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

<b>set </b><font color="Maroon">A</font> = <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>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> is  <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> &amp; ( for <font color="Olive">y</font> being <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>  st <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> holds <br/>(  <a href="relat_1.html#K1">dom</a> <font color="Olive">y</font> is <a href="finset_1.html#V1">finite</a> &amp; ( for <font color="Olive">v</font> being   <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Olive">y</font> holds <br/> (  <a href="modal_1.html#K5" target="_self">branchdeg</a> <font color="Olive">v</font> <a href="xxreal_0.html#R1">&lt;=</a> 2 &amp; (  not  <a href="modal_1.html#K5" target="_self">branchdeg</a> <font color="Olive">v</font> <a href="hidden.html#R1">=</a> 0 or <font color="Olive">y</font> <a href="trees_2.html#K3">.</a> <font color="Olive">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="Olive">y</font> <a href="trees_2.html#K3">.</a> <font color="Olive">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> ) &amp; (  not  <a href="modal_1.html#K5" target="_self">branchdeg</a> <font color="Olive">v</font> <a href="hidden.html#R1">=</a> 1 or <font color="Olive">y</font> <a href="trees_2.html#K3">.</a> <font color="Olive">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="Olive">y</font> <a href="trees_2.html#K3">.</a> <font color="Olive">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> ) &amp; (  <a href="modal_1.html#K5" target="_self">branchdeg</a> <font color="Olive">v</font> <a href="hidden.html#R1">=</a> 2 implies <font color="Olive">y</font> <a href="trees_2.html#K3">.</a> <font color="Olive">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> ) ) ) ) ) );<br/>
<b>consider </b><font color="Maroon">Y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:40_1_1"/><i><font color="Green">E37</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> iff ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="partfun1.html#K4">PFuncs</a> <span class="p1">(<span class="default"><a href="numbers.html#K5">NAT</a>  <a href="finseq_2.html#K3">*</a> </span>)</span>,<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> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1">XBOOLE_0:sch 1</a>();<br/></i>
<a NAME="E3:40_1_1"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">x</font> being <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> holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> iff (  <a href="relat_1.html#K1">dom</a> <font color="Olive">x</font> is <a href="finset_1.html#V1">finite</a> &amp; ( for <font color="Olive">v</font> being   <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Olive">x</font> holds <br/> (  <a href="modal_1.html#K5" target="_self">branchdeg</a> <font color="Olive">v</font> <a href="xxreal_0.html#R1">&lt;=</a> 2 &amp; (  not  <a href="modal_1.html#K5" target="_self">branchdeg</a> <font color="Olive">v</font> <a href="hidden.html#R1">=</a> 0 or <font color="Olive">x</font> <a href="trees_2.html#K3">.</a> <font color="Olive">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="Olive">x</font> <a href="trees_2.html#K3">.</a> <font color="Olive">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> ) &amp; (  not  <a href="modal_1.html#K5" target="_self">branchdeg</a> <font color="Olive">v</font> <a href="hidden.html#R1">=</a> 1 or <font color="Olive">x</font> <a href="trees_2.html#K3">.</a> <font color="Olive">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="Olive">x</font> <a href="trees_2.html#K3">.</a> <font color="Olive">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> ) &amp; (  <a href="modal_1.html#K5" target="_self">branchdeg</a> <font color="Olive">v</font> <a href="hidden.html#R1">=</a> 2 implies <font color="Olive">x</font> <a href="trees_2.html#K3">.</a> <font color="Olive">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="40_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be  <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>;<br/>

<b>thus </b><a NAME="E1:40_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> implies (  <a href="relat_1.html#K1">dom</a> <font color="Maroon">x</font> is <a href="finset_1.html#V1">finite</a> &amp; ( for <font color="Olive">v</font> being   <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">x</font> holds <br/> (  <a href="modal_1.html#K5" target="_self">branchdeg</a> <font color="Olive">v</font> <a href="xxreal_0.html#R1">&lt;=</a> 2 &amp; (  not  <a href="modal_1.html#K5" target="_self">branchdeg</a> <font color="Olive">v</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">x</font> <a href="trees_2.html#K3">.</a> <font color="Olive">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="trees_2.html#K3">.</a> <font color="Olive">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> ) &amp; (  not  <a href="modal_1.html#K5" target="_self">branchdeg</a> <font color="Olive">v</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">x</font> <a href="trees_2.html#K3">.</a> <font color="Olive">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="trees_2.html#K3">.</a> <font color="Olive">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> ) &amp; (  <a href="modal_1.html#K5" target="_self">branchdeg</a> <font color="Olive">v</font> <a href="hidden.html#R1">=</a> 2 implies <font color="Maroon">x</font> <a href="trees_2.html#K3">.</a> <font color="Olive">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/>;<br/>

