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

<b>set </b><font color="Maroon">c<sub>1</sub></font> =  <a href="necklace.html#K4">Necklace</a> 4;<br/>
<b>set </b><font color="Maroon">c<sub>2</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="necklace.html#K3">ComplRelStr</a> <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span>;<br/>
<a NAME="E1:21"/><i><font color="Green">E10</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span> <a href="hidden.html#R1">=</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>
 
<b>by </b><i><a class="ref" href="necklace.html#T2">NECKLACE:2</a>, <a class="ref" href="necklace.html#T21">NECKLACE:21</a></i>;<br/>
<b>thus </b><a NAME="E2:21"/>
the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K3">ComplRelStr</a> <span class="p2">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="enumset1.html#K4">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,2</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">2,0</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,3</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">3,0</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">1,3</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">3,1</span><a href="tarski.html#K4">]</a></span></span><a href="enumset1.html#K4">}</a></span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></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="E1:21_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K3">ComplRelStr</a> <span class="p2">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:21_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <span class="p2">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="necklace.html#D9">NECKLACE:def 9</a></i>;<br/>
<a NAME="E3:21_1"/><b>then </b><i><font color="Green">E11</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span> <a href="subset_1.html#K3">`</a>  &amp; not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E4:21_1"/><b>then </b><i><font color="Green">E12</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="subset_1.html#K6">\</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span> &amp; not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span> )
 
<b>by </b><i><a class="ref" href="subset_1.html#D5">SUBSET_1:def 5</a></i>;<br/>
<a NAME="E5:21_1"/><b>then </b>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span> &amp; not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:21_1"/><i><font color="Green">E13</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:21_1_1"/>
( ( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 0 ) or ( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 1 ) or ( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 2 ) or ( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 3 ) or ( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 0 ) or ( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 2 &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 0 ) or ( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 3 &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 0 ) or ( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 1 ) or ( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 2 ) or ( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 3 ) or ( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 2 &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 2 ) or ( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 2 &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 1 ) or ( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 2 &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 3 ) or ( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 3 &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 3 ) or ( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 3 &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 1 ) or ( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 3 &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 2 ) )
 </span><b>by </b><i><a class="txt" href="neckla_3.html#E1:21"><i><font color="Green">E10</font></i></a>, <a class="txt" href="neckla_3.html#E7:21_1"><i><font color="Green">E13</font></i></a>, <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="21_1_1_1"><b>suppose </b></a><a NAME="E1:21_1_1_1"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 0 )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:21_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="enumset1.html#K4">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,2</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">2,0</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,3</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">3,0</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">1,3</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">3,1</span><a href="tarski.html#K4">]</a></span></span><a href="enumset1.html#K4">}</a></span>
 <b>by </b><i><a class="txt" href="neckla_3.html#E3:21_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="neckla_3.html#E7:21_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="relat_1.html#D10">RELAT_1:def 10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1_2"><b>suppose </b></a><a NAME="E1:21_1_1_2"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 1 )
 ;<br/><div class="add"><a NAME="E2:21_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span>
 <b>by </b><i><a class="txt" href="neckla_3.html#E7:21_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="enumset1.html#D4">ENUMSET1:def 4</a>, <a class="ref" href="neckla_2.html#T2">NECKLA_2:2</a></i>;<br/><b>hence </b><a NAME="E3:21_1_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="enumset1.html#K4">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,2</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">2,0</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,3</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">3,0</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">1,3</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">3,1</span><a href="tarski.html#K4">]</a></span></span><a href="enumset1.html#K4">}</a></span>
