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<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="96_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:96_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">k9</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">k9</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T6" title="NAT_1:th.6">NAT_1:6</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="96_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:96_1_1_1_1"/><span class="lab"><font color="Green" title="E64">A1</font></span>: 
<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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="chain_1.html#NK8" title="CHAIN_1:NK.8">+</a>  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></span><br/><div class="add"><a NAME="E2:96_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A2</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c4">C1</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="chain_1.html#T68" target="_self" title="CHAIN_1:th.68">Th68</a></span>;<br/><a NAME="E3:96_1_1_1_1"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c5">C2</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:96_1_1_1_1"><span class="lab"><font color="Green" title="E64">A1</font></span></a>, <a class="ref" href="chain_1.html#T68" target="_self" title="CHAIN_1:th.68">Th68</a></span>;<br/><a NAME="E4:96_1_1_1_1"/><span class="kw">then </span>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">C1</font> <a href="chain_1.html#K7" title="CHAIN_1:func.7">+</a> <font color="Maroon" title="c5">C2</font></span>)</span> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:96_1_1_1_1"><span class="lab"><font color="Green" title="E65">A2</font></span></a>, <a class="ref" href="chain_1.html#T9" target="_self" title="CHAIN_1:th.9">Th9</a></span>;<br/><span class="kw">hence </span><a NAME="E5:96_1_1_1_1"/>
<font color="Maroon" title="c4">C1</font> <a href="chain_1.html#K7" title="CHAIN_1:func.7">+</a> <font color="Maroon" title="c5">C2</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:96_1_1_1_1"><span class="lab"><font color="Green" title="E64">A1</font></span></a>, <a class="ref" href="chain_1.html#T68" target="_self" title="CHAIN_1:th.68">Th68</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="96_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:96_1_1_1_2"/>
 ex <font color="Olive" title="b1">k9</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">k9</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="chain_1.html#NK8" title="CHAIN_1:NK.8">+</a>  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></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">k9</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E3:96_1_1_1_2"/><span class="lab"><font color="Green" title="E64">A3</font></span>: 
<font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">k9</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c7">k9</font> = <font color="Maroon" title="c6">k9</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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c8">C1</font> = <font color="Maroon" title="c4">C1</font>, <font color="Maroon" title="c9">C2</font> = <font color="Maroon" title="c5">C2</font> as    <a href="chain_1.html#M3" title="CHAIN_1:mode.3">Cycle</a> of <font color="Maroon" title="c7">k9</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1,<font color="Maroon" title="c2">G</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:96_1_1_1_2"><span class="lab"><font color="Green" title="E64">A3</font></span></a></span>;<br/><a NAME="E6:96_1_1_1_2"/><span class="lab"><font color="Green" title="E65">A4</font></span>: 
 <a href="chain_1.html#K10" title="CHAIN_1:func.10">del</a> <font color="Maroon" title="c8">C1</font> <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="c7">k9</font>,<font color="Maroon" title="c2">G</font>
 ;<br/><a NAME="E7:96_1_1_1_2"/>
 <a href="chain_1.html#K10" title="CHAIN_1:func.10">del</a> <font color="Maroon" title="c9">C2</font> <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="c7">k9</font>,<font color="Maroon" title="c2">G</font>
 ;<br/><span class="kw">then </span> <a href="chain_1.html#K10" title="CHAIN_1:func.10">del</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">C1</font> <a href="chain_1.html#K7" title="CHAIN_1:func.7">+</a> <font color="Maroon" title="c9">C2</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="chain_1.html#K11" title="CHAIN_1:func.11">0_</a> <font color="Maroon" title="c7">k9</font>,<font color="Maroon" title="c2">G</font></span>)</span> <a href="chain_1.html#K7" title="CHAIN_1:func.7">+</a> <span class="p1">(<span class="default"><a href="chain_1.html#K11" title="CHAIN_1:func.11">0_</a> <font color="Maroon" title="c7">k9</font>,<font color="Maroon" title="c2">G</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:96_1_1_1_2"><span class="lab"><font color="Green" title="E65">A4</font></span></a>, <a class="ref" href="chain_1.html#T63" target="_self" title="CHAIN_1:th.63">Th63</a></span>
<br/>.= 
 <a href="chain_1.html#K11" title="CHAIN_1:func.11">0_</a> <font color="Maroon" title="c7">k9</font>,<font color="Maroon" title="c2">G</font>

;<br/><span class="kw">hence </span><a NAME="E9:96_1_1_1_2"/>
 <a href="chain_1.html#NK8" title="CHAIN_1:NK.8">+</a>  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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:96_1_1_1_2"><span class="lab"><font color="Green" title="E64">A3</font></span></a>, <a class="ref" href="chain_1.html#T66" target="_self" title="CHAIN_1:th.66">Th66</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

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