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

<b>set </b><font color="Maroon">c<sub>1</sub></font> = <span class="p1"><a href="enumset1.html#K4">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,1</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">1,0</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">1,2</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">2,1</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">2,3</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">3,2</span><a href="tarski.html#K4">]</a></span></span><a href="enumset1.html#K4">}</a></span>;<br/>
<b>set </b><font color="Maroon">c<sub>2</sub></font> = 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/>
<a NAME="E1:2"/><i><font color="Green">E3</font></i>: 
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="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 4 </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Olive">b<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 4 </span>}</span> 
 
<b>by </b><i><a class="ref" href="necklace.html#T19">NECKLACE:19</a></i>;<br/>
<a NAME="E2:2"/><i><font color="Green">E4</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<span class="p2">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 4 </span>}</span> 
 
;<br/>
<a NAME="E3:2"/><i><font color="Green">E5</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span>,0</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Olive">b<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 4 </span>}</span> 
 
;<br/>
<a NAME="E4:2"/><i><font color="Green">E6</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 4 </span>}</span> 
 
;<br/>
<a NAME="E5:2"/><i><font color="Green">E7</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Olive">b<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 4 </span>}</span> 
 
;<br/>
<a NAME="E6:2"/><i><font color="Green">E8</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default">2 <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 4 </span>}</span> 
 
;<br/>
<a NAME="E7:2"/><i><font color="Green">E9</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Olive">b<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 4 </span>}</span> 
 
;<br/>
<a NAME="E8:2"/><i><font color="Green">E10</font></i>: 
<span class="p1"><a href="enumset1.html#K4">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,1</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">1,0</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">1,2</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">2,1</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">2,3</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">3,2</span><a href="tarski.html#K4">]</a></span></span><a href="enumset1.html#K4">}</a></span> <a href="tarski.html#R1">c=</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="2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</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="E1:2_1"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>3</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">0,1</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">1,0</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">1,2</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">2,1</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">2,3</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">3,2</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="2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_1_1"/>
( <font color="Maroon">c<sub>3</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>3</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>3</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>3</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>3</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>3</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_2.html#E1:2_1"><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="2_1_1_1"><b>suppose </b></a><a NAME="E1:2_1_1_1"/>
<font color="Maroon">c<sub>3</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="E2:2_1_1_1"/>
<font color="Maroon">c<sub>3</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_2.html#E1:2"><i><font color="Green">E3</font></i></a>, <a class="txt" href="neckla_2.html#E2:2"><i><font color="Green">E4</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2"><b>suppose </b></a><a NAME="E1:2_1_1_2"/>
<font color="Maroon">c<sub>3</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="E2:2_1_1_2"/>
<font color="Maroon">c<sub>3</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_2.html#E1:2"><i><font color="Green">E3</font></i></a>, <a class="txt" href="neckla_2.html#E3:2"><i><font color="Green">E5</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_3"><b>suppose </b></a><a NAME="E1:2_1_1_3"/>
<font color="Maroon">c<sub>3</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="E2:2_1_1_3"/>
<font color="Maroon">c<sub>3</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_2.html#E1:2"><i><font color="Green">E3</font></i></a>, <a class="txt" href="neckla_2.html#E4:2"><i><font color="Green">E6</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_4"><b>suppose </b></a><a NAME="E1:2_1_1_4"/>
<font color="Maroon">c<sub>3</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="E2:2_1_1_4"/>
<font color="Maroon">c<sub>3</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_2.html#E1:2"><i><font color="Green">E3</font></i></a>, <a class="txt" href="neckla_2.html#E5:2"><i><font color="Green">E7</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_5"><b>suppose </b></a><a NAME="E1:2_1_1_5"/>
<font color="Maroon">c<sub>3</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="E2:2_1_1_5"/>
<font color="Maroon">c<sub>3</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_2.html#E1:2"><i><font color="Green">E3</font></i></a>, <a class="txt" href="neckla_2.html#E6:2"><i><font color="Green">E8</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_6"><b>suppose </b></a><a NAME="E1:2_1_1_6"/>
<font color="Maroon">c<sub>3</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="E2:2_1_1_6"/>
<font color="Maroon">c<sub>3</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_2.html#E1:2"><i><font color="Green">E3</font></i></a>, <a class="txt" href="neckla_2.html#E7:2"><i><font color="Green">E9</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E9:2"/>
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="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">0,1</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">1,0</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">1,2</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">2,1</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">2,3</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">3,2</span><a href="tarski.html#K4">]</a></span></span><a href="enumset1.html#K4">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</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="E1:2_2"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>3</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/>

