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

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>for <font color="Olive">A</font> being  <a href="modal_1.html#NM5">MP-wff</a>  st  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Olive">A</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a<sub>1</sub></font> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive">A</font>];<br/>
<a NAME="E1:67_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:67_1_1"/><i><font color="Green">E62</font></i>: 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 0
 ;<br/>

<a NAME="E2:67_1_1"/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/>
<b>hence </b><a NAME="E3:67_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">A</font>]
 <b>by </b><i><a class="ref" href="card_2.html#T59">CARD_2:59</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:67_1"/><i><font color="Green">E63</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:67_1_2"/><i><font color="Green">E64</font></i>: 
for <font color="Olive">A</font> being  <a href="modal_1.html#NM5">MP-wff</a>  st  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Olive">A</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive">A</font>]
 ;<br/>

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

<b>assume </b><a NAME="E2:67_1_2"/><i><font color="Green">E69</font></i>: 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/>

<b>set </b><font color="Maroon">a</font> =  <a href="modal_1.html#K1" target="_self">Root</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">b</font> =  <a href="modal_1.html#K5" target="_self">branchdeg</a> <span class="p1">(<span class="default"><a href="modal_1.html#K1" target="_self">Root</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font></span>)</span></span>)</span>;<br/>
<a NAME="E3:67_1_2"/><i><font color="Green">E70</font></i>: 
 <a href="modal_1.html#K5" target="_self">branchdeg</a> <span class="p1">(<span class="default"><a href="modal_1.html#K1" target="_self">Root</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 2
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_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="E1:67_1_2_1_1"/>
(  <a href="modal_1.html#K5" target="_self">branchdeg</a> <span class="p1">(<span class="default"><a href="modal_1.html#K1" target="_self">Root</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0 or  <a href="modal_1.html#K5" target="_self">branchdeg</a> <span class="p1">(<span class="default"><a href="modal_1.html#K1" target="_self">Root</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1 or  <a href="modal_1.html#K5" target="_self">branchdeg</a> <span class="p1">(<span class="default"><a href="modal_1.html#K1" target="_self">Root</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 2 )
 </span><b>by </b><i>, <a class="ref" href="cqc_the1.html#T3">CQC_THE1:3</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:67_1_2_1_1_1"/>
 <a href="modal_1.html#K5" target="_self">branchdeg</a> <span class="p1">(<span class="default"><a href="modal_1.html#K1" target="_self">Root</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:67_1_2_1_1_1"/><b>then </b><i><font color="Green">E71</font></i>: 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i/>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_2_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_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="E1:67_1_2_1_1_1_1_1"/>
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="modal_1.html#K17" target="_self">VERUM</a>  or  ex <font color="Olive">p</font> being  <a href="modal_1.html#NM1">MP-variable</a> st <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="modal_1.html#K16" target="_self">@</a> <font color="Olive">p</font> )
 </span><b>by </b><i><a class="txt" href="modal_1.html#E1:67_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="modal_1.html#T4" target="_self">Th4</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_2_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:67_1_2_1_1_1_1_1_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="modal_1.html#K17" target="_self">VERUM</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:67_1_2_1_1_1_1_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">A</font>]
 <b>by </b><i><a class="ref" href="modal_1.html#T10" target="_self">Th10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_2_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:67_1_2_1_1_1_1_1_2"/>
 ex <font color="Olive">p</font> being  <a href="modal_1.html#NM1">MP-variable</a> st <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="modal_1.html#K16" target="_self">@</a> <font color="Olive">p</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:67_1_2_1_1_1_1_1_2"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">A</font>]
 <b>by </b><i><a class="ref" href="modal_1.html#T11" target="_self">Th11</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:67_1_2_1_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">A</font>]