 <b>by </b><i><a class="txt" href="neckla_3.html#E4:21_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1_3"><b>suppose </b></a><a NAME="E1:21_1_1_3"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 2 )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:21_1_1_3"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="enumset1.html#K4">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,2</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">2,0</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,3</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">3,0</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">1,3</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">3,1</span><a href="tarski.html#K4">]</a></span></span><a href="enumset1.html#K4">}</a></span>
 <b>by </b><i><a class="txt" href="neckla_3.html#E7:21_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="enumset1.html#D4">ENUMSET1:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1_4"><b>suppose </b></a><a NAME="E1:21_1_1_4"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 3 )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:21_1_1_4"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="enumset1.html#K4">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,2</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">2,0</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,3</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">3,0</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">1,3</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">3,1</span><a href="tarski.html#K4">]</a></span></span><a href="enumset1.html#K4">}</a></span>
 <b>by </b><i><a class="txt" href="neckla_3.html#E7:21_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="enumset1.html#D4">ENUMSET1:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1_5"><b>suppose </b></a><a NAME="E1:21_1_1_5"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 0 )
 ;<br/><div class="add"><a NAME="E2:21_1_1_5"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span>
 <b>by </b><i><a class="txt" href="neckla_3.html#E7:21_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="enumset1.html#D4">ENUMSET1:def 4</a>, <a class="ref" href="neckla_2.html#T2">NECKLA_2:2</a></i>;<br/><b>hence </b><a NAME="E3:21_1_1_5"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="enumset1.html#K4">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,2</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">2,0</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,3</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">3,0</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">1,3</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">3,1</span><a href="tarski.html#K4">]</a></span></span><a href="enumset1.html#K4">}</a></span>
 <b>by </b><i><a class="txt" href="neckla_3.html#E4:21_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1_6"><b>suppose </b></a><a NAME="E1:21_1_1_6"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 2 &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 0 )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:21_1_1_6"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="enumset1.html#K4">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,2</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">2,0</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,3</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">3,0</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">1,3</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">3,1</span><a href="tarski.html#K4">]</a></span></span><a href="enumset1.html#K4">}</a></span>
 <b>by </b><i><a class="txt" href="neckla_3.html#E7:21_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="enumset1.html#D4">ENUMSET1:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1_7"><b>suppose </b></a><a NAME="E1:21_1_1_7"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 3 &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 0 )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:21_1_1_7"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="enumset1.html#K4">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,2</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">2,0</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,3</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">3,0</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">1,3</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">3,1</span><a href="tarski.html#K4">]</a></span></span><a href="enumset1.html#K4">}</a></span>
 <b>by </b><i><a class="txt" href="neckla_3.html#E7:21_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="enumset1.html#D4">ENUMSET1:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1_8"><b>suppose </b></a><a NAME="E1:21_1_1_8"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 1 )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:21_1_1_8"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="enumset1.html#K4">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,2</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">2,0</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,3</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">3,0</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">1,3</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">3,1</span><a href="tarski.html#K4">]</a></span></span><a href="enumset1.html#K4">}</a></span>
 <b>by </b><i><a class="txt" href="neckla_3.html#E3:21_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="neckla_3.html#E7:21_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="relat_1.html#D10">RELAT_1:def 10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1_9"><b>suppose </b></a><a NAME="E1:21_1_1_9"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 2 )
 ;<br/><div class="add"><a NAME="E2:21_1_1_9"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span>
 <b>by </b><i><a class="txt" href="neckla_3.html#E7:21_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="enumset1.html#D4">ENUMSET1:def 4</a>, <a class="ref" href="neckla_2.html#T2">NECKLA_2:2</a></i>;<br/><b>hence </b><a NAME="E3:21_1_1_9"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="enumset1.html#K4">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,2</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">2,0</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,3</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">3,0</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">1,3</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">3,1</span><a href="tarski.html#K4">]</a></span></span><a href="enumset1.html#K4">}</a></span>
