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

<b>set </b><font color="Maroon">c<sub>1</sub></font> =  <a href="necklace.html#K4" target="_self">Necklace</a> 1;<br/>
<b>set </b><font color="Maroon">c<sub>2</sub></font> =  <a href="necklace.html#K3" target="_self">ComplRelStr</a> <span class="p1">(<span class="default"><a href="necklace.html#K4" target="_self">Necklace</a> 1</span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>3</sub></font> = 0 <a href="cqc_lang.html#K3">.--&gt;</a> 0;<br/>
<i><font color="Green">E30</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">0 <a href="cqc_lang.html#K3">.--&gt;</a> 0</span>)</span> = 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4" target="_self">Necklace</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="necklace.html#T21" target="_self">Th21</a>, <a class="ref" href="card_5.html#T1">CARD_5:1</a></i>
;<br/>
<a NAME="E2:53"/><i><font color="Green">E31</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4" target="_self">Necklace</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="necklace.html#T21" target="_self">Th21</a>, <a class="ref" href="card_5.html#T1">CARD_5:1</a></i>;<br/>
<a NAME="E3:53"/><i><font color="Green">E32</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4" target="_self">Necklace</a> 1</span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="necklace.html#K3" target="_self">ComplRelStr</a> <span class="p2">(<span class="default"><a href="necklace.html#K4" target="_self">Necklace</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="necklace.html#D9" target="_self">Def9</a></i>;<br/>
0 <a href="cqc_lang.html#K3">.--&gt;</a> 0 = 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> <a href="funcop_1.html#K2">--&gt;</a> 0

<br/>.= 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i><a class="ref" href="funcop_1.html#D2">FUNCOP_1:def 2</a></i>
;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = 0 <a href="cqc_lang.html#K3">.--&gt;</a> 0 as   <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4" target="_self">Necklace</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="necklace.html#K3" target="_self">ComplRelStr</a> <span class="p2">(<span class="default"><a href="necklace.html#K4" target="_self">Necklace</a> 1</span>)</span></span>)</span> <b>by </b><i><a class="txt" href="necklace.html#E1:53"><i><font color="Green">E30</font></i></a>, <a class="txt" href="necklace.html#E2:53"><i><font color="Green">E31</font></i></a>, <a class="txt" href="necklace.html#E3:53"><i><font color="Green">E32</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#D1">RELSET_1:def 1</a></i>;<br/>
<a NAME="E6:53"/><i><font color="Green">E33</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>;<br/>
<a NAME="E7:53"/><i><font color="Green">E34</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>;<br/>
<a NAME="E8:53"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>4</sub></font> is <a href="orders_3.html#V5">monotone</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4" target="_self">Necklace</a> 1</span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="orders_3.html#D5">ORDERS_3:def 5</a><br/>

<b>assume </b><a NAME="E1:53_3"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<a NAME="E2:53_3"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4" target="_self">Necklace</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="necklace.html#T21" target="_self">Th21</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>6</sub></font> as  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  <b>by </b><i><a class="ref" href="necklace.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E4:53_3"/>
<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> <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" target="_self">Necklace</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="necklace.html#E1:53_3"><i><font color="Green">E36</font></i></a>, <a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>
<a NAME="E5:53_3"/><b>then </b><i><font color="Green">E37</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="xcmplx_0.html#K2">+</a> 1 or <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="xcmplx_0.html#K2">+</a> 1 )
 
<b>by </b><i><a class="ref" href="necklace.html#T25" target="_self">Th25</a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="necklace.html#K3" target="_self">ComplRelStr</a> <span class="p2">(<span class="default"><a href="necklace.html#K4" target="_self">Necklace</a> 1</span>)</span></span>)</span>;<br/>

<b>assume </b><a NAME="E6:53_3"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/>

<a NAME="E7:53_3"/><i><font color="Green">E38</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4" target="_self">Necklace</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="necklace.html#T21" target="_self">Th21</a>, <a class="ref" href="card_5.html#T1">CARD_5:1</a></i>;<br/>
<a NAME="E8:53_3"/><b>then </b><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E9:53_3"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="txt" href="necklace.html#E7:53_3"><i><font color="Green">E38</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E10:53_3"/>
<font color="Maroon">c<sub>9</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="necklace.html#E5:53_3"><i><font color="Green">E37</font></i></a>, <a class="txt" href="necklace.html#E8:53_3"><i><font color="Green">E39</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:53"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K2">"</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_4"><b>proof </b></a><div class="add">