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:67_1_2_1_1_2"/><i><font color="Green">E76</font></i>: 
 <a href="modal_1.html#K5" target="_self">branchdeg</a> <span class="p1">(<span class="default"><a href="modal_1.html#K1" target="_self">Root</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:67_1_2_1_1_2"/><b>then </b><i><font color="Green">E77</font></i>: 
 <a href="trees_2.html#K1">succ</a> <span class="p1">(<span class="default"><a href="modal_1.html#K1" target="_self">Root</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i/>;<br/><a NAME="E3:67_1_2_1_1_2"/><b>then </b>
<span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R2">in</a>  <a href="trees_2.html#K1">succ</a> <span class="p1">(<span class="default"><a href="modal_1.html#K1" target="_self">Root</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>then reconsider </b><font color="Maroon">o</font> = <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span> as    <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font> ;<br/><a NAME="E5:67_1_2_1_1_2"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="trees_1.html#K2">elementary_tree</a> 1</span>)</span> <a href="trees_2.html#K9">--&gt;</a> <span class="p2">(<span class="default"><a href="modal_1.html#K2" target="_self">Root</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="trees_2.html#K8">with-replacement</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span>,<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="modal_1.html#K7" target="_self">|</a> <font color="Maroon">o</font></span>)</span>
 <b>by </b><i>, </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_2_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_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="E1:67_1_2_1_1_2_1_1"/>
( <font color="Maroon">A</font> <a href="trees_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="modal_1.html#K1" target="_self">Root</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default">1,0</span><a href="domain_1.html#K1">]</a></span> or <font color="Maroon">A</font> <a href="trees_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="modal_1.html#K1" target="_self">Root</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default">1,1</span><a href="domain_1.html#K1">]</a></span> )
 </span><b>by </b><i><a class="txt" href="modal_1.html#E1:67_1"><i><font color="Green">E61</font></i></a>, </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_2_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:67_1_2_1_1_2_1_1_1"/>
<font color="Maroon">A</font> <a href="trees_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="modal_1.html#K1" target="_self">Root</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default">1,0</span><a href="domain_1.html#K1">]</a></span>
 ;<br/><div class="add"><a NAME="E2:67_1_2_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E79</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="modal_1.html#K10" target="_self">'not'</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="modal_1.html#K7" target="_self">|</a> <font color="Maroon">o</font></span>)</span>
 <b>by </b><i><a class="txt" href="modal_1.html#E2:67_1"><i><font color="Green">E63</font></i></a></i>;<br/><a NAME="E3:67_1_2_1_1_2_1_1_1"/><i><font color="Green">E80</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="modal_1.html#K7" target="_self">|</a> <font color="Maroon">o</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font></span>)</span> <a href="trees_1.html#K4">|</a> <font color="Maroon">o</font>
 <b>by </b><i><a class="ref" href="trees_2.html#D11">TREES_2:def 11</a></i>;<br/><a NAME="E4:67_1_2_1_1_2_1_1_1"/>
<font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="modal_1.html#K1" target="_self">Root</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font></span>)</span>
 ;<br/><a NAME="E5:67_1_2_1_1_2_1_1_1"/><b>then </b>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font></span>)</span> <a href="trees_1.html#K4">|</a> <font color="Maroon">o</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E6:67_1_2_1_1_2_1_1_1"/><b>then </b>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="modal_1.html#K7" target="_self">|</a> <font color="Maroon">o</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="modal_1.html#T17" target="_self">Th17</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E7:67_1_2_1_1_2_1_1_1"/><b>then </b>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="modal_1.html#K7" target="_self">|</a> <font color="Maroon">o</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E8:67_1_2_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">A</font> <a href="modal_1.html#K7" target="_self">|</a> <font color="Maroon">o</font>]
 <b>by </b><i><a class="ref" href="modal_1.html#T16" target="_self">Th16</a></i>;<br/><b>hence </b><a NAME="E9:67_1_2_1_1_2_1_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">A</font>]
 <b>by </b><i><a class="ref" href="modal_1.html#T12" target="_self">Th12</a>, <a class="txt" href="modal_1.html#E1:67_1_2"><i><font color="Green">E64</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_2_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:67_1_2_1_1_2_1_1_2"/>