 <b>by </b><i><a class="txt" href="neckla_3.html#E4:21_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1_10"><b>suppose </b></a><a NAME="E1:21_1_1_10"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 3 )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:21_1_1_10"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="enumset1.html#K4">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,2</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">2,0</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,3</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">3,0</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">1,3</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">3,1</span><a href="tarski.html#K4">]</a></span></span><a href="enumset1.html#K4">}</a></span>
 <b>by </b><i><a class="txt" href="neckla_3.html#E7:21_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="enumset1.html#D4">ENUMSET1:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1_11"><b>suppose </b></a><a NAME="E1:21_1_1_11"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 2 &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 2 )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:21_1_1_11"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="enumset1.html#K4">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,2</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">2,0</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,3</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">3,0</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">1,3</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">3,1</span><a href="tarski.html#K4">]</a></span></span><a href="enumset1.html#K4">}</a></span>
 <b>by </b><i><a class="txt" href="neckla_3.html#E3:21_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="neckla_3.html#E7:21_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="relat_1.html#D10">RELAT_1:def 10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1_12"><b>suppose </b></a><a NAME="E1:21_1_1_12"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 2 &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 1 )
 ;<br/><div class="add"><a NAME="E2:21_1_1_12"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span>
 <b>by </b><i><a class="txt" href="neckla_3.html#E7:21_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="enumset1.html#D4">ENUMSET1:def 4</a>, <a class="ref" href="neckla_2.html#T2">NECKLA_2:2</a></i>;<br/><b>hence </b><a NAME="E3:21_1_1_12"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="enumset1.html#K4">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,2</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">2,0</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,3</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">3,0</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">1,3</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">3,1</span><a href="tarski.html#K4">]</a></span></span><a href="enumset1.html#K4">}</a></span>
 <b>by </b><i><a class="txt" href="neckla_3.html#E4:21_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1_13"><b>suppose </b></a><a NAME="E1:21_1_1_13"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 2 &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 3 )
 ;<br/><div class="add"><a NAME="E2:21_1_1_13"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span>
 <b>by </b><i><a class="txt" href="neckla_3.html#E7:21_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="enumset1.html#D4">ENUMSET1:def 4</a>, <a class="ref" href="neckla_2.html#T2">NECKLA_2:2</a></i>;<br/><b>hence </b><a NAME="E3:21_1_1_13"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="enumset1.html#K4">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,2</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">2,0</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,3</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">3,0</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">1,3</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">3,1</span><a href="tarski.html#K4">]</a></span></span><a href="enumset1.html#K4">}</a></span>
 <b>by </b><i><a class="txt" href="neckla_3.html#E4:21_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1_14"><b>suppose </b></a><a NAME="E1:21_1_1_14"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 3 &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 3 )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:21_1_1_14"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="enumset1.html#K4">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,2</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">2,0</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,3</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">3,0</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">1,3</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">3,1</span><a href="tarski.html#K4">]</a></span></span><a href="enumset1.html#K4">}</a></span>
 <b>by </b><i><a class="txt" href="neckla_3.html#E3:21_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="neckla_3.html#E7:21_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="relat_1.html#D10">RELAT_1:def 10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1_15"><b>suppose </b></a><a NAME="E1:21_1_1_15"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 3 &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 1 )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:21_1_1_15"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="enumset1.html#K4">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,2</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">2,0</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,3</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">3,0</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">1,3</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">3,1</span><a href="tarski.html#K4">]</a></span></span><a href="enumset1.html#K4">}</a></span>
 <b>by </b><i><a class="txt" href="neckla_3.html#E7:21_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="enumset1.html#D4">ENUMSET1:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1_16"><b>suppose </b></a><a NAME="E1:21_1_1_16"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 3 &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 2 )