<b>then consider </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:2_2"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>3</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>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#D1">RELAT_1:def 1</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_2_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 4 </span>}</span>  or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Olive">b<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 4 </span>}</span>  )
 </span><b>by </b><i><a class="txt" href="neckla_2.html#E1:2"><i><font color="Green">E3</font></i></a>, <a class="txt" href="neckla_2.html#E1:2_2"><i><font color="Green">E11</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_1"><b>suppose </b></a><a NAME="E1:2_2_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 4 </span>}</span> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:2_2_1_1"/><i><font color="Green">E13</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 4 )
 <b>by </b><i><a class="txt" href="neckla_2.html#E3:2_2"><i><font color="Green">E12</font></i></a></i>;<br/><a NAME="E4:2_2_1_1"/><i><font color="Green">E14</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="neckla_2.html#E3:2_2_1_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><a NAME="E5:2_2_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="nat_1.html#K1">+</a> 3
 <b>by </b><i><a class="txt" href="neckla_2.html#E3:2_2_1_1"><i><font color="Green">E13</font></i></a></i>;<br/><a NAME="E6:2_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E7:2_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 2
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E8:2_2_1_1"/><b>then </b>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 2 )
 <b>by </b><i><a class="txt" href="neckla_2.html#E4:2_2_1_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="cqc_the1.html#T3">CQC_THE1:3</a></i>;<br/><b>hence </b><a NAME="E9:2_2_1_1"/>
<font color="Maroon">c<sub>3</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">0,1</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">1,0</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">1,2</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">2,1</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">2,3</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">3,2</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_2.html#E3:2_2"><i><font color="Green">E12</font></i></a>, <a class="txt" href="neckla_2.html#E4:2_2_1_1"><i><font color="Green">E14</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="2_2_1_2"><b>suppose </b></a><a NAME="E1:2_2_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Olive">b<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 4 </span>}</span> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:2_2_1_2"/><i><font color="Green">E13</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 4 )
 <b>by </b><i><a class="txt" href="neckla_2.html#E3:2_2"><i><font color="Green">E12</font></i></a></i>;<br/><a NAME="E4:2_2_1_2"/><i><font color="Green">E14</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="neckla_2.html#E3:2_2_1_2"><i><font color="Green">E13</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><a NAME="E5:2_2_1_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="nat_1.html#K1">+</a> 3
 <b>by </b><i><a class="txt" href="neckla_2.html#E3:2_2_1_2"><i><font color="Green">E13</font></i></a></i>;<br/><a NAME="E6:2_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E7:2_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 2
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E8:2_2_1_2"/><b>then </b>
( <font color="Maroon">c<sub>5</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 or <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 2 )
 <b>by </b><i><a class="txt" href="neckla_2.html#E4:2_2_1_2"><i><font color="Green">E14</font></i></a>, <a class="ref" href="cqc_the1.html#T3">CQC_THE1:3</a></i>;<br/><b>hence </b><a NAME="E9:2_2_1_2"/>
<font color="Maroon">c<sub>3</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">0,1</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">1,0</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">1,2</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">2,1</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">2,3</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">3,2</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_2.html#E3:2_2"><i><font color="Green">E12</font></i></a>, <a class="txt" href="neckla_2.html#E4:2_2_1_2"><i><font color="Green">E14</font></i></a>, <a class="ref" href="enumset1.html#D4">ENUMSET1:def 4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E10:2"/>
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="hidden.html#R1">=</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">0,1</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">1,0</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">1,2</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">2,1</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">2,3</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">3,2</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_2.html#E8:2"><i><font color="Green">E10</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div>
