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

<b>set </b><font color="Maroon">x</font> = <font color="Maroon">A</font> <a href="trees_2.html#K7">|</a> <font color="Maroon">a</font>;<br/>
<a NAME="E1:43_1_1"/><i><font color="Green">E37</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="trees_2.html#K7">|</a> <font color="Maroon">a</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">a</font>
 
<b>by </b><i><a class="ref" href="trees_2.html#D11">TREES_2:def 11</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">db</font> =  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="trees_2.html#K7">|</a> <font color="Maroon">a</font></span>)</span> as  <a href="finset_1.html#V1">finite</a> <a href="trees_1.html#NM1">Tree</a> ;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">A</font> <a href="trees_2.html#K7">|</a> <font color="Maroon">a</font> as  <a href="finset_1.html#V1">finite</a> <a href="trees_2.html#NM6">DecoratedTree</a> of <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><a href="numbers.html#K5">NAT</a> ,<a href="numbers.html#K5">NAT</a> </span><a href="mcart_1.html#K12">:]</a></span> <b>by </b><i>, </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1_1"><b>now </b></a><div class="add"><b>set </b><font color="Maroon">da</font> =  <a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font>;<br/><b>thus </b><a NAME="E1:43_1_1_1"/>
<font color="Maroon">db</font> is <a href="finset_1.html#V1">finite</a>
 ;<br/><b>let </b><font color="Maroon">v</font> be    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">db</font>;<br/><a NAME="E2:43_1_1_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">db</font>
 ;<br/><a NAME="E3:43_1_1_1"/><b>then </b><i><font color="Green">E38</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</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">a</font>
 <b>by </b><i><a class="ref" href="trees_2.html#D11">TREES_2:def 11</a></i>;<br/><b>then reconsider </b><font color="Maroon">w</font> = <font color="Maroon">a</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">v</font> as    <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font> <b>by </b><i><a class="ref" href="trees_1.html#D9">TREES_1:def 9</a></i>;<br/><b>reconsider </b><font color="Maroon">v'</font> = <font color="Maroon">v</font> as    <a href="trees_1.html#M1">Element</a> of <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">a</font> <b>by </b><i><a class="ref" href="trees_2.html#D11">TREES_2:def 11</a></i>;<br/><a NAME="E6:43_1_1_1"/><i><font color="Green">E39</font></i>: 
 <a href="trees_2.html#K1">succ</a> <font color="Maroon">w</font>, <a href="trees_2.html#K1">succ</a> <font color="Maroon">v'</font> <a href="tarski.html#R2">are_equipotent</a> 
 <b>by </b><i/>;<br/><a NAME="E7:43_1_1_1"/>
 <a href="trees_2.html#K1">succ</a> <font color="Maroon">v'</font> <a href="hidden.html#R1">=</a>  <a href="trees_2.html#K1">succ</a> <font color="Maroon">v</font>
 <b>by </b><i><a class="ref" href="trees_2.html#D11">TREES_2:def 11</a></i>;<br/><a NAME="E8:43_1_1_1"/><b>then </b>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="trees_2.html#K1">succ</a> <font color="Maroon">v</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="trees_2.html#K1">succ</a> <font color="Maroon">w</font></span>)</span>
 <b>by </b><i><a class="txt" href="modal_1.html#E1"><i><font color="Green">Lemma33</font></i></a>, <a class="ref" href="card_1.html#T81">CARD_1:81</a></i>;<br/><a NAME="E9:43_1_1_1"/><b>then </b>
 <a href="modal_1.html#K5" target="_self">branchdeg</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="trees_2.html#K1">succ</a> <font color="Maroon">w</font></span>)</span>
 <b>by </b><i><a class="ref" href="trees_2.html#D13">TREES_2:def 13</a></i>;<br/><a NAME="E10:43_1_1_1"/><b>then </b><i><font color="Green">E41</font></i>: 
 <a href="modal_1.html#K5" target="_self">branchdeg</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="modal_1.html#K5" target="_self">branchdeg</a> <font color="Maroon">w</font>
 <b>by </b><i><a class="ref" href="trees_2.html#D13">TREES_2:def 13</a></i>;<br/><b>hence </b><a NAME="E11:43_1_1_1"/>
 <a href="modal_1.html#K5" target="_self">branchdeg</a> <font color="Maroon">v</font> <a href="xxreal_0.html#R1">&lt;=</a> 2
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E12:43_1_1_1"/>
(  not  <a href="modal_1.html#K5" target="_self">branchdeg</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default">0,0</span><a href="domain_1.html#K1">]</a></span> or  ex <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">x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default">3,<font color="Olive">k</font></span><a href="domain_1.html#K1">]</a></span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:43_1_1_1_1"/><i><font color="Green">E44</font></i>: 
 <a href="modal_1.html#K5" target="_self">branchdeg</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_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="E1:43_1_1_1_1_1"/>
( <font color="Maroon">A</font> <a href="trees_2.html#K3">.</a> <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default">0,0</span><a href="domain_1.html#K1">]</a></span> or  ex <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">A</font> <a href="trees_2.html#K3">.</a> <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default">3,<font color="Olive">k</font></span><a href="domain_1.html#K1">]</a></span> )
 </span><b>by </b><i>, , </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:43_1_1_1_1_1_1"/>
