<?xml version="1.0"?>
<div class="add">

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <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>means </b>for <font color="Olive" title="b1">A</font> being  <a href="modal_1.html#NM5" title="MODAL_1:NM.5">MP-wff</a>  st  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b1">A</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> $1 holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">A</font>];<br/>
<a NAME="E1:60_1"/><i><font color="Green" title="E49">A6</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c1">A</font> be   <a href="modal_1.html#NM5" title="MODAL_1:NM.5">MP-wff</a>;<br/>

<b>assume </b><a NAME="E1:60_1_1"/><i><font color="Green" title="E50">A7</font></i>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/>

<a NAME="E2:60_1_1"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="modal_1.html#E1:60_1_1"><i><font color="Green" title="E50">A7</font></i></a>, <a class="ref" href="nat_1.html#T2" title="NAT_1:th.2">NAT_1:2</a></i>;<br/>
<b>hence </b><a NAME="E3:60_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">A</font>]
 <b>by </b><i><a class="ref" href="card_2.html#T59" title="CARD_2:th.59">CARD_2:59</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:60_1"/><i><font color="Green" title="E50">A8</font></i>: 
for <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 <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c1">k</font> be    <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> ;<br/>

<b>assume </b><a NAME="E1:60_1_2"/><i><font color="Green" title="E51">A9</font></i>: 
for <font color="Olive" title="b1">A</font> being  <a href="modal_1.html#NM5" title="MODAL_1:NM.5">MP-wff</a>  st  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b1">A</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">k</font> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">A</font>]
 ;<br/>

<b>let </b><font color="Maroon" title="c2">A</font> be   <a href="modal_1.html#NM5" title="MODAL_1:NM.5">MP-wff</a>;<br/>

<b>assume </b><a NAME="E2:60_1_2"/><i><font color="Green" title="E52">A10</font></i>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ;<br/>

<b>set </b><font color="Maroon" title="c3">a</font> =  <a href="modal_1.html#K1" title="MODAL_1:func.1">Root</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">A</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c4">b</font> =  <a href="modal_1.html#K5" title="MODAL_1:func.5">branchdeg</a> <span class="p1">(<span class="default"><a href="modal_1.html#K1" title="MODAL_1:func.1">Root</a> <span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span>;<br/>
<a NAME="E3:60_1_2"/><i><font color="Green" title="E53">A11</font></i>: 
 <a href="modal_1.html#K5" title="MODAL_1:func.5">branchdeg</a> <span class="p1">(<span class="default"><a href="modal_1.html#K1" title="MODAL_1:func.1">Root</a> <span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2
 
