Li</font>,<font color="Maroon">Pl</font>,<font color="Maroon">i1</font>,<font color="Maroon">i2</font>,<font color="Maroon">i3</font> #) ;<br/>
<b>take </b>
<font color="Maroon">A</font>
;<br/>

<a NAME="E7:28_1_1_4_7"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="incsp_1.html#T12" target="_self">Th12</a>, <a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/>
<a NAME="E8:28_1_1_4_7"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">p</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">i2</font>
 
;<br/>
<b>hence </b><a NAME="E9:28_1_1_4_7"/>
<font color="Maroon">A</font> <a href="incsp_1.html#R2" target="_self">on</a> <font color="Maroon">P</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E8:28_1_1_4"/>
for <font color="Olive">A</font>, <font color="Olive">B</font>, <font color="Olive">C</font> being  <a href="incsp_1.html#NM1">POINT</a> of <a href="incsp_1.html#G2" target="_self">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>,<font color="Maroon">Li</font>,<font color="Maroon">Pl</font>,<font color="Maroon">i1</font>,<font color="Maroon">i2</font>,<font color="Maroon">i3</font> #)  ex <font color="Olive">P</font> being  <a href="incsp_1.html#NM3">PLANE</a> of <a href="incsp_1.html#G2" target="_self">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>,<font color="Maroon">Li</font>,<font color="Maroon">Pl</font>,<font color="Maroon">i1</font>,<font color="Maroon">i2</font>,<font color="Maroon">i3</font> #) st <span class="p1"><a href="domain_1.html#K8">{</a><span class="default"><font color="Olive">A</font>,<font color="Olive">B</font>,<font color="Olive">C</font></span><a href="domain_1.html#K8">}</a></span> <a href="incsp_1.html#R5" target="_self">on</a> <font color="Olive">P</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_4_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">A</font> be   <a href="incsp_1.html#NM1">POINT</a> of <a href="incsp_1.html#G2" target="_self">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>,<font color="Maroon">Li</font>,<font color="Maroon">Pl</font>,<font color="Maroon">i1</font>,<font color="Maroon">i2</font>,<font color="Maroon">i3</font> #), <font color="Maroon">B</font> be   <a href="incsp_1.html#NM1">POINT</a> of <a href="incsp_1.html#G2" target="_self">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>,<font color="Maroon">Li</font>,<font color="Maroon">Pl</font>,<font color="Maroon">i1</font>,<font color="Maroon">i2</font>,<font color="Maroon">i3</font> #), <font color="Maroon">C</font> be   <a href="incsp_1.html#NM1">POINT</a> of <a href="incsp_1.html#G2" target="_self">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>,<font color="Maroon">Li</font>,<font color="Maroon">Pl</font>,<font color="Maroon">i1</font>,<font color="Maroon">i2</font>,<font color="Maroon">i3</font> #);<br/>

<b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">A</font>, <font color="Maroon">b</font> = <font color="Maroon">B</font>, <font color="Maroon">c</font> = <font color="Maroon">C</font> as    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span> ;<br/>
<div><i><font color="Green">E89</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_4_8_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:28_1_1_4_8_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c</font> )
 ;<br/><a NAME="E2:28_1_1_4_8_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K8">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font></span><a href="domain_1.html#K8">}</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">Pl</font>
 ;<br/><b>then consider </b><font color="Maroon">p</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Pl</font><b> such that </b><br/><a NAME="E4:28_1_1_4_8_1"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K8">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font></span><a href="domain_1.html#K8">}</a></span>
 ;<br/><b>reconsider </b><font color="Maroon">P</font> = <font color="Maroon">p</font> as   <a href="incsp_1.html#NM3">PLANE</a> of <a href="incsp_1.html#G2" target="_self">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>,<font color="Maroon">Li</font>,<font color="Maroon">Pl</font>,<font color="Maroon">i1</font>,<font color="Maroon">i2</font>,<font color="Maroon">i3</font> #) ;<br/><a NAME="E6:28_1_1_4_8_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">p</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">p</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">p</font> )
 <b>by </b><i><a class="ref" href="incsp_1.html#T13" target="_self">Th13</a>, <a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/><a NAME="E7:28_1_1_4_8_1"/><b>then </b>
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">p</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">i2</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">p</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">i2</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">p</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">i2</font> )
 ;<br/><a NAME="E8:28_1_1_4_8_1"/><b>then </b>
( <font color="Maroon">A</font> <a href="incsp_1.html#R2" target="_self">on</a> <font color="Maroon">P</font> &amp; <font color="Maroon">B</font> <a href="incsp_1.html#R2" target="_self">on</a> <font color="Maroon">P</font> &amp; <font color="Maroon">C</font> <a href="incsp_1.html#R2" target="_self">on</a> <font color="Maroon">P</font> )
 <b>by </b><i/>;<br/><a NAME="E9:28_1_1_4_8_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K8">{</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font>,<font color="Maroon">C</font></span><a href="domain_1.html#K8">}</a></span> <a href="incsp_1.html#R5" target="_self">on</a> <font color="Maroon">P</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E10:28_1_1_4_8_1"/>
 ex <font color="Olive">P</font> being  <a href="incsp_1.html#NM3">PLANE</a> of <a href="incsp_1.html#G2" target="_self">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>,<font color="Maroon">Li</font>,<font color="Maroon">Pl</font>,<font color="Maroon">i1</font>,<font color="Maroon">i2</font>,<font color="Maroon">i3</font> #) st <span class="p1"><a href="domain_1.html#K8">{</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font>,<font color="Maroon">C</font></span><a href="domain_1.html#K8">}</a></span> <a href="incsp_1.html#R5" target="_self">on</a> <font color="Olive">P</font>
 ;<br/></div><b>end;</b></div>
<div><i><font color="Green">E91</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_4_8_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:28_1_1_4_8_2"/><i><font color="Green">E92</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> )
 ;<br/><a NAME="E2:28_1_1_4_8_2"/>
for <font color="Olive">a</font> being   <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>  ex <font color="Olive">b</font>, <font color="Olive">c</font> being   <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span> st <br/>( <font color="Olive">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b</font> &amp; <font color="Olive">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">c</font> &amp; <font color="Olive">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">c</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_4_8_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>;<br/>

