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

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means for <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></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">b<sub>1</sub></font>];<br/>
<a NAME="E1:67_1"/><i><font color="Green">E56</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">c<sub>1</sub></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">E57</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">c<sub>1</sub></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">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="txt" href="modal_1.html#E1:67_1_1"><i><font color="Green">E57</font></i></a>, <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">c<sub>1</sub></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">E57</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></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">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:67_1_2"/><i><font color="Green">E58</font></i>: 
for <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 ;<br/>

<b>let </b><font color="Maroon">c<sub>2</sub></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">E59</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">c<sub>2</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1
 ;<br/>

<b>set </b><font color="Maroon">c<sub>3</sub></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">c<sub>2</sub></font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></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">c<sub>2</sub></font></span>)</span></span>)</span>;<br/>
<a NAME="E3:67_1_2"/><i><font color="Green">E60</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">c<sub>2</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 2
 
<b>by </b><i><a class="ref" href="modal_1.html#D6" target="_self">Def6</a></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">c<sub>2</sub></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">c<sub>2</sub></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">c<sub>2</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 2 )
 </span><b>by </b><i><a class="txt" href="modal_1.html#E3:67_1_2"><i><font color="Green">E60</font></i></a>, <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">c<sub>2</sub></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">E61</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">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="modal_1.html#T22" target="_self">Th22</a></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">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="modal_1.html#K17" target="_self">VERUM</a>  or ex <font color="Olive">b<sub>1</sub></font> being  <a href="modal_1.html#NM1">MP-variable</a> st <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="modal_1.html#K16" target="_self">@</a> <font color="Olive">b<sub>1</sub></font> )
 </span><b>by </b><i><a class="txt" href="modal_1.html#E2:67_1_2_1_1_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="modal_1.html#T42" target="_self">Th42</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">c<sub>2</sub></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">c<sub>2</sub></font>]
 <b>by </b><i><a class="txt" href="modal_1.html#E1:67"><i><font color="Green">E51</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_1_1_1_2"><b>suppose </b></a><a NAME="E1:67_1_2_1_1_1_1_1_2"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="modal_1.html#NM1">MP-variable</a> st <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="modal_1.html#K16" target="_self">@</a> <font color="Olive">b<sub>1</sub></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">c<sub>2</sub></font>]
 <b>by </b><i><a class="txt" href="modal_1.html#E2:67"><i><font color="Green">E52</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="E4:67_1_2_1_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></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">E61</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">c<sub>2</sub></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">E62</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">c<sub>2</sub></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><a class="ref" href="modal_1.html#T23" target="_self">Th23</a></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">c<sub>2</sub></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">c<sub>5</sub></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">c<sub>2</sub></font> ;<br/><a NAME="E5:67_1_2_1_1_2"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">c<sub>2</sub></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">c<sub>2</sub></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">c<sub>2</sub></font> <a href="modal_1.html#K7" target="_self">|</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="modal_1.html#E2:67_1_2_1_1_2"><i><font color="Green">E62</font></i></a>, <a class="ref" href="modal_1.html#T28" target="_self">Th28</a></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">c<sub>2</sub></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">c<sub>2</sub></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">c<sub>2</sub></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">c<sub>2</sub></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_2_1_1_2"><i><font color="Green">E61</font></i></a>, <a class="ref" href="modal_1.html#D6" target="_self">Def6</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">c<sub>2</sub></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">c<sub>2</sub></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">E64</font></i>: 