<b>by </b><i><a class="ref" href="modal_1.html#D6" target="_self" title="MODAL_1:def.6">Def6</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:60_1_2_1_1"/>
(  <a href="modal_1.html#K5" title="MODAL_1:func.5">branchdeg</a> <span class="p1">(<span class="default"><a href="modal_1.html#K1" title="MODAL_1:func.1">Root</a> <span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or  <a href="modal_1.html#K5" title="MODAL_1:func.5">branchdeg</a> <span class="p1">(<span class="default"><a href="modal_1.html#K1" title="MODAL_1:func.1">Root</a> <span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or  <a href="modal_1.html#K5" title="MODAL_1:func.5">branchdeg</a> <span class="p1">(<span class="default"><a href="modal_1.html#K1" title="MODAL_1:func.1">Root</a> <span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 </span><b>by </b><i><a class="txt" href="modal_1.html#E3:60_1_2"><i><font color="Green" title="E53">A11</font></i></a>, <a class="ref" href="nat_1.html#T27" title="NAT_1:th.27">NAT_1:27</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:60_1_2_1_1_1"/>
 <a href="modal_1.html#K5" title="MODAL_1:func.5">branchdeg</a> <span class="p1">(<span class="default"><a href="modal_1.html#K1" title="MODAL_1:func.1">Root</a> <span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span> <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"><a NAME="E2:60_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E54">A12</font></i>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="ref" href="modal_1.html#T22" target="_self" title="MODAL_1:th.22">Th22</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_2_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_1_2_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:60_1_2_1_1_1_1_1"/>
( <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modal_1.html#K17" title="MODAL_1:func.17">VERUM</a>  or  ex <font color="Olive" title="b1">p</font> being  <a href="modal_1.html#NM1" title="MODAL_1:NM.1">MP-variable</a> st <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modal_1.html#K16" title="MODAL_1:func.16">@</a> <font color="Olive" title="b1">p</font> )
 </span><b>by </b><i><a class="txt" href="modal_1.html#E2:60_1_2_1_1_1"><i><font color="Green" title="E54">A12</font></i></a>, <a class="ref" href="modal_1.html#T42" target="_self" title="MODAL_1:th.42">Th42</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_1_2_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:60_1_2_1_1_1_1_1_1"/>
<font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modal_1.html#K17" title="MODAL_1:func.17">VERUM</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:60_1_2_1_1_1_1_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">A</font>]
 <b>by </b><i><a class="txt" href="modal_1.html#E1:60"><i><font color="Green" title="E44">A1</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_1_2_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:60_1_2_1_1_1_1_1_2"/>
 ex <font color="Olive" title="b1">p</font> being  <a href="modal_1.html#NM1" title="MODAL_1:NM.1">MP-variable</a> st <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modal_1.html#K16" title="MODAL_1:func.16">@</a> <font color="Olive" title="b1">p</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:60_1_2_1_1_1_1_1_2"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">A</font>]
 <b>by </b><i><a class="txt" href="modal_1.html#E2:60"><i><font color="Green" title="E45">A2</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:60_1_2_1_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">A</font>]
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:60_1_2_1_1_2"/><i><font color="Green" title="E54">A13</font></i>: 
 <a href="modal_1.html#K5" title="MODAL_1:func.5">branchdeg</a> <span class="p1">(<span class="default"><a href="modal_1.html#K1" title="MODAL_1:func.1">Root</a> <span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:60_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E55">A14</font></i>: 
 <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <span class="p1">(<span class="default"><a href="modal_1.html#K1" title="MODAL_1:func.1">Root</a> <span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="modal_1.html#T23" target="_self" title="MODAL_1:th.23">Th23</a></i>;<br/><a NAME="E3:60_1_2_1_1_2"/><b>then </b>
<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <span class="p1">(<span class="default"><a href="modal_1.html#K1" title="MODAL_1:func.1">Root</a> <span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c5">o</font> = <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> as    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">A</font> ;<br/><a NAME="E5:60_1_2_1_1_2"/><i><font color="Green" title="E56">A15</font></i>: 
<font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> 1</span>)</span> <a href="trees_2.html#K9" title="TREES_2:func.9">--&gt;</a> <span class="p2">(<span class="default"><a href="modal_1.html#K2" title="MODAL_1:func.2">Root</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span> <a href="trees_2.html#K8" title="TREES_2:func.8">with-replacement</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="modal_1.html#K7" title="MODAL_1:func.7">|</a> <font color="Maroon" title="c5">o</font></span>)</span>
