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

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="102_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:102_1_1_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0 or ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 </span><b>by </b><i><a class="ref" href="nat_1.html#T22">NAT_1:22</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="102_1_1_1_1"><b>suppose </b></a><a NAME="E1:102_1_1_1_1"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="chain_1.html#NM4">Chain</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>2</sub></font>] means  <a href="chain_1.html#K11" target="_self">del</a> <font color="Maroon">a<sub>1</sub></font> is    <a href="chain_1.html#M3" target="_self">Cycle</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/><a NAME="E2:102_1_1_1_1"/>
 <a href="chain_1.html#K11" target="_self">del</a> <span class="p1">(<span class="default"><a href="chain_1.html#K12" target="_self">0_</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="chain_1.html#K12" target="_self">0_</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 ;<br/><a NAME="E3:102_1_1_1_1"/><b>then </b><i><font color="Green">E69</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="chain_1.html#K12" target="_self">0_</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>2</sub></font>]
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="102_1_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="chain_1.html#NM3">Cell</a> of <span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>2</sub></font>;<br/><b>assume </b><a NAME="E1:102_1_1_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><a NAME="E2:102_1_1_1_1_1"/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="chain_1.html#K11" target="_self">del</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> 2 <a href="nat_1.html#K2">*</a> 1
 <b>by </b><i><a class="ref" href="chain_1.html#T56" target="_self">Th56</a></i>;<br/><b>hence </b><a NAME="E3:102_1_1_1_1_1"/>
 <a href="chain_1.html#K11" target="_self">del</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span> is    <a href="chain_1.html#M3" target="_self">Cycle</a> of 0,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="chain_1.html#D15" target="_self">Def15</a></i>;<br/></div><b>end;</b></div><a NAME="E5:102_1_1_1_1"/><b>then </b><i><font color="Green">E70</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="chain_1.html#NM3">Cell</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="domain_1.html#K6">}</a></span>]
 <b>by </b><i><a class="txt" href="chain_1.html#E1:102_1_1_1_1"><i><font color="Green">E68</font></i></a></i>;<br/><a NAME="E6:102_1_1_1_1"/><i><font color="Green">E71</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="chain_1.html#NM4">Chain</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="chain_1.html#K8" target="_self">+</a> <font color="Olive">b<sub>2</sub></font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="102_1_1_1_1_2"><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="chain_1.html#NM4">Chain</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>2</sub></font>;<br/>

<b>assume </b><a NAME="E1:102_1_1_1_1_2"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>5</sub></font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>6</sub></font>] )
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> =  <a href="chain_1.html#K11" target="_self">del</a> <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>8</sub></font> =  <a href="chain_1.html#K11" target="_self">del</a> <font color="Maroon">c<sub>6</sub></font> as    <a href="chain_1.html#M3" target="_self">Cycle</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> ;<br/>
<a NAME="E3:102_1_1_1_1_2"/>
 <a href="chain_1.html#K11" target="_self">del</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="chain_1.html#K8" target="_self">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="chain_1.html#K14" target="_self">+</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="ref" href="chain_1.html#T63" target="_self">Th63</a></i>;<br/>
<b>hence </b><a NAME="E4:102_1_1_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>5</sub></font> <a href="chain_1.html#K8" target="_self">+</a> <font color="Maroon">c<sub>6</sub></font>]
 ;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E7:102_1_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>4</sub></font>]
 <b>from </b><i><a class="ref" href="chain_1.html#S4" target="_self">CHAIN_1:sch 4</a>(<a class="txt" href="chain_1.html#E3:102_1_1_1_1"><i><font color="Green">E69</font></i></a>, <a class="txt" href="chain_1.html#E5:102_1_1_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="chain_1.html#E6:102_1_1_1_1"><i><font color="Green">E71</font></i></a>);</i><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="102_1_1_1_2"><b>suppose </b></a><a NAME="E1:102_1_1_1_2"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:102_1_1_1_2"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="chain_1.html#NM4">Chain</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="chain_1.html#E3:102_1_1_1_2"><i><font color="Green">E68</font></i></a></i>;<br/><a NAME="E5:102_1_1_1_2"/>
 <a href="chain_1.html#K11" target="_self">del</a> <span class="p1">(<span class="default"><a href="chain_1.html#K11" target="_self">del</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="chain_1.html#K12" target="_self">0_</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="chain_1.html#T65" target="_self">Th65</a></i>;<br/><b>hence </b><a NAME="E6:102_1_1_1_2"/>
 <a href="chain_1.html#K11" target="_self">del</a> <font color="Maroon">c<sub>4</sub></font> is    <a href="chain_1.html#M3" target="_self">Cycle</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="chain_1.html#E3:102_1_1_1_2"><i><font color="Green">E68</font></i></a>, <a class="ref" href="chain_1.html#T66" target="_self">Th66</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
