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<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="100_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:100_1_1_1"/>
( <font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or  ex <font color="Olive" title="b1">k'</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">k'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 </span><b>by </b><i><a class="ref" href="nat_1.html#T6" title="NAT_1:th.6">NAT_1:6</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="100_1_1_1_1"><b>suppose </b></a><a NAME="E1:100_1_1_1_1"/><i><font color="Green" title="E66">A1</font></i>: 
<font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="chain_1.html#NM4" title="CHAIN_1:NM.4">Chain</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c2">G</font>] <b>means </b> <a href="chain_1.html#K10" title="CHAIN_1:func.10">del</a> $1 is    <a href="chain_1.html#M3" title="CHAIN_1:mode.3">Cycle</a> of <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">G</font>;<br/><a NAME="E2:100_1_1_1_1"/>
 <a href="chain_1.html#K10" title="CHAIN_1:func.10">del</a> <span class="p1">(<span class="default"><a href="chain_1.html#K11" title="CHAIN_1:func.11">0_</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c2">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="chain_1.html#K11" title="CHAIN_1:func.11">0_</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">G</font>
 ;<br/><a NAME="E3:100_1_1_1_1"/><b>then </b><i><font color="Green" title="E67">A2</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="chain_1.html#K11" title="CHAIN_1:func.11">0_</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c2">G</font>]
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="100_1_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">B</font> be   <a href="chain_1.html#NM3" title="CHAIN_1:NM.3">Cell</a> of <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c2">G</font>;<br/><b>assume </b><a NAME="E1:100_1_1_1_1_1"/>
<font color="Maroon" title="c5">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font>
 ;<br/><a NAME="E2:100_1_1_1_1_1"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="chain_1.html#K10" title="CHAIN_1:func.10">del</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">B</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 1
 <b>by </b><i><a class="ref" href="chain_1.html#T56" target="_self" title="CHAIN_1:th.56">Th56</a></i>;<br/><b>hence </b><a NAME="E3:100_1_1_1_1_1"/>
 <a href="chain_1.html#K10" title="CHAIN_1:func.10">del</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">B</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> is    <a href="chain_1.html#M3" title="CHAIN_1:mode.3">Cycle</a> of  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c2">G</font>
 <b>by </b><i><a class="ref" href="chain_1.html#D15" target="_self" title="CHAIN_1:def.15">Def15</a></i>;<br/></div><b>end;</b></div><a NAME="E5:100_1_1_1_1"/><b>then </b><i><font color="Green" title="E68">A3</font></i>: 
for <font color="Olive" title="b1">A</font> being  <a href="chain_1.html#NM3" title="CHAIN_1:NM.3">Cell</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c2">G</font>  st <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b1">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>]
 <b>by </b><i><a class="txt" href="chain_1.html#E1:100_1_1_1_1"><i><font color="Green" title="E66">A1</font></i></a></i>;<br/><a NAME="E6:100_1_1_1_1"/><i><font color="Green" title="E69">A4</font></i>: 
for <font color="Olive" title="b1">C1</font>, <font color="Olive" title="b2">C2</font> being  <a href="chain_1.html#NM4" title="CHAIN_1:NM.4">Chain</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c2">G</font>  st <font color="Olive" title="b1">C1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Olive" title="b2">C2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">C1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">C2</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">C1</font> <a href="chain_1.html#K7" title="CHAIN_1:func.7">+</a> <font color="Olive" title="b2">C2</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="100_1_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">C1</font>, <font color="Maroon" title="c6">C2</font> be   <a href="chain_1.html#NM4" title="CHAIN_1:NM.4">Chain</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c2">G</font>;<br/>

<b>assume </b><a NAME="E1:100_1_1_1_1_2"/>
( <font color="Maroon" title="c5">C1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Maroon" title="c6">C2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">C1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">C2</font>] )
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c7">C1'</font> =  <a href="chain_1.html#K10" title="CHAIN_1:func.10">del</a> <font color="Maroon" title="c5">C1</font>, <font color="Maroon" title="c8">C2'</font> =  <a href="chain_1.html#K10" title="CHAIN_1:func.10">del</a> <font color="Maroon" title="c6">C2</font> as    <a href="chain_1.html#M3" title="CHAIN_1:mode.3">Cycle</a> of <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">G</font> ;<br/>
<a NAME="E3:100_1_1_1_1_2"/>
 <a href="chain_1.html#K10" title="CHAIN_1:func.10">del</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">C1</font> <a href="chain_1.html#K7" title="CHAIN_1:func.7">+</a> <font color="Maroon" title="c6">C2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">C1'</font> <a href="chain_1.html#K13" title="CHAIN_1:func.13">+</a> <font color="Maroon" title="c8">C2'</font>
 
<b>by </b><i><a class="ref" href="chain_1.html#T63" target="_self" title="CHAIN_1:th.63">Th63</a></i>;<br/>
<b>hence </b><a NAME="E4:100_1_1_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">C1</font> <a href="chain_1.html#K7" title="CHAIN_1:func.7">+</a> <font color="Maroon" title="c6">C2</font>]
 ;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E7:100_1_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">C</font>]
 <b>from </b><i><a class="ref" href="chain_1.html#S4" target="_self" title="CHAIN_1:sch.4">CHAIN_1:sch 4</a>(<a class="txt" href="chain_1.html#E3:100_1_1_1_1"><i><font color="Green" title="E67">A2</font></i></a>, <a class="txt" href="chain_1.html#E5:100_1_1_1_1"><i><font color="Green" title="E68">A3</font></i></a>, <a class="txt" href="chain_1.html#E6:100_1_1_1_1"><i><font color="Green" title="E69">A4</font></i></a>);</i><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="100_1_1_1_2"><b>suppose </b></a><a NAME="E1:100_1_1_1_2"/>
 ex <font color="Olive" title="b1">k'</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">k'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c5">k'</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E3:100_1_1_1_2"/><i><font color="Green" title="E66">A5</font></i>: 
<font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><b>reconsider </b><font color="Maroon" title="c6">k'</font> = <font color="Maroon" title="c5">k'</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c7">C</font> = <font color="Maroon" title="c4">C</font> as   <a href="chain_1.html#NM4" title="CHAIN_1:NM.4">Chain</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">k'</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c2">G</font> <b>by </b><i><a class="txt" href="chain_1.html#E3:100_1_1_1_2"><i><font color="Green" title="E66">A5</font></i></a></i>;<br/><a NAME="E6:100_1_1_1_2"/>
 <a href="chain_1.html#K10" title="CHAIN_1:func.10">del</a> <span class="p1">(<span class="default"><a href="chain_1.html#K10" title="CHAIN_1:func.10">del</a> <font color="Maroon" title="c7">C</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="chain_1.html#K11" title="CHAIN_1:func.11">0_</a> <font color="Maroon" title="c6">k'</font>,<font color="Maroon" title="c2">G</font>
 <b>by </b><i><a class="ref" href="chain_1.html#T65" target="_self" title="CHAIN_1:th.65">Th65</a></i>;<br/><b>hence </b><a NAME="E7:100_1_1_1_2"/>
 <a href="chain_1.html#K10" title="CHAIN_1:func.10">del</a> <font color="Maroon" title="c4">C</font> is    <a href="chain_1.html#M3" title="CHAIN_1:mode.3">Cycle</a> of <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">G</font>
 <b>by </b><i><a class="txt" href="chain_1.html#E3:100_1_1_1_2"><i><font color="Green" title="E66">A5</font></i></a>, <a class="ref" href="chain_1.html#T66" target="_self" title="CHAIN_1:th.66">Th66</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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