<div><i><font color="Green">E93</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_4_8_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:28_1_1_4_8_2_1_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 1 )
 ;<br/><a NAME="E2:28_1_1_4_8_2_1_1"/><b>then </b>
( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">Two</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">Three</font> &amp; <font color="Maroon">Two</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">Three</font> )
 ;<br/><b>hence </b><a NAME="E3:28_1_1_4_8_2_1_1"/>
 ex <font color="Olive">b</font>, <font color="Olive">c</font> being   <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span> st <br/>( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">c</font> &amp; <font color="Olive">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">c</font> )
 ;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_4_8_2_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:28_1_1_4_8_2_1_2"/>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 2 or <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 3 )
 ;<br/><a NAME="E2:28_1_1_4_8_2_1_2"/><b>then </b>
( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">Zero1</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">One</font> &amp; <font color="Maroon">Zero1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">One</font> )
 ;<br/><b>hence </b><a NAME="E3:28_1_1_4_8_2_1_2"/>
 ex <font color="Olive">b</font>, <font color="Olive">c</font> being   <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span> st <br/>( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">c</font> &amp; <font color="Olive">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">c</font> )
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:28_1_1_4_8_2_1"/>
 ex <font color="Olive">b</font>, <font color="Olive">c</font> being   <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span> st <br/>( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">c</font> &amp; <font color="Olive">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">c</font> )
 <b>by </b><i><a class="ref" href="incsp_1.html#T15" target="_self">Th15</a>, <a class="ref" href="enumset1.html#D2">ENUMSET1:def 2</a></i>;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>, <font color="Maroon">y</font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span><b> such that </b><br/><a NAME="E4:28_1_1_4_8_2"/><i><font color="Green">E95</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font> )
 ;<br/><a NAME="E5:28_1_1_4_8_2"/>
<span class="p1"><a href="domain_1.html#K8">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K8">}</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">Pl</font>
 <b>by </b><i><a class="ref" href="incsp_1.html#T15" target="_self">Th15</a></i>;<br/><b>then consider </b><font color="Maroon">p</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Pl</font><b> such that </b><br/><a NAME="E7:28_1_1_4_8_2"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K8">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K8">}</a></span>
 ;<br/><b>reconsider </b><font color="Maroon">P</font> = <font color="Maroon">p</font> as   <a href="incsp_1.html#NM3">PLANE</a> of <a href="incsp_1.html#G2" target="_self">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>,<font color="Maroon">Li</font>,<font color="Maroon">Pl</font>,<font color="Maroon">i1</font>,<font color="Maroon">i2</font>,<font color="Maroon">i3</font> #) ;<br/><a NAME="E9:28_1_1_4_8_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">p</font>
 <b>by </b><i>, <a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/><a NAME="E10:28_1_1_4_8_2"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">p</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">i2</font>
 ;<br/><a NAME="E11:28_1_1_4_8_2"/><b>then </b>
<font color="Maroon">A</font> <a href="incsp_1.html#R2" target="_self">on</a> <font color="Maroon">P</font>
 <b>by </b><i/>;<br/><a NAME="E12:28_1_1_4_8_2"/><b>then </b>
<span class="p1"><a href="domain_1.html#K8">{</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font>,<font color="Maroon">C</font></span><a href="domain_1.html#K8">}</a></span> <a href="incsp_1.html#R5" target="_self">on</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="incsp_1.html#T14" target="_self">Th14</a>, </i>;<br/><b>hence </b><a NAME="E13:28_1_1_4_8_2"/>
 ex <font color="Olive">P</font> being  <a href="incsp_1.html#NM3">PLANE</a> of <a href="incsp_1.html#G2" target="_self">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>,<font color="Maroon">Li</font>,<font color="Maroon">Pl</font>,<font color="Maroon">i1</font>,<font color="Maroon">i2</font>,<font color="Maroon">i3</font> #) st <span class="p1"><a href="domain_1.html#K8">{</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font>,<font color="Maroon">C</font></span><a href="domain_1.html#K8">}</a></span> <a href="incsp_1.html#R5" target="_self">on</a> <font color="Olive">P</font>
 ;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_4_8_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:28_1_1_4_8_3"/><i><font color="Green">E97</font></i>: 
( ( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c</font> ) or ( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> ) or ( <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> ) )
 ;<br/><b>then consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>, <font color="Maroon">y</font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span><b> such that </b><br/><a NAME="E3:28_1_1_4_8_3"/><i><font color="Green">E98</font></i>: 
( ( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> ) &amp; ( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> or <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> or <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> ) )
 <b>and </b><br/><a NAME="E4:28_1_1_4_8_3"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font>
 ;<br/><a NAME="E5:28_1_1_4_8_3"/>
for <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>  ex <font color="Olive">c</font> being   <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span> st <br/>( <font color="Olive">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">c</font> &amp; <font color="Olive">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">c</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_4_8_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>;<br/><b>let </b><font color="Maroon">b</font> be    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>;<br/>