 ;<br/><div class="add"><a NAME="E2:21_1_1_16"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span>
 <b>by </b><i><a class="txt" href="neckla_3.html#E7:21_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="enumset1.html#D4">ENUMSET1:def 4</a>, <a class="ref" href="neckla_2.html#T2">NECKLA_2:2</a></i>;<br/><b>hence </b><a NAME="E3:21_1_1_16"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="enumset1.html#K4">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,2</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">2,0</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,3</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">3,0</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">1,3</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">3,1</span><a href="tarski.html#K4">]</a></span></span><a href="enumset1.html#K4">}</a></span>
 <b>by </b><i><a class="txt" href="neckla_3.html#E4:21_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>

<b>let </b><font color="Maroon">c<sub>4</sub></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="E3:21"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="enumset1.html#K4">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,2</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">2,0</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,3</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">3,0</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">1,3</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">3,1</span><a href="tarski.html#K4">]</a></span></span><a href="enumset1.html#K4">}</a></span>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:21_2"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,2</span><a href="tarski.html#K4">]</a></span> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,0</span><a href="tarski.html#K4">]</a></span> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,3</span><a href="tarski.html#K4">]</a></span> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">3,0</span><a href="tarski.html#K4">]</a></span> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,3</span><a href="tarski.html#K4">]</a></span> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">3,1</span><a href="tarski.html#K4">]</a></span> )
 </span><b>by </b><i><a class="txt" href="neckla_3.html#E3:21"><i><font color="Green">E11</font></i></a>, <a class="ref" href="enumset1.html#D4">ENUMSET1:def 4</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_2_1"><b>suppose </b></a><a NAME="E1:21_2_1"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,2</span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><a NAME="E2:21_2_1"/><i><font color="Green">E13</font></i>: 
not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:21_2_1_1"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:21_2_1_1_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,1</span><a href="tarski.html#K4">]</a></span> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,0</span><a href="tarski.html#K4">]</a></span> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,2</span><a href="tarski.html#K4">]</a></span> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,1</span><a href="tarski.html#K4">]</a></span> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,3</span><a href="tarski.html#K4">]</a></span> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">3,2</span><a href="tarski.html#K4">]</a></span> )
 </span><b>by </b><i><a class="txt" href="neckla_3.html#E1:21_2_1_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="enumset1.html#D4">ENUMSET1:def 4</a>, <a class="ref" href="neckla_2.html#T2">NECKLA_2:2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:21_2_1_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,1</span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:21_2_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:21_2_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:21_2_1_1_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,0</span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:21_2_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:21_2_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_2_1_1_1_3"><b>suppose </b></a><a NAME="E1:21_2_1_1_1_3"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,2</span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:21_2_1_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:21_2_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_2_1_1_1_4"><b>suppose </b></a><a NAME="E1:21_2_1_1_1_4"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,1</span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:21_2_1_1_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:21_2_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_2_1_1_1_5"><b>suppose </b></a><a NAME="E1:21_2_1_1_1_5"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,3</span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:21_2_1_1_1_5"/>
contradiction
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:21_2_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_2_1_1_1_6"><b>suppose </b></a><a NAME="E1:21_2_1_1_1_6"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">3,2</span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:21_2_1_1_1_6"/>
contradiction
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:21_2_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E3:21_2_1"/>
( 0 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span> &amp; 2 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span> )
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:21"><i><font color="Green">E10</font></i></a>, <a class="ref" href="enumset1.html#D2">ENUMSET1:def 2</a></i>;<br/><a NAME="E4:21_2_1"/><b>then </b>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span> &amp; not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span> )
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:21_2_1"><i><font color="Green">E12</font></i></a>, <a class="txt" href="neckla_3.html#E2:21_2_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="relat_1.html#D10">RELAT_1:def 10</a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E5:21_2_1"/><b>then </b>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="subset_1.html#K6">\</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span> &amp; not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E6:21_2_1"/><b>then </b>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span> <a href="subset_1.html#K3">`</a>  &amp; not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span> )
