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<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="92_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:92_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="92_1_1_1_1"><b>suppose </b></a><a NAME="E1:92_1_1_1_1"/><i><font color="Green" title="E63">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>take </b><font color="Maroon" title="c4">C</font> =  <a href="chain_1.html#K6" title="CHAIN_1:func.6">0_</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">G</font>;<br/><a NAME="E2:92_1_1_1_1"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c4">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="card_1.html#T78" title="CARD_1:th.78">CARD_1:78</a></i>;<br/><b>hence </b><a NAME="E3:92_1_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>  &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c4">C</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> ) or  ex <font color="Olive" title="b1">k'</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <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#K2" title="NAT_1:func.2">+</a> 1 &amp;  ex <font color="Olive" title="b2">C</font> being  <a href="chain_1.html#NM4" title="CHAIN_1:NM.4">Chain</a> of <span class="p1">(<span class="default"><font color="Olive" title="b1">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 <br/>( <font color="Olive" title="b2">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">C</font> &amp;  <a href="chain_1.html#K11" title="CHAIN_1:func.11">del</a> <font color="Olive" title="b2">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="chain_1.html#K6" title="CHAIN_1:func.6">0_</a> <font color="Olive" title="b1">k'</font>,<font color="Maroon" title="c2">G</font> ) ) )
 <b>by </b><i><a class="txt" href="chain_1.html#E1:92_1_1_1_1"><i><font color="Green" title="E63">A1</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="92_1_1_1_2"><b>suppose </b></a><a NAME="E1:92_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="c4">k'</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E3:92_1_1_1_2"/><i><font color="Green" title="E63">A2</font></i>: 
<font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">k'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><b>reconsider </b><font color="Maroon" title="c5">k'</font> = <font color="Maroon" title="c4">k'</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">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>take </b><font color="Maroon" title="c6">C'</font> =  <a href="chain_1.html#K6" title="CHAIN_1:func.6">0_</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">G</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1_1_1_2_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon" title="c7">k'</font> = <font color="Maroon" title="c5">k'</font>;<br/><b>thus </b><a NAME="E1:92_1_1_1_2_1"/>
<font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">k'</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="chain_1.html#E3:92_1_1_1_2"><i><font color="Green" title="E63">A2</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c8">C</font> = <font color="Maroon" title="c6">C'</font> as   <a href="chain_1.html#NM4" title="CHAIN_1:NM.4">Chain</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c7">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>by </b><i><a class="txt" href="chain_1.html#E3:92_1_1_1_2"><i><font color="Green" title="E63">A2</font></i></a></i>;<br/><b>take </b><font color="Maroon" title="c9">C</font> = <font color="Maroon" title="c8">C</font>;<br/><b>thus </b><a NAME="E3:92_1_1_1_2_1"/>
( <font color="Maroon" title="c9">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">C'</font> &amp;  <a href="chain_1.html#K11" title="CHAIN_1:func.11">del</a> <font color="Maroon" title="c9">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="chain_1.html#K6" title="CHAIN_1:func.6">0_</a> <font color="Maroon" title="c7">k'</font>,<font color="Maroon" title="c2">G</font> )
 <b>by </b><i><a class="txt" href="chain_1.html#E3:92_1_1_1_2"><i><font color="Green" title="E63">A2</font></i></a>, <a class="ref" href="chain_1.html#T61" target="_self" title="CHAIN_1:th.61">Th61</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:92_1_1_1_2"/>
( ( <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>  &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c6">C'</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> ) or  ex <font color="Olive" title="b1">k'</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <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#K2" title="NAT_1:func.2">+</a> 1 &amp;  ex <font color="Olive" title="b2">C</font> being  <a href="chain_1.html#NM4" title="CHAIN_1:NM.4">Chain</a> of <span class="p1">(<span class="default"><font color="Olive" title="b1">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 <br/>( <font color="Olive" title="b2">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">C'</font> &amp;  <a href="chain_1.html#K11" title="CHAIN_1:func.11">del</a> <font color="Olive" title="b2">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="chain_1.html#K6" title="CHAIN_1:func.6">0_</a> <font color="Olive" title="b1">k'</font>,<font color="Maroon" title="c2">G</font> ) ) )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

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