<div><i><font color="Green">E100</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_4_8_3_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:28_1_1_4_8_3_1_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0 &amp; ( <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 2 ) )
 ;<br/><a NAME="E2:28_1_1_4_8_3_1_1"/><b>then </b>
( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">Three</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">Three</font> )
 ;<br/><b>hence </b><a NAME="E3:28_1_1_4_8_3_1_1"/>
 ex <font color="Olive">c</font> being   <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span> st <br/>( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">c</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">c</font> )
 ;<br/></div><b>end;</b></div>
<div><i><font color="Green">E101</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_4_8_3_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:28_1_1_4_8_3_1_2"/>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 3 )
 ;<br/><a NAME="E2:28_1_1_4_8_3_1_2"/><b>then </b>
( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">One</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">One</font> )
 ;<br/><b>hence </b><a NAME="E3:28_1_1_4_8_3_1_2"/>
 ex <font color="Olive">c</font> being   <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span> st <br/>( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">c</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">c</font> )
 ;<br/></div><b>end;</b></div>
<div><i><font color="Green">E102</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_4_8_3_1_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:28_1_1_4_8_3_1_3"/>
( ( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 2 or <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 3 ) &amp; ( <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 2 or <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 3 ) )
 ;<br/><a NAME="E2:28_1_1_4_8_3_1_3"/><b>then </b>
( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">Zero1</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">Zero1</font> )
 ;<br/><b>hence </b><a NAME="E3:28_1_1_4_8_3_1_3"/>
 ex <font color="Olive">c</font> being   <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span> st <br/>( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">c</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">c</font> )
 ;<br/></div><b>end;</b></div>
<div><i><font color="Green">E103</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_4_8_3_1_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:28_1_1_4_8_3_1_4"/>
( ( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 2 ) &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 0 )
 ;<br/><a NAME="E2:28_1_1_4_8_3_1_4"/><b>then </b>
( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">Three</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">Three</font> )
 ;<br/><b>hence </b><a NAME="E3:28_1_1_4_8_3_1_4"/>
 ex <font color="Olive">c</font> being   <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span> st <br/>( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">c</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">c</font> )
 ;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_4_8_3_1_5"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:28_1_1_4_8_3_1_5"/>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 3 &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 0 )
 ;<br/><a NAME="E2:28_1_1_4_8_3_1_5"/><b>then </b>
( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">Two</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">Two</font> )
 ;<br/><b>hence </b><a NAME="E3:28_1_1_4_8_3_1_5"/>
 ex <font color="Olive">c</font> being   <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span> st <br/>( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">c</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">c</font> )
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:28_1_1_4_8_3_1"/>
 ex <font color="Olive">c</font> being   <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span> st <br/>( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">c</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">c</font> )
 <b>by </b><i><a class="ref" href="incsp_1.html#T16" target="_self">Th16</a>, <a class="ref" href="incsp_1.html#T17" target="_self">Th17</a>, <a class="ref" href="incsp_1.html#T18" target="_self">Th18</a>, <a class="ref" href="incsp_1.html#T19" target="_self">Th19</a>, <a class="ref" href="enumset1.html#D2">ENUMSET1:def 2</a></i>;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon">z</font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span><b> such that </b><br/><a NAME="E7:28_1_1_4_8_3"/><i><font color="Green">E105</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">z</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">z</font> )
 ;<br/><a NAME="E8:28_1_1_4_8_3"/>
<span class="p1"><a href="domain_1.html#K8">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K8">}</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">Pl</font>
 <b>by </b><i>, <a class="ref" href="incsp_1.html#T16" target="_self">Th16</a></i>;<br/><b>then consider </b><font color="Maroon">p</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Pl</font><b> such that </b><br/><a NAME="E10:28_1_1_4_8_3"/><i><font color="Green">E106</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K8">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K8">}</a></span>
 ;<br/><b>reconsider </b><font color="Maroon">P</font> = <font color="Maroon">p</font> as   <a href="incsp_1.html#NM3">PLANE</a> of <a href="incsp_1.html#G2" target="_self">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>,<font color="Maroon">Li</font>,<font color="Maroon">Pl</font>,<font color="Maroon">i1</font>,<font color="Maroon">i2</font>,<font color="Maroon">i3</font> #) ;<br/><a NAME="E12:28_1_1_4_8_3"/>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">p</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">p</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">p</font> )
 <b>by </b><i><a class="ref" href="incsp_1.html#T14" target="_self">Th14</a>, <a class="ref" href="incsp_1.html#T15" target="_self">Th15</a>, , <a class="ref" href="incsp_1.html#T17" target="_self">Th17</a>, <a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/><a NAME="E13:28_1_1_4_8_3"/><b>then </b>
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">p</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">i2</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">p</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">i2</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">p</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">i2</font> )
 ;<br/><a NAME="E14:28_1_1_4_8_3"/><b>then </b>
( <font color="Maroon">A</font> <a href="incsp_1.html#R2" target="_self">on</a> <font color="Maroon">P</font> &amp; <font color="Maroon">B</font> <a href="incsp_1.html#R2" target="_self">on</a> <font color="Maroon">P</font> &amp; <font color="Maroon">C</font> <a href="incsp_1.html#R2" target="_self">on</a> <font color="Maroon">P</font> )
 <b>by </b><i/>;<br/><a NAME="E15:28_1_1_4_8_3"/><b>then </b>
<span class="p1"><a href="domain_1.html#K8">{</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font>,<font color="Maroon">C</font></span><a href="domain_1.html#K8">}</a></span> <a href="incsp_1.html#R5" target="_self">on</a> <font color="Maroon">P</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E16:28_1_1_4_8_3"/>
 ex <font color="Olive">P</font> being  <a href="incsp_1.html#NM3">PLANE</a> of <a href="incsp_1.html#G2" target="_self">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>,<font color="Maroon">Li</font>,<font color="Maroon">Pl</font>,<font color="Maroon">i1</font>,<font color="Maroon">i2</font>,<font color="Maroon">i3</font> #) st <span class="p1"><a href="domain_1.html#K8">{</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font>,<font color="Maroon">C</font></span><a href="domain_1.html#K8">}</a></span> <a href="incsp_1.html#R5" target="_self">on</a> <font color="Olive">P</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:28_1_1_4_8"/>
 ex <font color="Olive">P</font> being  <a href="incsp_1.html#NM3">PLANE</a> of <a href="incsp_1.html#G2" target="_self">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>,<font color="Maroon">Li</font>,<font color="Maroon">Pl</font>,<font color="Maroon">i1</font>,<font color="Maroon">i2</font>,<font color="Maroon">i3</font> #) st <span class="p1"><a href="domain_1.html#K8">{</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font>,<font color="Maroon">C</font></span><a href="domain_1.html#K8">}</a></span> <a href="incsp_1.html#R5" target="_self">on</a> <font color="Olive">P</font>
 <b>by </b><i><a class="ref" href="incsp_1.html#T12" target="_self">Th12</a>, <a class="ref" href="incsp_1.html#T13" target="_self">Th13</a></i>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E9:28_1_1_4"/>
for <font color="Olive">A</font>, <font color="Olive">B</font>, <font color="Olive">C</font> being  <a href="incsp_1.html#NM1">POINT</a> of <a href="incsp_1.html#G2" target="_self">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>,<font color="Maroon">Li</font>,<font color="Maroon">Pl</font>,<font color="Maroon">i1</font>,<font color="Maroon">i2</font>,<font color="Maroon">i3</font> #)<br/> for <font color="Olive">P</font>, <font color="Olive">Q</font> being  <a href="incsp_1.html#NM3">PLANE</a> of <a href="incsp_1.html#G2" target="_self">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>,<font color="Maroon">Li</font>,<font color="Maroon">Pl</font>,<font color="Maroon">i1</font>,<font color="Maroon">i2</font>,<font color="Maroon">i3</font> #)  st not <span class="p1"><a href="domain_1.html#K8">{</a><span class="default"><font color="Olive">A</font>,<font color="Olive">B</font>,<font color="Olive">C</font></span><a href="domain_1.html#K8">}</a></span> <a href="incsp_1.html#NR6" target="_self">is_collinear</a>  &amp; <span class="p1"><a href="domain_1.html#K8">{</a><span class="default"><font color="Olive">A</font>,<font color="Olive">B</font>,<font color="Olive">C</font></span><a href="domain_1.html#K8">}</a></span> <a href="incsp_1.html#R5" target="_self">on</a> <font color="Olive">P</font> &amp; <span class="p1"><a href="domain_1.html#K8">{</a><span class="default"><font color="Olive">A</font>,<font color="Olive">B</font>,<font color="Olive">C</font></span><a href="domain_1.html#K8">}</a></span> <a href="incsp_1.html#R5" target="_self">on</a> <font color="Olive">Q</font> holds <br/><font color="Olive">P</font> <a href="hidden.html#R1">=</a> <font color="Olive">Q</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_4_9"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">A</font> be   <a href="incsp_1.html#NM1">POINT</a> of <a href="incsp_1.html#G2" target="_self">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>,<font color="Maroon">Li</font>,<font color="Maroon">Pl</font>,<font color="Maroon">i1</font>,<font color="Maroon">i2</font>,<font color="Maroon">i3</font> #), <font color="Maroon">B</font> be   <a href="incsp_1.html#NM1">POINT</a> of <a href="incsp_1.html#G2" target="_self">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>,<font color="Maroon">Li</font>,<font color="Maroon">Pl</font>,<font color="Maroon">i1</font>,<font color="Maroon">i2</font>,<font color="Maroon">i3</font> #), <font color="Maroon">C</font> be   <a href="incsp_1.html#NM1">POINT</a> of <a href="incsp_1.html#G2" target="_self">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>,<font color="Maroon">Li</font>,<font color="Maroon">Pl</font>,<font color="Maroon">i1</font>,<font color="Maroon">i2</font>,<font color="Maroon">i3</font> #);<br/><b>let </b><font color="Maroon">P</font> be   <a href="incsp_1.html#NM3">PLANE</a> of <a href="incsp_1.html#G2" target="_self">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>,<font color="Maroon">Li</font>,<font color="Maroon">Pl</font>,<font color="Maroon">i1</font>,<font color="Maroon">i2</font>,<font color="Maroon">i3</font> #), <font color="Maroon">Q</font> be   <a href="incsp_1.html#NM3">PLANE</a> of <a href="incsp_1.html#G2" target="_self">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>,<font color="Maroon">Li</font>,<font color="Maroon">Pl</font>,<font color="Maroon">i1</font>,<font color="Maroon">i2</font>,<font color="Maroon">i3</font> #);<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:28_1_1_4_9"/><i><font color="Green">E107</font></i>: 
not <span class="p1"><a href="domain_1.html#K8">{</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font>,<font color="Maroon">C</font></span><a href="domain_1.html#K8">}</a></span> <a href="incsp_1.html#NR6" target="_self">is_collinear</a> 
 <b>and </b><br/><a NAME="E2:28_1_1_4_9"/><i><font color="Green">E108</font></i>: 
<span class="p1"><a href="domain_1.html#K8">{</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font>,<font color="Maroon">C</font></span><a href="domain_1.html#K8">}</a></span> <a href="incsp_1.html#R5" target="_self">on</a> <font color="Maroon">P</font>
 <b>and </b><br/><a NAME="E3:28_1_1_4_9"/><i><font color="Green">E109</font></i>: 
<span class="p1"><a href="domain_1.html#K8">{</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font>,<font color="Maroon">C</font></span><a href="domain_1.html#K8">}</a></span> <a href="incsp_1.html#R5" target="_self">on</a> <font color="Maroon">Q</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">P</font>, <font color="Maroon">q</font> = <font color="Maroon">Q</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Pl</font> ;<br/>
<div><i><font color="Green">E110</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_4_9_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:28_1_1_4_9_1"/><i><font color="Green">E111</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> or <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> or <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> )
 ;<br/><b>consider </b><font color="Maroon">K</font> being   <a href="incsp_1.html#NM2">LINE</a> of <a href="incsp_1.html#G2" target="_self">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>,<font color="Maroon">Li</font>,<font color="Maroon">Pl</font>,<font color="Maroon">i1</font>,<font color="Maroon">i2</font>,<font color="Maroon">i3</font> #)<b> such that </b><br/><a NAME="E3:28_1_1_4_9_1"/><i><font color="Green">E112</font></i>: 
<span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="domain_1.html#K7">}</a></span> <a href="incsp_1.html#R4" target="_self">on</a> <font color="Maroon">K</font>
 <b>by </b><i><a class="txt" href="incsp_1.html#E1:28_1_1_4"><i><font color="Green">E24</font></i></a></i>;<br/><b>consider </b><font color="Maroon">L</font> being   <a href="incsp_1.html#NM2">LINE</a> of <a href="incsp_1.html#G2" target="_self">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>,<font color="Maroon">Li</font>,<font color="Maroon">Pl</font>,<font color="Maroon">i1</font>,<font color="Maroon">i2</font>,<font color="Maroon">i3</font> #)<b> such that </b><br/><a NAME="E5:28_1_1_4_9_1"/><i><font color="Green">E113</font></i>: 
<span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">C</font></span><a href="domain_1.html#K7">}</a></span> <a href="incsp_1.html#R4" target="_self">on</a> <font color="Maroon">L</font>
 <b>by </b><i><a class="txt" href="incsp_1.html#E1:28_1_1_4"><i><font color="Green">E24</font></i></a></i>;<br/><a NAME="E6:28_1_1_4_9_1"/><i><font color="Green">E114</font></i>: 
( <font color="Maroon">A</font> <a href="incsp_1.html#R1" target="_self">on</a> <font color="Maroon">K</font> &amp; <font color="Maroon">B</font> <a href="incsp_1.html#R1" target="_self">on</a> <font color="Maroon">K</font> &amp; <font color="Maroon">A</font> <a href="incsp_1.html#R1" target="_self">on</a> <font color="Maroon">L</font> &amp; <font color="Maroon">C</font> <a href="incsp_1.html#R1" target="_self">on</a> <font color="Maroon">L</font> )
 <b>by </b><i><a class="ref" href="incsp_1.html#T16" target="_self">Th16</a>, <a class="ref" href="incsp_1.html#T17" target="_self">Th17</a>, </i>;<br/><a NAME="E7:28_1_1_4_9_1"/>
( not <span class="p1"><a href="domain_1.html#K8">{</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font>,<font color="Maroon">C</font></span><a href="domain_1.html#K8">}</a></span> <a href="incsp_1.html#R4" target="_self">on</a> <font color="Maroon">K</font> &amp; not <span class="p1"><a href="domain_1.html#K8">{</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font>,<font color="Maroon">C</font></span><a href="domain_1.html#K8">}</a></span> <a href="incsp_1.html#R4" target="_self">on</a> <font color="Maroon">L</font> )
 <b>by </b><i><a class="ref" href="incsp_1.html#T12" target="_self">Th12</a>, </i>;<br/><b>hence </b><a NAME="E8:28_1_1_4_9_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="incsp_1.html#T18" target="_self">Th18</a>, </i>;<br/></div><b>end;</b></div>
<a NAME="E6:28_1_1_4_9"/><i><font color="Green">E115</font></i>: 
( <font color="Maroon">A</font> <a href="incsp_1.html#R2" target="_self">on</a> <font color="Maroon">P</font> &amp; <font color="Maroon">B</font> <a href="incsp_1.html#R2" target="_self">on</a> <font color="Maroon">P</font> &amp; <font color="Maroon">C</font> <a href="incsp_1.html#R2" target="_self">on</a> <font color="Maroon">P</font> )
 