<font color="Maroon">c<sub>2</sub></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">c<sub>2</sub></font> <a href="modal_1.html#K7" target="_self">|</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="modal_1.html#E5:67_1_2_1_1_2"><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">E65</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="modal_1.html#K7" target="_self">|</a> <font color="Maroon">c<sub>5</sub></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">c<sub>2</sub></font></span>)</span> <a href="trees_1.html#K4">|</a> <font color="Maroon">c<sub>5</sub></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">c<sub>5</sub></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">c<sub>2</sub></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">c<sub>2</sub></font></span>)</span> <a href="trees_1.html#K4">|</a> <font color="Maroon">c<sub>5</sub></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">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="modal_1.html#T26" target="_self">Th26</a></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">c<sub>2</sub></font> <a href="modal_1.html#K7" target="_self">|</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="modal_1.html#E2:67_1_2"><i><font color="Green">E59</font></i></a>, <a class="txt" href="modal_1.html#E3:67_1_2_1_1_2_1_1_1"><i><font color="Green">E65</font></i></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">c<sub>2</sub></font> <a href="modal_1.html#K7" target="_self">|</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font> <a href="modal_1.html#K7" target="_self">|</a> <font color="Maroon">c<sub>5</sub></font>]
 <b>by </b><i><a class="txt" href="modal_1.html#E1:67_1_2"><i><font color="Green">E58</font></i></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">c<sub>2</sub></font>]
 <b>by </b><i><a class="txt" href="modal_1.html#E3:67"><i><font color="Green">E53</font></i></a>, <a class="txt" href="modal_1.html#E2:67_1_2_1_1_2_1_1_1"><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">c<sub>2</sub></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">c<sub>2</sub></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">E64</font></i>: 
<font color="Maroon">c<sub>2</sub></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">c<sub>2</sub></font> <a href="modal_1.html#K7" target="_self">|</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="modal_1.html#E5:67_1_2_1_1_2"><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">E65</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="modal_1.html#K7" target="_self">|</a> <font color="Maroon">c<sub>5</sub></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">c<sub>2</sub></font></span>)</span> <a href="trees_1.html#K4">|</a> <font color="Maroon">c<sub>5</sub></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">c<sub>5</sub></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">c<sub>2</sub></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">c<sub>2</sub></font></span>)</span> <a href="trees_1.html#K4">|</a> <font color="Maroon">c<sub>5</sub></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">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="modal_1.html#T26" target="_self">Th26</a></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">c<sub>2</sub></font> <a href="modal_1.html#K7" target="_self">|</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="modal_1.html#E2:67_1_2"><i><font color="Green">E59</font></i></a>, <a class="txt" href="modal_1.html#E3:67_1_2_1_1_2_1_1_2"><i><font color="Green">E65</font></i></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">c<sub>2</sub></font> <a href="modal_1.html#K7" target="_self">|</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font> <a href="modal_1.html#K7" target="_self">|</a> <font color="Maroon">c<sub>5</sub></font>]
 <b>by </b><i><a class="txt" href="modal_1.html#E1:67_1_2"><i><font color="Green">E58</font></i></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">c<sub>2</sub></font>]
 <b>by </b><i><a class="txt" href="modal_1.html#E4:67"><i><font color="Green">E54</font></i></a>, <a class="txt" href="modal_1.html#E2:67_1_2_1_1_2_1_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">c<sub>2</sub></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">E61</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">c<sub>2</sub></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">E62</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">c<sub>2</sub></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><a class="ref" href="modal_1.html#T24" target="_self">Th24</a></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">c<sub>2</sub></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">c<sub>2</sub></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">c<sub>5</sub></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">c<sub>6</sub></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">c<sub>2</sub></font> ;<br/><a NAME="E5:67_1_2_1_1_3"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">c<sub>2</sub></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">c<sub>2</sub></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">c<sub>2</sub></font> <a href="modal_1.html#K7" target="_self">|</a> <font color="Maroon">c<sub>5</sub></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">c<sub>2</sub></font> <a href="modal_1.html#K7" target="_self">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="modal_1.html#E2:67_1_2_1_1_3"><i><font color="Green">E62</font></i></a>, <a class="ref" href="modal_1.html#T30" target="_self">Th30</a></i>;<br/> <a href="modal_1.html#K2" target="_self">Root</a> <font color="Maroon">c<sub>2</sub></font> = 
<font color="Maroon">c<sub>2</sub></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">c<sub>2</sub></font></span>)</span></span>)</span>