 <b>by </b><i><a class="txt" href="modal_1.html#E2:60_1_2_1_1_2"><i><font color="Green" title="E55">A14</font></i></a>, <a class="ref" href="modal_1.html#T28" target="_self" title="MODAL_1:th.28">Th28</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_2_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_1_2_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:60_1_2_1_1_2_1_1"/>
( <font color="Maroon" title="c2">A</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p1">(<span class="default"><a href="modal_1.html#K1" title="MODAL_1:func.1">Root</a> <span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> or <font color="Maroon" title="c2">A</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p1">(<span class="default"><a href="modal_1.html#K1" title="MODAL_1:func.1">Root</a> <span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">1,1</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> )
 </span><b>by </b><i><a class="txt" href="modal_1.html#E1:60_1_2_1_1_2"><i><font color="Green" title="E54">A13</font></i></a>, <a class="ref" href="modal_1.html#D6" target="_self" title="MODAL_1:def.6">Def6</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_1_2_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:60_1_2_1_1_2_1_1_1"/>
<font color="Maroon" title="c2">A</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p1">(<span class="default"><a href="modal_1.html#K1" title="MODAL_1:func.1">Root</a> <span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 ;<br/><div class="add"><a NAME="E2:60_1_2_1_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E57">A16</font></i>: 
<font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modal_1.html#K10" title="MODAL_1:func.10">'not'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="modal_1.html#K7" title="MODAL_1:func.7">|</a> <font color="Maroon" title="c5">o</font></span>)</span>
 <b>by </b><i><a class="txt" href="modal_1.html#E5:60_1_2_1_1_2"><i><font color="Green" title="E56">A15</font></i></a></i>;<br/><a NAME="E3:60_1_2_1_1_2_1_1_1"/><i><font color="Green" title="E58">A17</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="modal_1.html#K7" title="MODAL_1:func.7">|</a> <font color="Maroon" title="c5">o</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="trees_1.html#K4" title="TREES_1:func.4">|</a> <font color="Maroon" title="c5">o</font>
 <b>by </b><i><a class="ref" href="trees_2.html#D11" title="TREES_2:def.11">TREES_2:def 11</a></i>;<br/><a NAME="E4:60_1_2_1_1_2_1_1_1"/>
<font color="Maroon" title="c5">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="modal_1.html#K1" title="MODAL_1:func.1">Root</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">A</font></span>)</span>
 ;<br/><a NAME="E5:60_1_2_1_1_2_1_1_1"/><b>then </b>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="modal_1.html#K7" title="MODAL_1:func.7">|</a> <font color="Maroon" title="c5">o</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="modal_1.html#E2:60_1_2"><i><font color="Green" title="E52">A10</font></i></a>, <a class="txt" href="modal_1.html#E3:60_1_2_1_1_2_1_1_1"><i><font color="Green" title="E58">A17</font></i></a>, <a class="ref" href="modal_1.html#T26" target="_self" title="MODAL_1:th.26">Th26</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E6:60_1_2_1_1_2_1_1_1"/><b>then </b>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="modal_1.html#K7" title="MODAL_1:func.7">|</a> <font color="Maroon" title="c5">o</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>hence </b><a NAME="E7:60_1_2_1_1_2_1_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">A</font>]
 <b>by </b><i><a class="txt" href="modal_1.html#E3:60"><i><font color="Green" title="E46">A3</font></i></a>, <a class="txt" href="modal_1.html#E1:60_1_2"><i><font color="Green" title="E51">A9</font></i></a>, <a class="txt" href="modal_1.html#E2:60_1_2_1_1_2_1_1_1"><i><font color="Green" title="E57">A16</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_1_2_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:60_1_2_1_1_2_1_1_2"/>
<font color="Maroon" title="c2">A</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p1">(<span class="default"><a href="modal_1.html#K1" title="MODAL_1:func.1">Root</a> <span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">1,1</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 ;<br/><div class="add"><a NAME="E2:60_1_2_1_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E57">A18</font></i>: 
<font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modal_1.html#K11" title="MODAL_1:func.11">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="modal_1.html#K7" title="MODAL_1:func.7">|</a> <font color="Maroon" title="c5">o</font></span>)</span>
 <b>by </b><i><a class="txt" href="modal_1.html#E5:60_1_2_1_1_2"><i><font color="Green" title="E56">A15</font></i></a></i>;<br/><a NAME="E3:60_1_2_1_1_2_1_1_2"/><i><font color="Green" title="E58">A19</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="modal_1.html#K7" title="MODAL_1:func.7">|</a> <font color="Maroon" title="c5">o</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="trees_1.html#K4" title="TREES_1:func.4">|</a> <font color="Maroon" title="c5">o</font>