 <b>by </b><i><a class="ref" href="subset_1.html#D5">SUBSET_1:def 5</a></i>;<br/><a NAME="E7:21_2_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <span class="p2">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E8:21_2_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K3">ComplRelStr</a> <span class="p2">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="necklace.html#D9">NECKLACE:def 9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_2_2"><b>suppose </b></a><a NAME="E1:21_2_2"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,0</span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><a NAME="E2:21_2_2"/><i><font color="Green">E13</font></i>: 
not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:21_2_2_1"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_2_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:21_2_2_1_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,1</span><a href="tarski.html#K4">]</a></span> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,0</span><a href="tarski.html#K4">]</a></span> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,2</span><a href="tarski.html#K4">]</a></span> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,1</span><a href="tarski.html#K4">]</a></span> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,3</span><a href="tarski.html#K4">]</a></span> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">3,2</span><a href="tarski.html#K4">]</a></span> )
 </span><b>by </b><i><a class="txt" href="neckla_3.html#E1:21_2_2_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="enumset1.html#D4">ENUMSET1:def 4</a>, <a class="ref" href="neckla_2.html#T2">NECKLA_2:2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_2_2_1_1_1"><b>suppose </b></a><a NAME="E1:21_2_2_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,1</span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:21_2_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:21_2_2"><i><font color="Green">E12</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_2_2_1_1_2"><b>suppose </b></a><a NAME="E1:21_2_2_1_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,0</span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:21_2_2_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:21_2_2"><i><font color="Green">E12</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_2_2_1_1_3"><b>suppose </b></a><a NAME="E1:21_2_2_1_1_3"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,2</span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:21_2_2_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:21_2_2"><i><font color="Green">E12</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_2_2_1_1_4"><b>suppose </b></a><a NAME="E1:21_2_2_1_1_4"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,1</span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:21_2_2_1_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:21_2_2"><i><font color="Green">E12</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_2_2_1_1_5"><b>suppose </b></a><a NAME="E1:21_2_2_1_1_5"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,3</span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:21_2_2_1_1_5"/>
contradiction
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:21_2_2"><i><font color="Green">E12</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_2_2_1_1_6"><b>suppose </b></a><a NAME="E1:21_2_2_1_1_6"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">3,2</span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:21_2_2_1_1_6"/>
contradiction
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:21_2_2"><i><font color="Green">E12</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E3:21_2_2"/>
( 0 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span> &amp; 2 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span> )
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:21"><i><font color="Green">E10</font></i></a>, <a class="ref" href="enumset1.html#D2">ENUMSET1:def 2</a></i>;<br/><a NAME="E4:21_2_2"/><b>then </b>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span> &amp; not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span> )
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:21_2_2"><i><font color="Green">E12</font></i></a>, <a class="txt" href="neckla_3.html#E2:21_2_2"><i><font color="Green">E13</font></i></a>, <a class="ref" href="relat_1.html#D10">RELAT_1:def 10</a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E5:21_2_2"/><b>then </b>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="subset_1.html#K6">\</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span> &amp; not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E6:21_2_2"/><b>then </b>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span> <a href="subset_1.html#K3">`</a>  &amp; not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span> )
 <b>by </b><i><a class="ref" href="subset_1.html#D5">SUBSET_1:def 5</a></i>;<br/><a NAME="E7:21_2_2"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <span class="p2">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E8:21_2_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K3">ComplRelStr</a> <span class="p2">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="necklace.html#D9">NECKLACE:def 9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_2_3"><b>suppose </b></a><a NAME="E1:21_2_3"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,3</span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><a NAME="E2:21_2_3"/><i><font color="Green">E13</font></i>: 
not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:21_2_3_1"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_2_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:21_2_3_1_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,1</span><a href="tarski.html#K4">]</a></span> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,0</span><a href="tarski.html#K4">]</a></span> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,2</span><a href="tarski.html#K4">]</a></span> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,1</span><a href="tarski.html#K4">]</a></span> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,3</span><a href="tarski.html#K4">]</a></span> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">3,2</span><a href="tarski.html#K4">]</a></span> )