<a NAME="E1:53_4"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> iff ( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<a NAME="E1:53_4_1"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> implies ( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_4_1_1"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:53_4_1_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E2:53_4_1_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<a NAME="E3:53_4_1_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="txt" href="necklace.html#E7:53"><i><font color="Green">E34</font></i></a>, <a class="txt" href="necklace.html#E1:53_4_1_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E4:53_4_1_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="txt" href="necklace.html#E2:53_4_1_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="cqc_lang.html#T6">CQC_LANG:6</a></i>;<br/>
<b>hence </b><a NAME="E5:53_4_1_1"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="necklace.html#E6:53"><i><font color="Green">E33</font></i></a>, <a class="txt" href="necklace.html#E3:53_4_1_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="cqc_lang.html#T6">CQC_LANG:6</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:53_4_1"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> implies ( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_4_1_2"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:53_4_1_2"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E2:53_4_1_2"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<a NAME="E3:53_4_1_2"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4" target="_self">Necklace</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="necklace.html#T21" target="_self">Th21</a>, <a class="ref" href="card_5.html#T1">CARD_5:1</a></i>;<br/>
<a NAME="E4:53_4_1_2"/><b>then </b><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="txt" href="necklace.html#E1:53_4_1_2"><i><font color="Green">E37</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E5:53_4_1_2"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="txt" href="necklace.html#E2:53_4_1_2"><i><font color="Green">E38</font></i></a>, <a class="ref" href="cqc_lang.html#T6">CQC_LANG:6</a></i>;<br/>
<b>hence </b><a NAME="E6:53_4_1_2"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="necklace.html#E7:53"><i><font color="Green">E34</font></i></a>, <a class="txt" href="necklace.html#E4:53_4_1_2"><i><font color="Green">E39</font></i></a>, <a class="ref" href="cqc_lang.html#T6">CQC_LANG:6</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:53_4_1"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> iff ( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> ) )
 <b>by </b><i><a class="txt" href="necklace.html#E7:53"><i><font color="Green">E34</font></i></a>, <a class="txt" href="necklace.html#E1:53_4_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:53_4"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K2">"</a> 
 <b>by </b><i><a class="txt" href="necklace.html#E6:53"><i><font color="Green">E33</font></i></a>, <a class="txt" href="necklace.html#E7:53"><i><font color="Green">E34</font></i></a>, <a class="ref" href="funct_1.html#T54">FUNCT_1:54</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K2">"</a>  as   <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><a href="necklace.html#K3" target="_self">ComplRelStr</a> <span class="p2">(<span class="default"><a href="necklace.html#K4" target="_self">Necklace</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="necklace.html#K4" target="_self">Necklace</a> 1</span>)</span> <b>by </b><i><a class="txt" href="necklace.html#E3:53"><i><font color="Green">E32</font></i></a></i>;<br/>
<a NAME="E11:53"/>
<font color="Maroon">c<sub>5</sub></font> is <a href="orders_3.html#V5">monotone</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="necklace.html#K3" target="_self">ComplRelStr</a> <span class="p2">(<span class="default"><a href="necklace.html#K4" target="_self">Necklace</a> 1</span>)</span></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="orders_3.html#D5">ORDERS_3:def 5</a><br/>

<b>assume </b><a NAME="E1:53_5"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/>

<a NAME="E2:53_5"/><i><font color="Green">E37</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="necklace.html#K3" target="_self">ComplRelStr</a> <span class="p2">(<span class="default"><a href="necklace.html#K4" target="_self">Necklace</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="txt" href="necklace.html#E3:53"><i><font color="Green">E32</font></i></a>, <a class="ref" href="necklace.html#T21" target="_self">Th21</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>7</sub></font> as  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  <b>by </b><i><a class="ref" href="necklace.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E4:53_5"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span> <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" target="_self">ComplRelStr</a> <span class="p2">(<span class="default"><a href="necklace.html#K4" target="_self">Necklace</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="necklace.html#E1:53_5"><i><font color="Green">E36</font></i></a>, <a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>
<a NAME="E5:53_5"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span> <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" target="_self">Necklace</a> 1</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" target="_self">Necklace</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="necklace.html#D9" target="_self">Def9</a></i>;<br/>
<a NAME="E6:53_5"/><b>then </b>
not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span> <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" target="_self">Necklace</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E7:53_5"/><b>then </b><i><font color="Green">E38</font></i>: 
(  not <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" target="_self">Necklace</a> 1</span>)</span> or <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font> )
 