<font color="Maroon">A</font> <a href="trees_2.html#K3">.</a> <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default">0,0</span><a href="domain_1.html#K1">]</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:43_1_1_1_1_1_1"/>
( <font color="Maroon">x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default">0,0</span><a href="domain_1.html#K1">]</a></span> or  ex <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">x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default">3,<font color="Olive">k</font></span><a href="domain_1.html#K1">]</a></span> )
 <b>by </b><i>, <a class="ref" href="trees_2.html#D11">TREES_2:def 11</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:43_1_1_1_1_1_2"/>
 ex <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">A</font> <a href="trees_2.html#K3">.</a> <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default">3,<font color="Olive">k</font></span><a href="domain_1.html#K1">]</a></span>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:43_1_1_1_1_1_2"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">A</font> <a href="trees_2.html#K3">.</a> <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default">3,<font color="Maroon">k</font></span><a href="domain_1.html#K1">]</a></span>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1_1_1_1_2_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon">k</font> = <font color="Maroon">k</font>;<br/><b>thus </b><a NAME="E1:43_1_1_1_1_1_2_1"/>
<font color="Maroon">x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default">3,<font color="Maroon">k</font></span><a href="domain_1.html#K1">]</a></span>
 <b>by </b><i>, <a class="ref" href="modal_1.html#T3" target="_self">Th3</a>, <a class="ref" href="trees_2.html#D11">TREES_2:def 11</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:43_1_1_1_1_1_2"/>
( <font color="Maroon">x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default">0,0</span><a href="domain_1.html#K1">]</a></span> or  ex <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">x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default">3,<font color="Olive">k</font></span><a href="domain_1.html#K1">]</a></span> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>thus </b><a NAME="E13:43_1_1_1"/>
(  not  <a href="modal_1.html#K5" target="_self">branchdeg</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> <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">x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> <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> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:43_1_1_1_2"/><i><font color="Green">E62</font></i>: 
 <a href="modal_1.html#K5" target="_self">branchdeg</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> 1
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:43_1_1_1_2_1"/>
( <font color="Maroon">A</font> <a href="trees_2.html#K3">.</a> <font color="Maroon">w</font> <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> <font color="Maroon">w</font> <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>, , </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:43_1_1_1_2_1_1"/>
<font color="Maroon">A</font> <a href="trees_2.html#K3">.</a> <font color="Maroon">w</font> <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"><b>hence </b><a NAME="E2:43_1_1_1_2_1_1"/>
( <font color="Maroon">x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> <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">x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> <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> )
 <b>by </b><i>, <a class="ref" href="trees_2.html#D11">TREES_2:def 11</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:43_1_1_1_2_1_2"/>
<font color="Maroon">A</font> <a href="trees_2.html#K3">.</a> <font color="Maroon">w</font> <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"><b>hence </b><a NAME="E2:43_1_1_1_2_1_2"/>
( <font color="Maroon">x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> <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">x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> <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> )
 <b>by </b><i>, <a class="ref" href="trees_2.html#D11">TREES_2:def 11</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>thus </b><a NAME="E14:43_1_1_1"/>
(  <a href="modal_1.html#K5" target="_self">branchdeg</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> 2 implies <font color="Maroon">x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</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> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1_1_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:43_1_1_1_3"/>
 <a href="modal_1.html#K5" target="_self">branchdeg</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> 2
 ;<br/>

<a NAME="E2:43_1_1_1_3"/><b>then </b>
<font color="Maroon">A</font> <a href="trees_2.html#K3">.</a> <font color="Maroon">w</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>, </i>;<br/>
<b>hence </b><a NAME="E3:43_1_1_1_3"/>
<font color="Maroon">x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</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="ref" href="trees_2.html#D11">TREES_2:def 11</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E5:43_1_1"/>
<font color="Maroon">A</font> <a href="trees_2.html#K5">|</a> <font color="Maroon">a</font> is   <a href="modal_1.html#NM5">MP-wff</a>
 <b>by </b><i/>;<br/>


</div>