 </span><b>by </b><i><a class="txt" href="neckla_3.html#E1:21_2_3_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="enumset1.html#D4">ENUMSET1:def 4</a>, <a class="ref" href="neckla_2.html#T2">NECKLA_2:2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_2_3_1_1_1"><b>suppose </b></a><a NAME="E1:21_2_3_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,1</span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:21_2_3_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:21_2_3"><i><font color="Green">E12</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_2_3_1_1_2"><b>suppose </b></a><a NAME="E1:21_2_3_1_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,0</span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:21_2_3_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:21_2_3"><i><font color="Green">E12</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_2_3_1_1_3"><b>suppose </b></a><a NAME="E1:21_2_3_1_1_3"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,2</span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:21_2_3_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:21_2_3"><i><font color="Green">E12</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_2_3_1_1_4"><b>suppose </b></a><a NAME="E1:21_2_3_1_1_4"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,1</span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:21_2_3_1_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:21_2_3"><i><font color="Green">E12</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_2_3_1_1_5"><b>suppose </b></a><a NAME="E1:21_2_3_1_1_5"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,3</span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:21_2_3_1_1_5"/>
contradiction
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:21_2_3"><i><font color="Green">E12</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_2_3_1_1_6"><b>suppose </b></a><a NAME="E1:21_2_3_1_1_6"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">3,2</span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:21_2_3_1_1_6"/>
contradiction
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:21_2_3"><i><font color="Green">E12</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E3:21_2_3"/>
( 0 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span> &amp; 3 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span> )
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:21"><i><font color="Green">E10</font></i></a>, <a class="ref" href="enumset1.html#D2">ENUMSET1:def 2</a></i>;<br/><a NAME="E4:21_2_3"/><b>then </b>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span> &amp; not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span> )
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:21_2_3"><i><font color="Green">E12</font></i></a>, <a class="txt" href="neckla_3.html#E2:21_2_3"><i><font color="Green">E13</font></i></a>, <a class="ref" href="relat_1.html#D10">RELAT_1:def 10</a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E5:21_2_3"/><b>then </b>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="subset_1.html#K6">\</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span> &amp; not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E6:21_2_3"/><b>then </b>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span> <a href="subset_1.html#K3">`</a>  &amp; not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span> )
 <b>by </b><i><a class="ref" href="subset_1.html#D5">SUBSET_1:def 5</a></i>;<br/><a NAME="E7:21_2_3"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <span class="p2">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E8:21_2_3"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K3">ComplRelStr</a> <span class="p2">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="necklace.html#D9">NECKLACE:def 9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_2_4"><b>suppose </b></a><a NAME="E1:21_2_4"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">3,0</span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><a NAME="E2:21_2_4"/><i><font color="Green">E13</font></i>: 
not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2_4_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:21_2_4_1"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_2_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:21_2_4_1_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,1</span><a href="tarski.html#K4">]</a></span> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,0</span><a href="tarski.html#K4">]</a></span> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,2</span><a href="tarski.html#K4">]</a></span> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,1</span><a href="tarski.html#K4">]</a></span> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,3</span><a href="tarski.html#K4">]</a></span> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">3,2</span><a href="tarski.html#K4">]</a></span> )
 </span><b>by </b><i><a class="txt" href="neckla_3.html#E1:21_2_4_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="enumset1.html#D4">ENUMSET1:def 4</a>, <a class="ref" href="neckla_2.html#T2">NECKLA_2:2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_2_4_1_1_1"><b>suppose </b></a><a NAME="E1:21_2_4_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,1</span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:21_2_4_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:21_2_4"><i><font color="Green">E12</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_2_4_1_1_2"><b>suppose </b></a><a NAME="E1:21_2_4_1_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,0</span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:21_2_4_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:21_2_4"><i><font color="Green">E12</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_2_4_1_1_3"><b>suppose </b></a><a NAME="E1:21_2_4_1_1_3"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,2</span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:21_2_4_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:21_2_4"><i><font color="Green">E12</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_2_4_1_1_4"><b>suppose </b></a><a NAME="E1:21_2_4_1_1_4"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,1</span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:21_2_4_1_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:21_2_4"><i><font color="Green">E12</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_2_4_1_1_5"><b>suppose </b></a><a NAME="E1:21_2_4_1_1_5"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,3</span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:21_2_4_1_1_5"/>