<b>by </b><i><a class="ref" href="incsp_1.html#T13" target="_self">Th13</a>, </i>;<br/>
<b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">A</font>, <font color="Maroon">b</font> = <font color="Maroon">B</font>, <font color="Maroon">c</font> = <font color="Maroon">C</font> as    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span> ;<br/>
<a NAME="E8:28_1_1_4_9"/>
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">p</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">i2</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">p</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">i2</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">p</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">i2</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E9:28_1_1_4_9"/><b>then </b><i><font color="Green">E116</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">p</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">p</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">p</font> )
 
<b>by </b><i><a class="ref" href="incsp_1.html#T11" target="_self">Th11</a></i>;<br/>
<a NAME="E10:28_1_1_4_9"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Pl</font>
 
;<br/>
<b>then consider </b><font color="Maroon">x1</font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>, <font color="Maroon">x2</font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>, <font color="Maroon">x3</font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span><b> such that </b><br/><a NAME="E12:28_1_1_4_9"/><i><font color="Green">E117</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K8">{</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font>,<font color="Maroon">x3</font></span><a href="domain_1.html#K8">}</a></span>
 <b>and </b><br/><a NAME="E13:28_1_1_4_9"/>
( <font color="Maroon">x1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x2</font> &amp; <font color="Maroon">x1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x3</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x3</font> )
 ;<br/>
<a NAME="E14:28_1_1_4_9"/><i><font color="Green">E118</font></i>: 
( ( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> or <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font> or <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x3</font> ) &amp; ( <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> or <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font> or <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x3</font> ) &amp; ( <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> or <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font> or <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x3</font> ) )
 
<b>by </b><i><a class="ref" href="incsp_1.html#T16" target="_self">Th16</a>, <a class="ref" href="incsp_1.html#T17" target="_self">Th17</a>, <a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/>
<a NAME="E15:28_1_1_4_9"/>
( <font color="Maroon">A</font> <a href="incsp_1.html#R2" target="_self">on</a> <font color="Maroon">Q</font> &amp; <font color="Maroon">B</font> <a href="incsp_1.html#R2" target="_self">on</a> <font color="Maroon">Q</font> &amp; <font color="Maroon">C</font> <a href="incsp_1.html#R2" target="_self">on</a> <font color="Maroon">Q</font> )
 
<b>by </b><i><a class="ref" href="incsp_1.html#T14" target="_self">Th14</a>, </i>;<br/>
<a NAME="E16:28_1_1_4_9"/><b>then </b>
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">q</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">i2</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">q</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">i2</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">q</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">i2</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E17:28_1_1_4_9"/><b>then </b><i><font color="Green">E119</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">q</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">q</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">q</font> )
 
<b>by </b><i><a class="ref" href="incsp_1.html#T11" target="_self">Th11</a></i>;<br/>
<a NAME="E18:28_1_1_4_9"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Pl</font>
 
;<br/>
<b>then consider </b><font color="Maroon">x1</font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>, <font color="Maroon">x2</font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>, <font color="Maroon">x3</font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span><b> such that </b><br/><a NAME="E20:28_1_1_4_9"/><i><font color="Green">E120</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K8">{</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font>,<font color="Maroon">x3</font></span><a href="domain_1.html#K8">}</a></span>
 <b>and </b><br/><a NAME="E21:28_1_1_4_9"/>
( <font color="Maroon">x1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x2</font> &amp; <font color="Maroon">x1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x3</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x3</font> )
 ;<br/>
<a NAME="E22:28_1_1_4_9"/>
( ( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> or <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font> or <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x3</font> ) &amp; ( <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> or <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font> or <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x3</font> ) &amp; ( <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> or <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font> or <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x3</font> ) )
 
<b>by </b><i><a class="ref" href="incsp_1.html#T19" target="_self">Th19</a>, <a class="ref" href="incsp_1.html#T20" target="_self">Th20</a>, <a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E23:28_1_1_4_9"/>
<font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Q</font>
 <b>by </b><i><a class="ref" href="incsp_1.html#T15" target="_self">Th15</a>, <a class="ref" href="incsp_1.html#T17" target="_self">Th17</a>, <a class="ref" href="incsp_1.html#T18" target="_self">Th18</a>, <a class="ref" href="incsp_1.html#T20" target="_self">Th20</a>, <a class="ref" href="enumset1.html#T98">ENUMSET1:98</a>, <a class="ref" href="enumset1.html#T99">ENUMSET1:99</a>, <a class="ref" href="enumset1.html#T100">ENUMSET1:100</a>, <a class="ref" href="enumset1.html#T102">ENUMSET1:102</a></i>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E10:28_1_1_4"/>
for <font color="Olive">L</font> being  <a href="incsp_1.html#NM2">LINE</a> of <a href="incsp_1.html#G2" target="_self">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>,<font color="Maroon">Li</font>,<font color="Maroon">Pl</font>,<font color="Maroon">i1</font>,<font color="Maroon">i2</font>,<font color="Maroon">i3</font> #)<br/> for <font color="Olive">P</font> being  <a href="incsp_1.html#NM3">PLANE</a> of <a href="incsp_1.html#G2" target="_self">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>,<font color="Maroon">Li</font>,<font color="Maroon">Pl</font>,<font color="Maroon">i1</font>,<font color="Maroon">i2</font>,<font color="Maroon">i3</font> #)  st  ex <font color="Olive">A</font>, <font color="Olive">B</font> being  <a href="incsp_1.html#NM1">POINT</a> of <a href="incsp_1.html#G2" target="_self">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>,<font color="Maroon">Li</font>,<font color="Maroon">Pl</font>,<font color="Maroon">i1</font>,<font color="Maroon">i2</font>,<font color="Maroon">i3</font> #) st <br/>( <font color="Olive">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">B</font> &amp; <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Olive">A</font>,<font color="Olive">B</font></span><a href="domain_1.html#K7">}</a></span> <a href="incsp_1.html#R4" target="_self">on</a> <font color="Olive">L</font> &amp; <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Olive">A</font>,<font color="Olive">B</font></span><a href="domain_1.html#K7">}</a></span> <a href="incsp_1.html#R5" target="_self">on</a> <font color="Olive">P</font> ) holds <br/><font color="Olive">L</font> <a href="incsp_1.html#R3" target="_self">on</a> <font color="Olive">P</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_4_10"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">L</font> be   <a href="incsp_1.html#NM2">LINE</a> of <a href="incsp_1.html#G2" target="_self">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>,<font color="Maroon">Li</font>,<font color="Maroon">Pl</font>,<font color="Maroon">i1</font>,<font color="Maroon">i2</font>,<font color="Maroon">i3</font> #);<br/><b>let </b><font color="Maroon">P</font> be   <a href="incsp_1.html#NM3">PLANE</a> of <a href="incsp_1.html#G2" target="_self">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>,<font color="Maroon">Li</font>,<font color="Maroon">Pl</font>,<font color="Maroon">i1</font>,<font color="Maroon">i2</font>,<font color="Maroon">i3</font> #);<br/>