<br/>.= 
<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_2_1_1_3"><i><font color="Green">E61</font></i></a>, <a class="ref" href="modal_1.html#D6" target="_self">Def6</a></i>
;<br/><a NAME="E7:67_1_2_1_1_3"/><b>then </b><i><font color="Green">E64</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="modal_1.html#K7" target="_self">|</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="modal_1.html#K12" target="_self">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="modal_1.html#K7" target="_self">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="modal_1.html#E5:67_1_2_1_1_3"><i><font color="Green">E63</font></i></a></i>;<br/><a NAME="E8:67_1_2_1_1_3"/><i><font color="Green">E65</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="modal_1.html#K7" target="_self">|</a> <font color="Maroon">c<sub>5</sub></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">c<sub>2</sub></font></span>)</span> <a href="trees_1.html#K4">|</a> <font color="Maroon">c<sub>5</sub></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">c<sub>5</sub></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">c<sub>2</sub></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">c<sub>2</sub></font></span>)</span> <a href="trees_1.html#K4">|</a> <font color="Maroon">c<sub>5</sub></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">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="modal_1.html#T26" target="_self">Th26</a></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">c<sub>2</sub></font> <a href="modal_1.html#K7" target="_self">|</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="modal_1.html#E2:67_1_2"><i><font color="Green">E59</font></i></a>, <a class="txt" href="modal_1.html#E8:67_1_2_1_1_3"><i><font color="Green">E65</font></i></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">c<sub>2</sub></font> <a href="modal_1.html#K7" target="_self">|</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></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">E66</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font> <a href="modal_1.html#K7" target="_self">|</a> <font color="Maroon">c<sub>5</sub></font>]
 <b>by </b><i><a class="txt" href="modal_1.html#E1:67_1_2"><i><font color="Green">E58</font></i></a></i>;<br/><a NAME="E14:67_1_2_1_1_3"/><i><font color="Green">E67</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="modal_1.html#K7" target="_self">|</a> <font color="Maroon">c<sub>6</sub></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">c<sub>2</sub></font></span>)</span> <a href="trees_1.html#K4">|</a> <font color="Maroon">c<sub>6</sub></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">c<sub>6</sub></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">c<sub>2</sub></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">c<sub>2</sub></font></span>)</span> <a href="trees_1.html#K4">|</a> <font color="Maroon">c<sub>6</sub></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">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="modal_1.html#T26" target="_self">Th26</a></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">c<sub>2</sub></font> <a href="modal_1.html#K7" target="_self">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="modal_1.html#E2:67_1_2"><i><font color="Green">E59</font></i></a>, <a class="txt" href="modal_1.html#E14:67_1_2_1_1_3"><i><font color="Green">E67</font></i></a>, <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">c<sub>2</sub></font> <a href="modal_1.html#K7" target="_self">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font> <a href="modal_1.html#K7" target="_self">|</a> <font color="Maroon">c<sub>6</sub></font>]
 <b>by </b><i><a class="txt" href="modal_1.html#E1:67_1_2"><i><font color="Green">E58</font></i></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">c<sub>2</sub></font>]
 <b>by </b><i><a class="txt" href="modal_1.html#E5:67"><i><font color="Green">E55</font></i></a>, <a class="txt" href="modal_1.html#E7:67_1_2_1_1_3"><i><font color="Green">E64</font></i></a>, <a class="txt" href="modal_1.html#E13:67_1_2_1_1_3"><i><font color="Green">E66</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">c<sub>2</sub></font>]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:67_1"/><i><font color="Green">E58</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="modal_1.html#E1:67_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="modal_1.html#E2:67_1"><i><font color="Green">E57</font></i></a>);<br/></i>
<b>let </b><font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></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">c<sub>1</sub></font></span>)</span>
 
;<br/>
<b>hence </b><a NAME="E5:67_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>1</sub></font>]
 <b>by </b><i><a class="txt" href="modal_1.html#E3:67_1"><i><font color="Green">E58</font></i></a></i>;<br/>


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