 <b>by </b><i><a class="ref" href="trees_2.html#D11" title="TREES_2:def.11">TREES_2:def 11</a></i>;<br/><a NAME="E4:60_1_2_1_1_2_1_1_2"/>
<font color="Maroon" title="c5">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="modal_1.html#K1" title="MODAL_1:func.1">Root</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">A</font></span>)</span>
 ;<br/><a NAME="E5:60_1_2_1_1_2_1_1_2"/><b>then </b>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="modal_1.html#K7" title="MODAL_1:func.7">|</a> <font color="Maroon" title="c5">o</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="modal_1.html#E2:60_1_2"><i><font color="Green" title="E52">A10</font></i></a>, <a class="txt" href="modal_1.html#E3:60_1_2_1_1_2_1_1_2"><i><font color="Green" title="E58">A19</font></i></a>, <a class="ref" href="modal_1.html#T26" target="_self" title="MODAL_1:th.26">Th26</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E6:60_1_2_1_1_2_1_1_2"/><b>then </b>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="modal_1.html#K7" title="MODAL_1:func.7">|</a> <font color="Maroon" title="c5">o</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>hence </b><a NAME="E7:60_1_2_1_1_2_1_1_2"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">A</font>]
 <b>by </b><i><a class="txt" href="modal_1.html#E4:60"><i><font color="Green" title="E47">A4</font></i></a>, <a class="txt" href="modal_1.html#E1:60_1_2"><i><font color="Green" title="E51">A9</font></i></a>, <a class="txt" href="modal_1.html#E2:60_1_2_1_1_2_1_1_2"><i><font color="Green" title="E57">A18</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E7:60_1_2_1_1_2"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">A</font>]
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_1_2_1_1_3"><b>suppose </b></a><a NAME="E1:60_1_2_1_1_3"/><i><font color="Green" title="E54">A20</font></i>: 
 <a href="modal_1.html#K5" title="MODAL_1:func.5">branchdeg</a> <span class="p1">(<span class="default"><a href="modal_1.html#K1" title="MODAL_1:func.1">Root</a> <span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ;<br/><div class="add"><a NAME="E2:60_1_2_1_1_3"/><b>then </b><i><font color="Green" title="E55">A21</font></i>: 
 <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <span class="p1">(<span class="default"><a href="modal_1.html#K1" title="MODAL_1:func.1">Root</a> <span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>,<span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 <b>by </b><i><a class="ref" href="modal_1.html#T24" target="_self" title="MODAL_1:th.24">Th24</a></i>;<br/><a NAME="E3:60_1_2_1_1_3"/><b>then </b>
( <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <span class="p1">(<span class="default"><a href="modal_1.html#K1" title="MODAL_1:func.1">Root</a> <span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span> &amp; <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <span class="p1">(<span class="default"><a href="modal_1.html#K1" title="MODAL_1:func.1">Root</a> <span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c5">o1</font> = <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>, <font color="Maroon" title="c6">o2</font> = <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> as    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">A</font> ;<br/><a NAME="E5:60_1_2_1_1_3"/>
<font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> 2</span>)</span> <a href="trees_2.html#K9" title="TREES_2:func.9">--&gt;</a> <span class="p3">(<span class="default"><a href="modal_1.html#K2" title="MODAL_1:func.2">Root</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span> <a href="trees_2.html#K8" title="TREES_2:func.8">with-replacement</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="modal_1.html#K7" title="MODAL_1:func.7">|</a> <font color="Maroon" title="c5">o1</font></span>)</span></span>)</span> <a href="trees_2.html#K8" title="TREES_2:func.8">with-replacement</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="modal_1.html#K7" title="MODAL_1:func.7">|</a> <font color="Maroon" title="c6">o2</font></span>)</span>
 <b>by </b><i><a class="txt" href="modal_1.html#E2:60_1_2_1_1_3"><i><font color="Green" title="E55">A21</font></i></a>, <a class="ref" href="modal_1.html#T30" target="_self" title="MODAL_1:th.30">Th30</a></i>;<br/><a NAME="E6:60_1_2_1_1_3"/><b>then </b><i><font color="Green" title="E56">A22</font></i>: 
<font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="modal_1.html#K7" title="MODAL_1:func.7">|</a> <font color="Maroon" title="c5">o1</font></span>)</span> <a href="modal_1.html#K12" title="MODAL_1:func.12">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="modal_1.html#K7" title="MODAL_1:func.7">|</a> <font color="Maroon" title="c6">o2</font></span>)</span>