<b>given </b><font color="Maroon">A</font> being   <a href="incsp_1.html#NM1">POINT</a> of <a href="incsp_1.html#G2" target="_self">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>,<font color="Maroon">Li</font>,<font color="Maroon">Pl</font>,<font color="Maroon">i1</font>,<font color="Maroon">i2</font>,<font color="Maroon">i3</font> #), <font color="Maroon">B</font> being   <a href="incsp_1.html#NM1">POINT</a> of <a href="incsp_1.html#G2" target="_self">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>,<font color="Maroon">Li</font>,<font color="Maroon">Pl</font>,<font color="Maroon">i1</font>,<font color="Maroon">i2</font>,<font color="Maroon">i3</font> #)<b> such that </b><a NAME="E1:28_1_1_4_10"/><i><font color="Green">E121</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">B</font>
 <b>and </b><br/><a NAME="E2:28_1_1_4_10"/><i><font color="Green">E122</font></i>: 
<span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="domain_1.html#K7">}</a></span> <a href="incsp_1.html#R4" target="_self">on</a> <font color="Maroon">L</font>
 <b>and </b><br/><a NAME="E3:28_1_1_4_10"/><i><font color="Green">E123</font></i>: 
<span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="domain_1.html#K7">}</a></span> <a href="incsp_1.html#R5" target="_self">on</a> <font color="Maroon">P</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">l</font> = <font color="Maroon">L</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Li</font> ;<br/>
<b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">P</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Pl</font> ;<br/>
<b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">A</font>, <font color="Maroon">b</font> = <font color="Maroon">B</font> as    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span> ;<br/>
<a NAME="E7:28_1_1_4_10"/>
( <font color="Maroon">A</font> <a href="incsp_1.html#R1" target="_self">on</a> <font color="Maroon">L</font> &amp; <font color="Maroon">B</font> <a href="incsp_1.html#R1" target="_self">on</a> <font color="Maroon">L</font> )
 
<b>by </b><i><a class="ref" href="incsp_1.html#T13" target="_self">Th13</a>, </i>;<br/>
<a NAME="E8:28_1_1_4_10"/><b>then </b>
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">l</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">i1</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">l</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">i1</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E9:28_1_1_4_10"/><b>then </b><i><font color="Green">E124</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">l</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">l</font> )
 
<b>by </b><i><a class="ref" href="incsp_1.html#D7" target="_self">Def7</a></i>;<br/>
<a NAME="E10:28_1_1_4_10"/>
( <font color="Maroon">A</font> <a href="incsp_1.html#R2" target="_self">on</a> <font color="Maroon">P</font> &amp; <font color="Maroon">B</font> <a href="incsp_1.html#R2" target="_self">on</a> <font color="Maroon">P</font> )
 
<b>by </b><i><a class="ref" href="incsp_1.html#T14" target="_self">Th14</a>, </i>;<br/>
<a NAME="E11:28_1_1_4_10"/><b>then </b>
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">p</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">i2</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">p</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">i2</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E12:28_1_1_4_10"/><b>then </b><i><font color="Green">E125</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">p</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">p</font> )
 
<b>by </b><i><a class="ref" href="incsp_1.html#T11" target="_self">Th11</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_4_10_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:28_1_1_4_10_1"/><i><font color="Green">E126</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">l</font>
 ;<br/><a NAME="E2:28_1_1_4_10_1"/>
<font color="Maroon">l</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Li</font>
 ;<br/><b>then consider </b><font color="Maroon">x1</font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>, <font color="Maroon">x2</font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span><b> such that </b><br/><a NAME="E4:28_1_1_4_10_1"/><i><font color="Green">E127</font></i>: 
<font color="Maroon">l</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="domain_1.html#K7">}</a></span>
 <b>and </b><br/><a NAME="E5:28_1_1_4_10_1"/>
<font color="Maroon">x1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x2</font>
 ;<br/><a NAME="E6:28_1_1_4_10_1"/>
( ( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> or <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font> ) &amp; ( <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> or <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font> ) )
 <b>by </b><i><a class="ref" href="incsp_1.html#T15" target="_self">Th15</a>, <a class="ref" href="incsp_1.html#T17" target="_self">Th17</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><b>hence </b><a NAME="E7:28_1_1_4_10_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="incsp_1.html#T12" target="_self">Th12</a>, , <a class="ref" href="incsp_1.html#T16" target="_self">Th16</a>, <a class="ref" href="incsp_1.html#T17" target="_self">Th17</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div>
<a NAME="E14:28_1_1_4_10"/><b>then </b>
<font color="Maroon">l</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E15:28_1_1_4_10"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">l</font>,<font color="Maroon">p</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">i3</font>
 
;<br/>
<b>hence </b><a NAME="E16:28_1_1_4_10"/>
<font color="Maroon">L</font> <a href="incsp_1.html#R3" target="_self">on</a> <font color="Maroon">P</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E11:28_1_1_4"/>
for <font color="Olive">A</font> being  <a href="incsp_1.html#NM1">POINT</a> of <a href="incsp_1.html#G2" target="_self">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>,<font color="Maroon">Li</font>,<font color="Maroon">Pl</font>,<font color="Maroon">i1</font>,<font color="Maroon">i2</font>,<font color="Maroon">i3</font> #)<br/> for <font color="Olive">P</font>, <font color="Olive">Q</font> being  <a href="incsp_1.html#NM3">PLANE</a> of <a href="incsp_1.html#G2" target="_self">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>,<font color="Maroon">Li</font>,<font color="Maroon">Pl</font>,<font color="Maroon">i1</font>,<font color="Maroon">i2</font>,<font color="Maroon">i3</font> #)  st <font color="Olive">A</font> <a href="incsp_1.html#R2" target="_self">on</a> <font color="Olive">P</font> &amp; <font color="Olive">A</font> <a href="incsp_1.html#R2" target="_self">on</a> <font color="Olive">Q</font> holds <br/> ex <font color="Olive">B</font> being  <a href="incsp_1.html#NM1">POINT</a> of <a href="incsp_1.html#G2" target="_self">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>,<font color="Maroon">Li</font>,<font color="Maroon">Pl</font>,<font color="Maroon">i1</font>,<font color="Maroon">i2</font>,<font color="Maroon">i3</font> #) st <br/>( <font color="Olive">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">B</font> &amp; <font color="Olive">B</font> <a href="incsp_1.html#R2" target="_self">on</a> <font color="Olive">P</font> &amp; <font color="Olive">B</font> <a href="incsp_1.html#R2" target="_self">on</a> <font color="Olive">Q</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_4_11"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">A</font> be   <a href="incsp_1.html#NM1">POINT</a> of <a href="incsp_1.html#G2" target="_self">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>,<font color="Maroon">Li</font>,<font color="Maroon">Pl</font>,<font color="Maroon">i1</font>,<font color="Maroon">i2</font>,<font color="Maroon">i3</font> #);<br/><b>let </b><font color="Maroon">P</font> be   <a href="incsp_1.html#NM3">PLANE</a> of <a href="incsp_1.html#G2" target="_self">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>,<font color="Maroon">Li</font>,<font color="Maroon">Pl</font>,<font color="Maroon">i1</font>,<font color="Maroon">i2</font>,<font color="Maroon">i3</font> #), <font color="Maroon">Q</font> be   <a href="incsp_1.html#NM3">PLANE</a> of <a href="incsp_1.html#G2" target="_self">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>,<font color="Maroon">Li</font>,<font color="Maroon">Pl</font>,<font color="Maroon">i1</font>,<font color="Maroon">i2</font>,<font color="Maroon">i3</font> #);<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:28_1_1_4_11"/><i><font color="Green">E128</font></i>: 
<font color="Maroon">A</font> <a href="incsp_1.html#R2" target="_self">on</a> <font color="Maroon">P</font>
 <b>and </b><br/><a NAME="E2:28_1_1_4_11"/><i><font color="Green">E129</font></i>: 
<font color="Maroon">A</font> <a href="incsp_1.html#R2" target="_self">on</a> <font color="Maroon">Q</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">A</font> as    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span> ;<br/>
<b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">P</font>, <font color="Maroon">q</font> = <font color="Maroon">Q</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Pl</font> ;<br/>
<a NAME="E5:28_1_1_4_11"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Pl</font>
 
