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<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="99_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:99_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="99_1_1_1_1"><b>suppose </b></a><a NAME="E1:99_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"><a NAME="E2:99_1_1_1_1"/><b>then </b>
(  <a href="card_1.html#K4">card</a> <font color="Maroon">C1</font> is <a href="abian.html#V1">even</a> &amp;  <a href="card_1.html#K4">card</a> <font color="Maroon">C2</font> is <a href="abian.html#V1">even</a> )
 <b>by </b><i><a class="ref" href="chain_1.html#T6" target="_self">Th6</a></i>;<br/><a NAME="E3:99_1_1_1_1"/><b>then </b>
 <a href="card_1.html#K4">card</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> is <a href="abian.html#V1">even</a>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:99_1_1_1_1"/>
<font color="Maroon">C1</font> <a href="chain_1.html#K8" target="_self">+</a> <font color="Maroon">C2</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>, <a class="ref" href="chain_1.html#T6" target="_self">Th6</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="99_1_1_1_2"><b>suppose </b></a><a NAME="E1:99_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="E3:99_1_1_1_2"/><i><font color="Green">E51</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">C1</font> = <font color="Maroon">C1</font>, <font color="Maroon">C2</font> = <font color="Maroon">C2</font> as    <a href="chain_1.html#M3" target="_self">Cycle</a> of <font color="Maroon">k'</font> <a href="nat_1.html#K1">+</a> 1,<font color="Maroon">G</font> <b>by </b><i/>;<br/><a NAME="E6:99_1_1_1_2"/>
(  <a href="chain_1.html#K11" target="_self">del</a> <font color="Maroon">C1</font> <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> &amp;  <a href="chain_1.html#K11" target="_self">del</a> <font color="Maroon">C2</font> <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/><b>then </b> <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> = 
<span class="p1">(<span class="default"><a href="chain_1.html#K12" target="_self">0_</a> <font color="Maroon">k'</font>,<font color="Maroon">G</font></span>)</span> <a href="chain_1.html#K8" target="_self">+</a> <span class="p1">(<span class="default"><a href="chain_1.html#K12" target="_self">0_</a> <font color="Maroon">k'</font>,<font color="Maroon">G</font></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="chain_1.html#K12" target="_self">0_</a> <font color="Maroon">k'</font>,<font color="Maroon">G</font>

;<br/><b>hence </b><a NAME="E8:99_1_1_1_2"/>
 <a href="chain_1.html#NK9" target="_self">+</a>  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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