<font color="Maroon">A</font> <a href="trees_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="modal_1.html#K1" target="_self">Root</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default">1,1</span><a href="domain_1.html#K1">]</a></span>
 ;<br/><div class="add"><a NAME="E2:67_1_2_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E81</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="modal_1.html#K11" target="_self">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="modal_1.html#K7" target="_self">|</a> <font color="Maroon">o</font></span>)</span>
 <b>by </b><i><a class="txt" href="modal_1.html#E2:67_1"><i><font color="Green">E63</font></i></a></i>;<br/><a NAME="E3:67_1_2_1_1_2_1_1_2"/><i><font color="Green">E82</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="modal_1.html#K7" target="_self">|</a> <font color="Maroon">o</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font></span>)</span> <a href="trees_1.html#K4">|</a> <font color="Maroon">o</font>
 <b>by </b><i><a class="ref" href="trees_2.html#D11">TREES_2:def 11</a></i>;<br/><a NAME="E4:67_1_2_1_1_2_1_1_2"/>
<font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="modal_1.html#K1" target="_self">Root</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font></span>)</span>
 ;<br/><a NAME="E5:67_1_2_1_1_2_1_1_2"/><b>then </b>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font></span>)</span> <a href="trees_1.html#K4">|</a> <font color="Maroon">o</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E6:67_1_2_1_1_2_1_1_2"/><b>then </b>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="modal_1.html#K7" target="_self">|</a> <font color="Maroon">o</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="modal_1.html#T17" target="_self">Th17</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E7:67_1_2_1_1_2_1_1_2"/><b>then </b>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="modal_1.html#K7" target="_self">|</a> <font color="Maroon">o</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E8:67_1_2_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">A</font> <a href="modal_1.html#K7" target="_self">|</a> <font color="Maroon">o</font>]
 <b>by </b><i><a class="ref" href="modal_1.html#T16" target="_self">Th16</a></i>;<br/><b>hence </b><a NAME="E9:67_1_2_1_1_2_1_1_2"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">A</font>]
 <b>by </b><i><a class="ref" href="modal_1.html#T13" target="_self">Th13</a>, <a class="txt" href="modal_1.html#E1:67_1_2"><i><font color="Green">E64</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="E7:67_1_2_1_1_2"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">A</font>]
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_2_1_1_3"><b>suppose </b></a><a NAME="E1:67_1_2_1_1_3"/><i><font color="Green">E83</font></i>: 
 <a href="modal_1.html#K5" target="_self">branchdeg</a> <span class="p1">(<span class="default"><a href="modal_1.html#K1" target="_self">Root</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 2
 ;<br/><div class="add"><a NAME="E2:67_1_2_1_1_3"/><b>then </b><i><font color="Green">E84</font></i>: 
 <a href="trees_2.html#K1">succ</a> <span class="p1">(<span class="default"><a href="modal_1.html#K1" target="_self">Root</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span>,<span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12">*&gt;</a></span></span><a href="domain_1.html#K7">}</a></span>
 <b>by </b><i/>;<br/><a NAME="E3:67_1_2_1_1_3"/><b>then </b>
( <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R2">in</a>  <a href="trees_2.html#K1">succ</a> <span class="p1">(<span class="default"><a href="modal_1.html#K1" target="_self">Root</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font></span>)</span></span>)</span> &amp; <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R2">in</a>  <a href="trees_2.html#K1">succ</a> <span class="p1">(<span class="default"><a href="modal_1.html#K1" target="_self">Root</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><b>then reconsider </b><font color="Maroon">o1</font> = <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span>, <font color="Maroon">o2</font> = <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12">*&gt;</a></span> as    <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font> ;<br/><a NAME="E5:67_1_2_1_1_3"/><i><font color="Green">E130</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</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">elementary_tree</a> 2</span>)</span> <a href="trees_2.html#K9">--&gt;</a> <span class="p3">(<span class="default"><a href="modal_1.html#K2" target="_self">Root</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="trees_2.html#K8">with-replacement</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span>,<span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="modal_1.html#K7" target="_self">|</a> <font color="Maroon">o1</font></span>)</span></span>)</span> <a href="trees_2.html#K8">with-replacement</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12">*&gt;</a></span>,<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="modal_1.html#K7" target="_self">|</a> <font color="Maroon">o2</font></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E6:67_1_2_1_1_3"/>