;<br/>
<b>then consider </b><font color="Maroon">x1</font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>, <font color="Maroon">x2</font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>, <font color="Maroon">x3</font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span><b> such that </b><br/><a NAME="E7:28_1_1_4_11"/><i><font color="Green">E130</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K8">{</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font>,<font color="Maroon">x3</font></span><a href="domain_1.html#K8">}</a></span>
 <b>and </b><br/><a NAME="E8:28_1_1_4_11"/><i><font color="Green">E131</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x2</font> &amp; <font color="Maroon">x1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x3</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x3</font> )
 ;<br/>
<a NAME="E9:28_1_1_4_11"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Pl</font>
 
;<br/>
<b>then consider </b><font color="Maroon">y1</font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>, <font color="Maroon">y2</font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>, <font color="Maroon">y3</font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span><b> such that </b><br/><a NAME="E11:28_1_1_4_11"/><i><font color="Green">E135</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K8">{</a><span class="default"><font color="Maroon">y1</font>,<font color="Maroon">y2</font>,<font color="Maroon">y3</font></span><a href="domain_1.html#K8">}</a></span>
 <b>and </b><br/><a NAME="E12:28_1_1_4_11"/><i><font color="Green">E136</font></i>: 
( <font color="Maroon">y1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y2</font> &amp; <font color="Maroon">y1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y3</font> &amp; <font color="Maroon">y2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y3</font> )
 ;<br/>
<a NAME="E13:28_1_1_4_11"/><i><font color="Green">E137</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">p</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">p</font> &amp; <font color="Maroon">x3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">p</font> )
 
<b>by </b><i><a class="ref" href="incsp_1.html#T14" target="_self">Th14</a>, <a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/>
<a NAME="E14:28_1_1_4_11"/><i><font color="Green">E138</font></i>: 
( <font color="Maroon">y1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">q</font> &amp; <font color="Maroon">y2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">q</font> &amp; <font color="Maroon">y3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">q</font> )
 
<b>by </b><i>, <a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/>
<a NAME="E15:28_1_1_4_11"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">p</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">i2</font>
 
<b>by </b><i><a class="ref" href="incsp_1.html#T12" target="_self">Th12</a>, </i>;<br/>
<a NAME="E16:28_1_1_4_11"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="incsp_1.html#T11" target="_self">Th11</a></i>;<br/>
<a NAME="E17:28_1_1_4_11"/><b>then </b>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> or <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font> or <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x3</font> )
 
<b>by </b><i><a class="ref" href="incsp_1.html#T14" target="_self">Th14</a>, <a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">z1</font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>, <font color="Maroon">z2</font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span><b> such that </b><br/><a NAME="E19:28_1_1_4_11"/><i><font color="Green">E141</font></i>: 
( <font color="Maroon">z1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">p</font> &amp; <font color="Maroon">z2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">p</font> )
 <b>and </b><br/><a NAME="E20:28_1_1_4_11"/><i><font color="Green">E142</font></i>: 
( <font color="Maroon">z1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font> &amp; <font color="Maroon">z2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font> )
 <b>and </b><br/><a NAME="E21:28_1_1_4_11"/><i><font color="Green">E143</font></i>: 
<font color="Maroon">z1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">z2</font>
 <b>by </b><i><a class="ref" href="incsp_1.html#T15" target="_self">Th15</a>, <a class="ref" href="incsp_1.html#T17" target="_self">Th17</a></i>;<br/>
<a NAME="E22:28_1_1_4_11"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">q</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">i2</font>
 
<b>by </b><i><a class="ref" href="incsp_1.html#T13" target="_self">Th13</a>, </i>;<br/>
<a NAME="E23:28_1_1_4_11"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">q</font>
 
<b>by </b><i><a class="ref" href="incsp_1.html#T11" target="_self">Th11</a></i>;<br/>
<a NAME="E24:28_1_1_4_11"/><b>then </b>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y1</font> or <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y2</font> or <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y3</font> )
 
