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<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:101_1_1_1"/>
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0 or  ex <font color="Olive">k'</font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Olive">k'</font> <a href="xcmplx_0.html#K2">+</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="101_1_1_1_1"><b>suppose </b></a><a NAME="E1:101_1_1_1_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">k</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">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">G</font>] <b>means </b> <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">k</font>,<font color="Maroon">G</font>;<br/><a NAME="E2:101_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">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">G</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="chain_1.html#K12" target="_self">0_</a> <font color="Maroon">k</font>,<font color="Maroon">G</font>
 ;<br/><a NAME="E3:101_1_1_1_1"/><b>then </b><i><font color="Green">E51</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">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">G</font>]
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_1_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">B</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">G</font>;<br/><b>assume </b><a NAME="E1:101_1_1_1_1_1"/>
<font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 ;<br/><a NAME="E2:101_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">B</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#T5" target="_self">Th5</a></i>;<br/><b>hence </b><a NAME="E3:101_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">B</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">G</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><a NAME="E5:101_1_1_1_1"/><b>then </b><i><font color="Green">E52</font></i>: 
for <font color="Olive">A</font> being  <a href="chain_1.html#NM3">Cell</a> of <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">G</font>  st <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</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">A</font></span><a href="domain_1.html#K6">}</a></span>]
 <b>by </b><i/>;<br/><a NAME="E6:101_1_1_1_1"/><i><font color="Green">E53</font></i>: 
for <font color="Olive">C1</font>, <font color="Olive">C2</font> being  <a href="chain_1.html#NM4">Chain</a> of <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">G</font>  st <font color="Olive">C1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font> &amp; <font color="Olive">C2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">C1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">C2</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">C1</font> <a href="chain_1.html#K8" target="_self">+</a> <font color="Olive">C2</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_1_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">C1</font> be   <a href="chain_1.html#NM4">Chain</a> of <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">G</font>, <font color="Maroon">C2</font> be   <a href="chain_1.html#NM4">Chain</a> of <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">G</font>;<br/>

<b>assume </b><a NAME="E1:101_1_1_1_1_2"/>
( <font color="Maroon">C1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font> &amp; <font color="Maroon">C2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">C1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">C2</font>] )
 ;<br/>

<b>then reconsider </b><font color="Maroon">C1'</font> =  <a href="chain_1.html#K11" target="_self">del</a> <font color="Maroon">C1</font>, <font color="Maroon">C2'</font> =  <a href="chain_1.html#K11" target="_self">del</a> <font color="Maroon">C2</font> as    <a href="chain_1.html#M3" target="_self">Cycle</a> of <font color="Maroon">k</font>,<font color="Maroon">G</font> ;<br/>
<a NAME="E3:101_1_1_1_1_2"/>
 <a href="chain_1.html#K11" target="_self">del</a> <span class="p1">(<span class="default"><font color="Maroon">C1</font> <a href="chain_1.html#K8" target="_self">+</a> <font color="Maroon">C2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">C1'</font> <a href="chain_1.html#K14" target="_self">+</a> <font color="Maroon">C2'</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:101_1_1_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">C1</font> <a href="chain_1.html#K8" target="_self">+</a> <font color="Maroon">C2</font>]
 ;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E7:101_1_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">C</font>]
 <b>from </b><i><a class="ref" href="chain_1.html#S4" target="_self">CHAIN_1:sch 4</a>(<a class="ref" href="chain_1.html#T7" target="_self">Th7</a>, <a class="ref" href="chain_1.html#T8" target="_self">Th8</a>, <a class="ref" href="chain_1.html#T9" target="_self">Th9</a>);</i><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_1_1_1_2"><b>suppose </b></a><a NAME="E1:101_1_1_1_2"/>
 ex <font color="Olive">k'</font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Olive">k'</font> <a href="xcmplx_0.html#K2">+</a> 1
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">k'</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:101_1_1_1_2"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k'</font> <a href="xcmplx_0.html#K2">+</a> 1
 ;<br/><b>reconsider </b><font color="Maroon">k'</font> = <font color="Maroon">k'</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><b>reconsider </b><font color="Maroon">C</font> = <font color="Maroon">C</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">k'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">G</font> <b>by </b><i/>;<br/><a NAME="E6:101_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</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="chain_1.html#K12" target="_self">0_</a> <font color="Maroon">k'</font>,<font color="Maroon">G</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E7:101_1_1_1_2"/>
 <a href="chain_1.html#K11" target="_self">del</a> <font color="Maroon">C</font> is    <a href="chain_1.html#M3" target="_self">Cycle</a> of <font color="Maroon">k</font>,<font color="Maroon">G</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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