 <a href="modal_1.html#K2" target="_self">Root</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default">2,0</span><a href="domain_1.html#K1">]</a></span>
 <b>by </b><i><a class="txt" href="modal_1.html#E1:67_1"><i><font color="Green">E61</font></i></a>, </i>;<br/><a NAME="E7:67_1_2_1_1_3"/><b>then </b><i><font color="Green">E131</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="modal_1.html#K7" target="_self">|</a> <font color="Maroon">o1</font></span>)</span> <a href="modal_1.html#K12" target="_self">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="modal_1.html#K7" target="_self">|</a> <font color="Maroon">o2</font></span>)</span>
 <b>by </b><i><a class="txt" href="modal_1.html#E2:67_1"><i><font color="Green">E63</font></i></a></i>;<br/><a NAME="E8:67_1_2_1_1_3"/><i><font color="Green">E132</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="modal_1.html#K7" target="_self">|</a> <font color="Maroon">o1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font></span>)</span> <a href="trees_1.html#K4">|</a> <font color="Maroon">o1</font>
 <b>by </b><i><a class="ref" href="trees_2.html#D11">TREES_2:def 11</a></i>;<br/><a NAME="E9:67_1_2_1_1_3"/>
<font color="Maroon">o1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="modal_1.html#K1" target="_self">Root</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font></span>)</span>
 ;<br/><a NAME="E10:67_1_2_1_1_3"/><b>then </b>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font></span>)</span> <a href="trees_1.html#K4">|</a> <font color="Maroon">o1</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E11:67_1_2_1_1_3"/><b>then </b>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="modal_1.html#K7" target="_self">|</a> <font color="Maroon">o1</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="modal_1.html#T17" target="_self">Th17</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E12:67_1_2_1_1_3"/><b>then </b>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="modal_1.html#K7" target="_self">|</a> <font color="Maroon">o1</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E13:67_1_2_1_1_3"/><b>then </b><i><font color="Green">E133</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">A</font> <a href="modal_1.html#K7" target="_self">|</a> <font color="Maroon">o1</font>]
 <b>by </b><i><a class="ref" href="modal_1.html#T16" target="_self">Th16</a></i>;<br/><a NAME="E14:67_1_2_1_1_3"/><i><font color="Green">E134</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="modal_1.html#K7" target="_self">|</a> <font color="Maroon">o2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font></span>)</span> <a href="trees_1.html#K4">|</a> <font color="Maroon">o2</font>
 <b>by </b><i><a class="ref" href="trees_2.html#D11">TREES_2:def 11</a></i>;<br/><a NAME="E15:67_1_2_1_1_3"/>
<font color="Maroon">o2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="modal_1.html#K1" target="_self">Root</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font></span>)</span>
 ;<br/><a NAME="E16:67_1_2_1_1_3"/><b>then </b>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font></span>)</span> <a href="trees_1.html#K4">|</a> <font color="Maroon">o2</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E17:67_1_2_1_1_3"/><b>then </b>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="modal_1.html#K7" target="_self">|</a> <font color="Maroon">o2</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E18:67_1_2_1_1_3"/><b>then </b>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="modal_1.html#K7" target="_self">|</a> <font color="Maroon">o2</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E19:67_1_2_1_1_3"/><b>then </b>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">A</font> <a href="modal_1.html#K7" target="_self">|</a> <font color="Maroon">o2</font>]
 <b>by </b><i><a class="ref" href="modal_1.html#T16" target="_self">Th16</a></i>;<br/><b>hence </b><a NAME="E20:67_1_2_1_1_3"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">A</font>]
 <b>by </b><i><a class="ref" href="modal_1.html#T15" target="_self">Th15</a>, <a class="txt" href="modal_1.html#E1:67_1_2"><i><font color="Green">E64</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="E5:67_1_2"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">A</font>]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:67_1"/><i><font color="Green">E136</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<b>let </b><font color="Maroon">A</font> be   <a href="modal_1.html#NM5">MP-wff</a>;<br/>

<a NAME="E4:67_1"/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font></span>)</span>
 
;<br/>
<b>hence </b><a NAME="E5:67_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">A</font>]
 <b>by </b><i><a class="ref" href="modal_1.html#T16" target="_self">Th16</a></i>;<br/>


</div>