<b>by </b><i>, <a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">z3</font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>, <font color="Maroon">z4</font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span><b> such that </b><br/><a NAME="E26:28_1_1_4_11"/><i><font color="Green">E146</font></i>: 
( <font color="Maroon">z3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">q</font> &amp; <font color="Maroon">z4</font> <a href="hidden.html#R2">in</a> <font color="Maroon">q</font> )
 <b>and </b><br/><a NAME="E27:28_1_1_4_11"/><i><font color="Green">E147</font></i>: 
( <font color="Maroon">z3</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font> &amp; <font color="Maroon">z4</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font> )
 <b>and </b><br/><a NAME="E28:28_1_1_4_11"/><i><font color="Green">E148</font></i>: 
<font color="Maroon">z3</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">z4</font>
 <b>by </b><i><a class="ref" href="incsp_1.html#T16" target="_self">Th16</a>, <a class="ref" href="incsp_1.html#T18" target="_self">Th18</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_4_11_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:28_1_1_4_11_1"/><i><font color="Green">E149</font></i>: 
( <font color="Maroon">z1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">z3</font> &amp; <font color="Maroon">z1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">z4</font> &amp; <font color="Maroon">z2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">z3</font> &amp; <font color="Maroon">z2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">z4</font> )
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_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="E1:28_1_1_4_11_1_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 2 or <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 3 )
 </span><b>by </b><i><a class="ref" href="enumset1.html#D2">ENUMSET1:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1_4_11_1_1_1"><b>suppose </b></a><a NAME="E1:28_1_1_4_11_1_1_1"/><i><font color="Green">E150</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:28_1_1_4_11_1_1_1"/><b>then </b><i><font color="Green">E151</font></i>: 
( ( <font color="Maroon">z1</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">z1</font> <a href="hidden.html#R1">=</a> 2 or <font color="Maroon">z1</font> <a href="hidden.html#R1">=</a> 3 ) &amp; ( <font color="Maroon">z2</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">z2</font> <a href="hidden.html#R1">=</a> 2 or <font color="Maroon">z2</font> <a href="hidden.html#R1">=</a> 3 ) )
 <b>by </b><i><a class="ref" href="incsp_1.html#T20" target="_self">Th20</a>, <a class="ref" href="enumset1.html#D2">ENUMSET1:def 2</a></i>;<br/><a NAME="E3:28_1_1_4_11_1_1_1"/>
( ( <font color="Maroon">z3</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">z3</font> <a href="hidden.html#R1">=</a> 2 or <font color="Maroon">z3</font> <a href="hidden.html#R1">=</a> 3 ) &amp; ( <font color="Maroon">z4</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">z4</font> <a href="hidden.html#R1">=</a> 2 or <font color="Maroon">z4</font> <a href="hidden.html#R1">=</a> 3 ) )
 <b>by </b><i>, , <a class="ref" href="enumset1.html#D2">ENUMSET1:def 2</a></i>;<br/><b>hence </b><a NAME="E4:28_1_1_4_11_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="incsp_1.html#T21" target="_self">Th21</a>, <a class="ref" href="incsp_1.html#D8" target="_self">Def8</a>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1_4_11_1_1_2"><b>suppose </b></a><a NAME="E1:28_1_1_4_11_1_1_2"/><i><font color="Green">E152</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:28_1_1_4_11_1_1_2"/><b>then </b><i><font color="Green">E153</font></i>: 
( ( <font color="Maroon">z1</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">z1</font> <a href="hidden.html#R1">=</a> 2 or <font color="Maroon">z1</font> <a href="hidden.html#R1">=</a> 3 ) &amp; ( <font color="Maroon">z2</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">z2</font> <a href="hidden.html#R1">=</a> 2 or <font color="Maroon">z2</font> <a href="hidden.html#R1">=</a> 3 ) )
 <b>by </b><i><a class="ref" href="incsp_1.html#T20" target="_self">Th20</a>, <a class="ref" href="enumset1.html#D2">ENUMSET1:def 2</a></i>;<br/><a NAME="E3:28_1_1_4_11_1_1_2"/>
( ( <font color="Maroon">z3</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">z3</font> <a href="hidden.html#R1">=</a> 2 or <font color="Maroon">z3</font> <a href="hidden.html#R1">=</a> 3 ) &amp; ( <font color="Maroon">z4</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">z4</font> <a href="hidden.html#R1">=</a> 2 or <font color="Maroon">z4</font> <a href="hidden.html#R1">=</a> 3 ) )
 <b>by </b><i>, , <a class="ref" href="enumset1.html#D2">ENUMSET1:def 2</a></i>;<br/><b>hence </b><a NAME="E4:28_1_1_4_11_1_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="incsp_1.html#T21" target="_self">Th21</a>, <a class="ref" href="incsp_1.html#D8" target="_self">Def8</a>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1_4_11_1_1_3"><b>suppose </b></a><a NAME="E1:28_1_1_4_11_1_1_3"/><i><font color="Green">E154</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 2
 ;<br/><div class="add"><a NAME="E2:28_1_1_4_11_1_1_3"/><b>then </b><i><font color="Green">E155</font></i>: 
( ( <font color="Maroon">z1</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">z1</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">z1</font> <a href="hidden.html#R1">=</a> 3 ) &amp; ( <font color="Maroon">z2</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">z2</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">z2</font> <a href="hidden.html#R1">=</a> 3 ) )
 <b>by </b><i><a class="ref" href="incsp_1.html#T20" target="_self">Th20</a>, <a class="ref" href="enumset1.html#D2">ENUMSET1:def 2</a></i>;<br/><a NAME="E3:28_1_1_4_11_1_1_3"/>
( ( <font color="Maroon">z3</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">z3</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">z3</font> <a href="hidden.html#R1">=</a> 3 ) &amp; ( <font color="Maroon">z4</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">z4</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">z4</font> <a href="hidden.html#R1">=</a> 3 ) )
 <b>by </b><i>, , <a class="ref" href="enumset1.html#D2">ENUMSET1:def 2</a></i>;<br/><b>hence </b><a NAME="E4:28_1_1_4_11_1_1_3"/>
contradiction
 <b>by </b><i><a class="ref" href="incsp_1.html#T21" target="_self">Th21</a>, <a class="ref" href="incsp_1.html#D8" target="_self">Def8</a>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1_4_11_1_1_4"><b>suppose </b></a><a NAME="E1:28_1_1_4_11_1_1_4"/><i><font color="Green">E156</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 3
 ;<br/><div class="add"><a NAME="E2:28_1_1_4_11_1_1_4"/><b>then </b><i><font color="Green">E157</font></i>: 
( ( <font color="Maroon">z1</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">z1</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">z1</font> <a href="hidden.html#R1">=</a> 2 ) &amp; ( <font color="Maroon">z2</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">z2</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">z2</font> <a href="hidden.html#R1">=</a> 2 ) )
 <b>by </b><i><a class="ref" href="incsp_1.html#T20" target="_self">Th20</a>, <a class="ref" href="enumset1.html#D2">ENUMSET1:def 2</a></i>;<br/><a NAME="E3:28_1_1_4_11_1_1_4"/>
( ( <font color="Maroon">z3</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">z3</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">z3</font> <a href="hidden.html#R1">=</a> 2 ) &amp; ( <font color="Maroon">z4</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">z4</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">z4</font> <a href="hidden.html#R1">=</a> 2 ) )
 <b>by </b><i>, , <a class="ref" href="enumset1.html#D2">ENUMSET1:def 2</a></i>;<br/><b>hence </b><a NAME="E4:28_1_1_4_11_1_1_4"/>
contradiction
 <b>by </b><i><a class="ref" href="incsp_1.html#T21" target="_self">Th21</a>, <a class="ref" href="incsp_1.html#D8" target="_self">Def8</a>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>then consider </b><font color="Maroon">z</font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span><b> such that </b><br/><a NAME="E31:28_1_1_4_11"/><i><font color="Green">E158</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">p</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">q</font> )
 <b>and </b><br/><a NAME="E32:28_1_1_4_11"/><i><font color="Green">E159</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="ref" href="incsp_1.html#T19" target="_self">Th19</a>, <a class="ref" href="incsp_1.html#T20" target="_self">Th20</a>, </i>;<br/>
<b>reconsider </b><font color="Maroon">B</font> = <font color="Maroon">z</font> as   <a href="incsp_1.html#NM1">POINT</a> of <a href="incsp_1.html#G2" target="_self">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>,<font color="Maroon">Li</font>,<font color="Maroon">Pl</font>,<font color="Maroon">i1</font>,<font color="Maroon">i2</font>,<font color="Maroon">i3</font> #) ;<br/>
<b>take </b>
<font color="Maroon">B</font>
;<br/>

<b>thus </b><a NAME="E34:28_1_1_4_11"/>
<font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">B</font>
 <b>by </b><i/>;<br/>

<a NAME="E35:28_1_1_4_11"/>
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">p</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">i2</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">q</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">i2</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E36:28_1_1_4_11"/>
( <font color="Maroon">B</font> <a href="incsp_1.html#R2" target="_self">on</a> <font color="Maroon">P</font> &amp; <font color="Maroon">B</font> <a href="incsp_1.html#R2" target="_self">on</a> <font color="Maroon">Q</font> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E12:28_1_1_4"/>
not for <font color="Olive">A</font>, <font color="Olive">B</font>, <font color="Olive">C</font>, <font color="Olive">D</font> being  <a href="incsp_1.html#NM1">POINT</a> of <a href="incsp_1.html#G2" target="_self">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>,<font color="Maroon">Li</font>,<font color="Maroon">Pl</font>,<font color="Maroon">i1</font>,<font color="Maroon">i2</font>,<font color="Maroon">i3</font> #) holds  <span class="p1"><a href="domain_1.html#K9">{</a><span class="default"><font color="Olive">A</font>,<font color="Olive">B</font>,<font color="Olive">C</font>,<font color="Olive">D</font></span><a href="domain_1.html#K9">}</a></span> <a href="incsp_1.html#NR7" target="_self">is_coplanar</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_4_12"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">Three</font> = 3 as    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span> <b>by </b><i><a class="ref" href="enumset1.html#D2">ENUMSET1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">A</font> = <font color="Maroon">Zero1</font>, <font color="Maroon">B</font> = <font color="Maroon">One</font>, <font color="Maroon">C</font> = <font color="Maroon">Two</font>, <font color="Maroon">D</font> = <font color="Maroon">Three</font> as   <a href="incsp_1.html#NM1">POINT</a> of <a href="incsp_1.html#G2" target="_self">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>,<font color="Maroon">Li</font>,<font color="Maroon">Pl</font>,<font color="Maroon">i1</font>,<font color="Maroon">i2</font>,<font color="Maroon">i3</font> #) ;<br/>
<b>take </b>
<font color="Maroon">A</font>
;<br/>

<b>take </b>
<font color="Maroon">B</font>
;<br/>

<b>take </b>
<font color="Maroon">C</font>
;<br/>

<b>take </b>
<font color="Maroon">D</font>
;<br/>

<b>assume </b><a NAME="E3:28_1_1_4_12"/>
<span class="p1"><a href="domain_1.html#K9">{</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font>,<font color="Maroon">C</font>,<font color="Maroon">D</font></span><a href="domain_1.html#K9">}</a></span> <a href="incsp_1.html#NR7" target="_self">is_coplanar</a> 
 ;<br/>