 <b>by </b><i><a class="txt" href="modal_1.html#E1:60_1_2_1_1_3"><i><font color="Green" title="E54">A20</font></i></a>, <a class="ref" href="modal_1.html#D6" target="_self" title="MODAL_1:def.6">Def6</a></i>;<br/><a NAME="E7:60_1_2_1_1_3"/><i><font color="Green" title="E57">A23</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="modal_1.html#K7" title="MODAL_1:func.7">|</a> <font color="Maroon" title="c5">o1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="trees_1.html#K4" title="TREES_1:func.4">|</a> <font color="Maroon" title="c5">o1</font>
 <b>by </b><i><a class="ref" href="trees_2.html#D11" title="TREES_2:def.11">TREES_2:def 11</a></i>;<br/><a NAME="E8:60_1_2_1_1_3"/>
<font color="Maroon" title="c5">o1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="modal_1.html#K1" title="MODAL_1:func.1">Root</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">A</font></span>)</span>
 ;<br/><a NAME="E9:60_1_2_1_1_3"/><b>then </b>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="modal_1.html#K7" title="MODAL_1:func.7">|</a> <font color="Maroon" title="c5">o1</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="modal_1.html#E2:60_1_2"><i><font color="Green" title="E52">A10</font></i></a>, <a class="txt" href="modal_1.html#E7:60_1_2_1_1_3"><i><font color="Green" title="E57">A23</font></i></a>, <a class="ref" href="modal_1.html#T26" target="_self" title="MODAL_1:th.26">Th26</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E10:60_1_2_1_1_3"/><b>then </b>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="modal_1.html#K7" title="MODAL_1:func.7">|</a> <font color="Maroon" title="c5">o1</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E11:60_1_2_1_1_3"/><b>then </b><i><font color="Green" title="E58">A24</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">A</font> <a href="modal_1.html#K7" title="MODAL_1:func.7">|</a> <font color="Maroon" title="c5">o1</font>]
 <b>by </b><i><a class="txt" href="modal_1.html#E1:60_1_2"><i><font color="Green" title="E51">A9</font></i></a></i>;<br/><a NAME="E12:60_1_2_1_1_3"/><i><font color="Green" title="E59">A25</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="modal_1.html#K7" title="MODAL_1:func.7">|</a> <font color="Maroon" title="c6">o2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="trees_1.html#K4" title="TREES_1:func.4">|</a> <font color="Maroon" title="c6">o2</font>
 <b>by </b><i><a class="ref" href="trees_2.html#D11" title="TREES_2:def.11">TREES_2:def 11</a></i>;<br/><a NAME="E13:60_1_2_1_1_3"/>
<font color="Maroon" title="c6">o2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="modal_1.html#K1" title="MODAL_1:func.1">Root</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">A</font></span>)</span>
 ;<br/><a NAME="E14:60_1_2_1_1_3"/><b>then </b>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="modal_1.html#K7" title="MODAL_1:func.7">|</a> <font color="Maroon" title="c6">o2</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="modal_1.html#E2:60_1_2"><i><font color="Green" title="E52">A10</font></i></a>, <a class="txt" href="modal_1.html#E12:60_1_2_1_1_3"><i><font color="Green" title="E59">A25</font></i></a>, <a class="ref" href="modal_1.html#T26" target="_self" title="MODAL_1:th.26">Th26</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E15:60_1_2_1_1_3"/><b>then </b>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="modal_1.html#K7" title="MODAL_1:func.7">|</a> <font color="Maroon" title="c6">o2</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E16:60_1_2_1_1_3"/><b>then </b>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">A</font> <a href="modal_1.html#K7" title="MODAL_1:func.7">|</a> <font color="Maroon" title="c6">o2</font>]
 <b>by </b><i><a class="txt" href="modal_1.html#E1:60_1_2"><i><font color="Green" title="E51">A9</font></i></a></i>;<br/><b>hence </b><a NAME="E17:60_1_2_1_1_3"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">A</font>]
 <b>by </b><i><a class="txt" href="modal_1.html#E5:60"><i><font color="Green" title="E48">A5</font></i></a>, <a class="txt" href="modal_1.html#E6:60_1_2_1_1_3"><i><font color="Green" title="E56">A22</font></i></a>, <a class="txt" href="modal_1.html#E11:60_1_2_1_1_3"><i><font color="Green" title="E58">A24</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E5:60_1_2"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">A</font>]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:60_1"/><i><font color="Green" title="E51">A26</font></i>: 
for <font color="Olive" title="b1">n</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>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="modal_1.html#E1:60_1"><i><font color="Green" title="E49">A6</font></i></a>, <a class="txt" href="modal_1.html#E2:60_1"><i><font color="Green" title="E50">A8</font></i></a>);<br/></i>
<b>let </b><font color="Maroon" title="c1">A</font> be   <a href="modal_1.html#NM5" title="MODAL_1:NM.5">MP-wff</a>;<br/>

<a NAME="E4:60_1"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">A</font></span>)</span>
 
;<br/>
<b>hence </b><a NAME="E5:60_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">A</font>]
 <b>by </b><i><a class="txt" href="modal_1.html#E3:60_1"><i><font color="Green" title="E51">A26</font></i></a></i>;<br/>


</div>