contradiction
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:21_2_4"><i><font color="Green">E12</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_2_4_1_1_6"><b>suppose </b></a><a NAME="E1:21_2_4_1_1_6"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">3,2</span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:21_2_4_1_1_6"/>
contradiction
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:21_2_4"><i><font color="Green">E12</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E3:21_2_4"/>
( 0 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span> &amp; 3 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span> )
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:21"><i><font color="Green">E10</font></i></a>, <a class="ref" href="enumset1.html#D2">ENUMSET1:def 2</a></i>;<br/><a NAME="E4:21_2_4"/><b>then </b>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span> &amp; not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span> )
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:21_2_4"><i><font color="Green">E12</font></i></a>, <a class="txt" href="neckla_3.html#E2:21_2_4"><i><font color="Green">E13</font></i></a>, <a class="ref" href="relat_1.html#D10">RELAT_1:def 10</a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E5:21_2_4"/><b>then </b>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="subset_1.html#K6">\</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span> &amp; not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E6:21_2_4"/><b>then </b>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span> <a href="subset_1.html#K3">`</a>  &amp; not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span> )
 <b>by </b><i><a class="ref" href="subset_1.html#D5">SUBSET_1:def 5</a></i>;<br/><a NAME="E7:21_2_4"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <span class="p2">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E8:21_2_4"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K3">ComplRelStr</a> <span class="p2">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="necklace.html#D9">NECKLACE:def 9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_2_5"><b>suppose </b></a><a NAME="E1:21_2_5"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,3</span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><a NAME="E2:21_2_5"/><i><font color="Green">E13</font></i>: 
not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2_5_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:21_2_5_1"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_2_5_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:21_2_5_1_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,1</span><a href="tarski.html#K4">]</a></span> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,0</span><a href="tarski.html#K4">]</a></span> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,2</span><a href="tarski.html#K4">]</a></span> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,1</span><a href="tarski.html#K4">]</a></span> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,3</span><a href="tarski.html#K4">]</a></span> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">3,2</span><a href="tarski.html#K4">]</a></span> )
 </span><b>by </b><i><a class="txt" href="neckla_3.html#E1:21_2_5_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="enumset1.html#D4">ENUMSET1:def 4</a>, <a class="ref" href="neckla_2.html#T2">NECKLA_2:2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_2_5_1_1_1"><b>suppose </b></a><a NAME="E1:21_2_5_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,1</span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:21_2_5_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:21_2_5"><i><font color="Green">E12</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_2_5_1_1_2"><b>suppose </b></a><a NAME="E1:21_2_5_1_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,0</span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:21_2_5_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:21_2_5"><i><font color="Green">E12</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_2_5_1_1_3"><b>suppose </b></a><a NAME="E1:21_2_5_1_1_3"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,2</span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:21_2_5_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:21_2_5"><i><font color="Green">E12</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_2_5_1_1_4"><b>suppose </b></a><a NAME="E1:21_2_5_1_1_4"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,1</span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:21_2_5_1_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:21_2_5"><i><font color="Green">E12</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_2_5_1_1_5"><b>suppose </b></a><a NAME="E1:21_2_5_1_1_5"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,3</span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:21_2_5_1_1_5"/>
contradiction
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:21_2_5"><i><font color="Green">E12</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_2_5_1_1_6"><b>suppose </b></a><a NAME="E1:21_2_5_1_1_6"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">3,2</span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:21_2_5_1_1_6"/>
contradiction
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:21_2_5"><i><font color="Green">E12</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E3:21_2_5"/>
( 1 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span> &amp; 3 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span> )