<b>then consider </b><font color="Maroon">P</font> being   <a href="incsp_1.html#NM3">PLANE</a> of <a href="incsp_1.html#G2" target="_self">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>,<font color="Maroon">Li</font>,<font color="Maroon">Pl</font>,<font color="Maroon">i1</font>,<font color="Maroon">i2</font>,<font color="Maroon">i3</font> #)<b> such that </b><br/><a NAME="E5:28_1_1_4_12"/><i><font color="Green">E160</font></i>: 
<span class="p1"><a href="domain_1.html#K9">{</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font>,<font color="Maroon">C</font>,<font color="Maroon">D</font></span><a href="domain_1.html#K9">}</a></span> <a href="incsp_1.html#R5" target="_self">on</a> <font color="Maroon">P</font>
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">P</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Pl</font> ;<br/>
<a NAME="E7:28_1_1_4_12"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Pl</font>
 
;<br/>
<b>then consider </b><font color="Maroon">a</font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>, <font color="Maroon">b</font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>, <font color="Maroon">c</font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span><b> such that </b><br/><a NAME="E9:28_1_1_4_12"/><i><font color="Green">E161</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K8">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font></span><a href="domain_1.html#K8">}</a></span>
 <b>and </b><br/><a NAME="E10:28_1_1_4_12"/>
( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c</font> )
 ;<br/>
<a NAME="E11:28_1_1_4_12"/>
( <font color="Maroon">A</font> <a href="incsp_1.html#R2" target="_self">on</a> <font color="Maroon">P</font> &amp; <font color="Maroon">B</font> <a href="incsp_1.html#R2" target="_self">on</a> <font color="Maroon">P</font> &amp; <font color="Maroon">C</font> <a href="incsp_1.html#R2" target="_self">on</a> <font color="Maroon">P</font> &amp; <font color="Maroon">D</font> <a href="incsp_1.html#R2" target="_self">on</a> <font color="Maroon">P</font> )
 
<b>by </b><i><a class="ref" href="incsp_1.html#T12" target="_self">Th12</a>, </i>;<br/>
<a NAME="E12:28_1_1_4_12"/><b>then </b>
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">Zero1</font>,<font color="Maroon">p</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">i2</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">One</font>,<font color="Maroon">p</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">i2</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">Two</font>,<font color="Maroon">p</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">i2</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">Three</font>,<font color="Maroon">p</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">i2</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E13:28_1_1_4_12"/><b>then </b>
( <font color="Maroon">Zero1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">p</font> &amp; <font color="Maroon">One</font> <a href="hidden.html#R2">in</a> <font color="Maroon">p</font> &amp; <font color="Maroon">Two</font> <a href="hidden.html#R2">in</a> <font color="Maroon">p</font> &amp; <font color="Maroon">Three</font> <a href="hidden.html#R2">in</a> <font color="Maroon">p</font> )
 
<b>by </b><i><a class="ref" href="incsp_1.html#T11" target="_self">Th11</a></i>;<br/>
<a NAME="E14:28_1_1_4_12"/><b>then </b>
( ( <font color="Maroon">Zero1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> or <font color="Maroon">Zero1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> or <font color="Maroon">Zero1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> ) &amp; ( <font color="Maroon">One</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> or <font color="Maroon">One</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> or <font color="Maroon">One</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> ) &amp; ( <font color="Maroon">Two</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> or <font color="Maroon">Two</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> or <font color="Maroon">Two</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> ) &amp; ( <font color="Maroon">Three</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> or <font color="Maroon">Three</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> or <font color="Maroon">Three</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> ) )
 
<b>by </b><i><a class="ref" href="incsp_1.html#T13" target="_self">Th13</a>, <a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E15:28_1_1_4_12"/>
contradiction
 ;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E13:28_1_1_4"/>
for <font color="Olive">A</font> being  <a href="incsp_1.html#NM1">POINT</a> of <a href="incsp_1.html#G2" target="_self">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>,<font color="Maroon">Li</font>,<font color="Maroon">Pl</font>,<font color="Maroon">i1</font>,<font color="Maroon">i2</font>,<font color="Maroon">i3</font> #)<br/> for <font color="Olive">L</font> being  <a href="incsp_1.html#NM2">LINE</a> of <a href="incsp_1.html#G2" target="_self">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>,<font color="Maroon">Li</font>,<font color="Maroon">Pl</font>,<font color="Maroon">i1</font>,<font color="Maroon">i2</font>,<font color="Maroon">i3</font> #)<br/> for <font color="Olive">P</font> being  <a href="incsp_1.html#NM3">PLANE</a> of <a href="incsp_1.html#G2" target="_self">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>,<font color="Maroon">Li</font>,<font color="Maroon">Pl</font>,<font color="Maroon">i1</font>,<font color="Maroon">i2</font>,<font color="Maroon">i3</font> #)  st <font color="Olive">A</font> <a href="incsp_1.html#R1" target="_self">on</a> <font color="Olive">L</font> &amp; <font color="Olive">L</font> <a href="incsp_1.html#R3" target="_self">on</a> <font color="Olive">P</font> holds <br/><font color="Olive">A</font> <a href="incsp_1.html#R2" target="_self">on</a> <font color="Olive">P</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_4_13"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">A</font> be   <a href="incsp_1.html#NM1">POINT</a> of <a href="incsp_1.html#G2" target="_self">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>,<font color="Maroon">Li</font>,<font color="Maroon">Pl</font>,<font color="Maroon">i1</font>,<font color="Maroon">i2</font>,<font color="Maroon">i3</font> #);<br/><b>let </b><font color="Maroon">L</font> be   <a href="incsp_1.html#NM2">LINE</a> of <a href="incsp_1.html#G2" target="_self">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>,<font color="Maroon">Li</font>,<font color="Maroon">Pl</font>,<font color="Maroon">i1</font>,<font color="Maroon">i2</font>,<font color="Maroon">i3</font> #);<br/><b>let </b><font color="Maroon">P</font> be   <a href="incsp_1.html#NM3">PLANE</a> of <a href="incsp_1.html#G2" target="_self">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>,<font color="Maroon">Li</font>,<font color="Maroon">Pl</font>,<font color="Maroon">i1</font>,<font color="Maroon">i2</font>,<font color="Maroon">i3</font> #);<br/>





<b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">A</font> as    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span> ;<br/>
<b>reconsider </b><font color="Maroon">l</font> = <font color="Maroon">L</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Li</font> ;<br/>
<b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">P</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Pl</font> ;<br/>
<b>assume </b><a NAME="E4:28_1_1_4_13"/>
( <font color="Maroon">A</font> <a href="incsp_1.html#R1" target="_self">on</a> <font color="Maroon">L</font> &amp; <font color="Maroon">L</font> <a href="incsp_1.html#R3" target="_self">on</a> <font color="Maroon">P</font> )
 ;<br/>

<a NAME="E5:28_1_1_4_13"/><b>then </b>
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">l</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">i1</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">l</font>,<font color="Maroon">p</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">i3</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E6:28_1_1_4_13"/><b>then </b>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">l</font> &amp; <font color="Maroon">l</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">p</font> )
 
<b>by </b><i><a class="ref" href="incsp_1.html#D7" target="_self">Def7</a>, <a class="txt" href="incsp_1.html#E12:28_1_1"><i><font color="Green">E23</font></i></a></i>;<br/>
<a NAME="E7:28_1_1_4_13"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">p</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">i2</font>
 
;<br/>
<b>hence </b><a NAME="E8:28_1_1_4_13"/>
<font color="Maroon">A</font> <a href="incsp_1.html#R2" target="_self">on</a> <font color="Maroon">P</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E13:28_1_1"/>
 <a href="incsp_1.html#G2" target="_self">IncStruct</a>(# <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>,<font color="Maroon">Li</font>,<font color="Maroon">Pl</font>,<font color="Maroon">i1</font>,<font color="Maroon">i2</font>,<font color="Maroon">i3</font> #) is <a href="incsp_1.html#V5" target="_self">IncSpace-like</a>
 
<b>by </b><i><a class="ref" href="incsp_1.html#D5" target="_self">Def5</a>, <a class="ref" href="incsp_1.html#D6" target="_self">Def6</a></i>;<br/>
<b>hence </b><a NAME="E14:28_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being   <a href="incsp_1.html#L2" target="_self">IncStruct</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="incsp_1.html#V2" target="_self">strict</a> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="incsp_1.html#V5" target="_self">IncSpace-like</a> )
 ;<br/>


</div>