<b>by </b><i><a class="ref" href="relat_1.html#D10">RELAT_1:def 10</a></i>;<br/>
<a NAME="E8:53_5"/><i><font color="Green">E39</font></i>: 
( <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#K3" target="_self">ComplRelStr</a> <span class="p2">(<span class="default"><a href="necklace.html#K4" target="_self">Necklace</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>7</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#K3" target="_self">ComplRelStr</a> <span class="p2">(<span class="default"><a href="necklace.html#K4" target="_self">Necklace</a> 1</span>)</span></span>)</span> )
 
;<br/>
<a NAME="E9:53_5"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="txt" href="necklace.html#E2:53_5"><i><font color="Green">E37</font></i></a>, <a class="ref" href="card_5.html#T1">CARD_5:1</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E10:53_5"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="txt" href="necklace.html#E2:53_5"><i><font color="Green">E37</font></i></a>, <a class="ref" href="card_5.html#T1">CARD_5:1</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E11:53_5"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>9</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_5_1"><b>proof </b></a><div class="add">

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:53_5_1_1"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> or <font color="Maroon">c<sub>9</sub></font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> )
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_5_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:53_5_1_1_1"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> or <font color="Maroon">c<sub>9</sub></font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> )
 </span><b>by </b><i><a class="txt" href="necklace.html#E1:53_5_1_1"><i><font color="Green">E43</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_5_1_1_1_1"><b>suppose </b></a><a NAME="E1:53_5_1_1_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:53_5_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="necklace.html#E2:53_5"><i><font color="Green">E37</font></i></a>, <a class="txt" href="necklace.html#E9:53_5"><i><font color="Green">E40</font></i></a>, <a class="ref" href="euler_1.html#T1">EULER_1:1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_5_1_1_1_2"><b>suppose </b></a><a NAME="E1:53_5_1_1_1_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:53_5_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="necklace.html#E2:53_5"><i><font color="Green">E37</font></i></a>, <a class="txt" href="necklace.html#E10:53_5"><i><font color="Green">E41</font></i></a>, <a class="ref" href="euler_1.html#T1">EULER_1:1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div>
<b>end;</b></div>

<b>thus </b><a NAME="E2:53_5_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="necklace.html#E7:53_5"><i><font color="Green">E38</font></i></a>, <a class="txt" href="necklace.html#E8:53_5"><i><font color="Green">E39</font></i></a>, <a class="ref" href="necklace.html#D9" target="_self">Def9</a></i>;<br/>


</div><b>end;</b></div>
<b>let </b><font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>11</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4" target="_self">Necklace</a> 1</span>)</span>;<br/>

<b>assume </b><a NAME="E12:53_5"/>
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font> )
 ;<br/>

<a NAME="E13:53_5"/><i><font color="Green">E43</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="necklace.html#K3" target="_self">ComplRelStr</a> <span class="p2">(<span class="default"><a href="necklace.html#K4" target="_self">Necklace</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="txt" href="necklace.html#E3:53"><i><font color="Green">E32</font></i></a>, <a class="ref" href="necklace.html#T21" target="_self">Th21</a>, <a class="ref" href="card_5.html#T1">CARD_5:1</a></i>;<br/>
<a NAME="E14:53_5"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="txt" href="necklace.html#E2:53_5"><i><font color="Green">E37</font></i></a>, <a class="ref" href="card_5.html#T1">CARD_5:1</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E15:53_5"/>
<font color="Maroon">c<sub>10</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="necklace.html#E11:53_5"><i><font color="Green">E42</font></i></a>, <a class="txt" href="necklace.html#E13:53_5"><i><font color="Green">E43</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:53"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> is <a href="waybel_0.html#V23">isomorphic</a>
 
<b>by </b><i><a class="txt" href="necklace.html#E8:53"><i><font color="Green">E35</font></i></a>, <a class="ref" href="waybel_0.html#D38">WAYBEL_0:def 38</a></i>;<br/>
<b>hence </b><a NAME="E13:53"/>
 <a href="necklace.html#K4" target="_self">Necklace</a> 1, <a href="necklace.html#K3" target="_self">ComplRelStr</a> <span class="p1">(<span class="default"><a href="necklace.html#K4" target="_self">Necklace</a> 1</span>)</span> <a href="waybel_1.html#R5">are_isomorphic</a> 
 <b>by </b><i><a class="ref" href="waybel_1.html#D8">WAYBEL_1:def 8</a></i>;<br/>


</div>