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:21"><i><font color="Green">E10</font></i></a>, <a class="ref" href="enumset1.html#D2">ENUMSET1:def 2</a></i>;<br/><a NAME="E4:21_2_5"/><b>then </b>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span> &amp; not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span> )
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:21_2_5"><i><font color="Green">E12</font></i></a>, <a class="txt" href="neckla_3.html#E2:21_2_5"><i><font color="Green">E13</font></i></a>, <a class="ref" href="relat_1.html#D10">RELAT_1:def 10</a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E5:21_2_5"/><b>then </b>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="subset_1.html#K6">\</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span> &amp; not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E6:21_2_5"/><b>then </b>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span> <a href="subset_1.html#K3">`</a>  &amp; not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span> )
 <b>by </b><i><a class="ref" href="subset_1.html#D5">SUBSET_1:def 5</a></i>;<br/><a NAME="E7:21_2_5"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <span class="p2">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E8:21_2_5"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K3">ComplRelStr</a> <span class="p2">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="necklace.html#D9">NECKLACE:def 9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_2_6"><b>suppose </b></a><a NAME="E1:21_2_6"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">3,1</span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><a NAME="E2:21_2_6"/><i><font color="Green">E13</font></i>: 
not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2_6_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:21_2_6_1"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_2_6_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:21_2_6_1_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,1</span><a href="tarski.html#K4">]</a></span> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,0</span><a href="tarski.html#K4">]</a></span> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,2</span><a href="tarski.html#K4">]</a></span> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,1</span><a href="tarski.html#K4">]</a></span> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,3</span><a href="tarski.html#K4">]</a></span> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">3,2</span><a href="tarski.html#K4">]</a></span> )
 </span><b>by </b><i><a class="txt" href="neckla_3.html#E1:21_2_6_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="enumset1.html#D4">ENUMSET1:def 4</a>, <a class="ref" href="neckla_2.html#T2">NECKLA_2:2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_2_6_1_1_1"><b>suppose </b></a><a NAME="E1:21_2_6_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,1</span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:21_2_6_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:21_2_6"><i><font color="Green">E12</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_2_6_1_1_2"><b>suppose </b></a><a NAME="E1:21_2_6_1_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,0</span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:21_2_6_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:21_2_6"><i><font color="Green">E12</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_2_6_1_1_3"><b>suppose </b></a><a NAME="E1:21_2_6_1_1_3"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,2</span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:21_2_6_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:21_2_6"><i><font color="Green">E12</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_2_6_1_1_4"><b>suppose </b></a><a NAME="E1:21_2_6_1_1_4"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,1</span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:21_2_6_1_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:21_2_6"><i><font color="Green">E12</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_2_6_1_1_5"><b>suppose </b></a><a NAME="E1:21_2_6_1_1_5"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,3</span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:21_2_6_1_1_5"/>
contradiction
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:21_2_6"><i><font color="Green">E12</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_2_6_1_1_6"><b>suppose </b></a><a NAME="E1:21_2_6_1_1_6"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">3,2</span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:21_2_6_1_1_6"/>
contradiction
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:21_2_6"><i><font color="Green">E12</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E3:21_2_6"/>
( 1 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span> &amp; 3 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span> )
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:21"><i><font color="Green">E10</font></i></a>, <a class="ref" href="enumset1.html#D2">ENUMSET1:def 2</a></i>;<br/><a NAME="E4:21_2_6"/><b>then </b>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span> &amp; not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span> )
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:21_2_6"><i><font color="Green">E12</font></i></a>, <a class="txt" href="neckla_3.html#E2:21_2_6"><i><font color="Green">E13</font></i></a>, <a class="ref" href="relat_1.html#D10">RELAT_1:def 10</a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E5:21_2_6"/><b>then </b>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="subset_1.html#K6">\</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span> &amp; not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E6:21_2_6"/><b>then </b>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span> <a href="subset_1.html#K3">`</a>  &amp; not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span> )
 <b>by </b><i><a class="ref" href="subset_1.html#D5">SUBSET_1:def 5</a></i>;<br/><a NAME="E7:21_2_6"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <span class="p2">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E8:21_2_6"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K3">ComplRelStr</a> <span class="p2">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="necklace.html#D9">NECKLACE:def 9</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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