<b>assume </b><a NAME="E2:40_1_1_1"/>
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">x</font> is <a href="finset_1.html#V1">finite</a> &amp; ( for <font color="Olive">v</font> being   <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">x</font> holds <br/> (  <a href="modal_1.html#K5" target="_self">branchdeg</a> <font color="Olive">v</font> <a href="xxreal_0.html#R1">&lt;=</a> 2 &amp; (  not  <a href="modal_1.html#K5" target="_self">branchdeg</a> <font color="Olive">v</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">x</font> <a href="trees_2.html#K3">.</a> <font color="Olive">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="trees_2.html#K3">.</a> <font color="Olive">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> ) &amp; (  not  <a href="modal_1.html#K5" target="_self">branchdeg</a> <font color="Olive">v</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">x</font> <a href="trees_2.html#K3">.</a> <font color="Olive">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="trees_2.html#K3">.</a> <font color="Olive">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> ) &amp; (  <a href="modal_1.html#K5" target="_self">branchdeg</a> <font color="Olive">v</font> <a href="hidden.html#R1">=</a> 2 implies <font color="Maroon">x</font> <a href="trees_2.html#K3">.</a> <font color="Olive">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> ) ) ) )
 ;<br/>

<a NAME="E3:40_1_1_1"/><b>then </b><i><font color="Green">E39</font></i>: 
for <font color="Olive">y</font> being <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>  st <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> holds <br/>(  <a href="relat_1.html#K1">dom</a> <font color="Olive">y</font> is <a href="finset_1.html#V1">finite</a> &amp; ( for <font color="Olive">v</font> being   <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Olive">y</font> holds <br/> (  <a href="modal_1.html#K5" target="_self">branchdeg</a> <font color="Olive">v</font> <a href="xxreal_0.html#R1">&lt;=</a> 2 &amp; (  not  <a href="modal_1.html#K5" target="_self">branchdeg</a> <font color="Olive">v</font> <a href="hidden.html#R1">=</a> 0 or <font color="Olive">y</font> <a href="trees_2.html#K3">.</a> <font color="Olive">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="Olive">y</font> <a href="trees_2.html#K3">.</a> <font color="Olive">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> ) &amp; (  not  <a href="modal_1.html#K5" target="_self">branchdeg</a> <font color="Olive">v</font> <a href="hidden.html#R1">=</a> 1 or <font color="Olive">y</font> <a href="trees_2.html#K3">.</a> <font color="Olive">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="Olive">y</font> <a href="trees_2.html#K3">.</a> <font color="Olive">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> ) &amp; (  <a href="modal_1.html#K5" target="_self">branchdeg</a> <font color="Olive">v</font> <a href="hidden.html#R1">=</a> 2 implies <font color="Olive">y</font> <a href="trees_2.html#K3">.</a> <font color="Olive">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> ) ) ) )
 
;<br/>
<a NAME="E4:40_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="partfun1.html#K4">PFuncs</a> <span class="p1">(<span class="default"><a href="numbers.html#K5">NAT</a>  <a href="finseq_2.html#K3">*</a> </span>)</span>,<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/>;<br/>
<b>hence </b><a NAME="E5:40_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>set </b><font color="Maroon">t</font> =  <a href="trees_1.html#K2">elementary_tree</a> 0;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K5">NAT</a> ,<a href="numbers.html#K5">NAT</a> </span><a href="zfmisc_1.html#K2">:]</a></span> = <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/>
<b>consider </b><font color="Maroon">T</font> being   <a href="trees_2.html#NM4">DecoratedTree</a><b> such that </b><br/><a NAME="E5:40_1_1"/><i><font color="Green">E41</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">T</font> <a href="hidden.html#R1">=</a>  <a href="trees_1.html#K2">elementary_tree</a> 0 &amp; ( for <font color="Olive">p</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">p</font> <a href="hidden.html#R2">in</a>  <a href="trees_1.html#K2">elementary_tree</a> 0 holds <br/><font color="Maroon">T</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">p</font>) ) )
 <b>from </b><i><a class="ref" href="trees_2.html#S7">TREES_2:sch 7</a>();<br/></i>
<a NAME="E6:40_1_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">T</font> <a href="tarski.html#R1">c=</a> <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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:40_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">T</font>
 ;<br/>

<b>then consider </b><font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:40_1_1_2"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">T</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">T</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:40_1_1_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, <a class="txt" href="modal_1.html#E1"><i><font color="Green">Lemma33</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="trees_1.html#T56">TREES_1:56</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">z</font> = <font color="Maroon">z</font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="finseq_1.html#T29">FINSEQ_1:29</a></i>;<br/>
<a NAME="E6:40_1_1_2"/>
<font color="Maroon">T</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</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>
 
<b>by </b><i>, <a class="txt" href="modal_1.html#E1"><i><font color="Green">Lemma33</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:40_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <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><a class="txt" href="modal_1.html#E1"><i><font color="Green">Lemma33</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">T</font> = <font color="Maroon">T</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>, , <a class="ref" href="trees_2.html#D9">TREES_2:def 9</a></i>;<br/>
<a NAME="E8:40_1_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">T</font> <a href="hidden.html#R2">in</a>  <a href="partfun1.html#K4">PFuncs</a> <span class="p1">(<span class="default"><a href="numbers.html#K5">NAT</a>  <a href="finseq_2.html#K3">*</a> </span>)</span>,<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/>;<br/>
<a NAME="E9:40_1_1"/>
for <font color="Olive">y</font> being <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>  st <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">T</font> holds <br/>(  <a href="relat_1.html#K1">dom</a> <font color="Olive">y</font> is <a href="finset_1.html#V1">finite</a> &amp; ( for <font color="Olive">v</font> being   <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Olive">y</font> holds <br/> (  <a href="modal_1.html#K5" target="_self">branchdeg</a> <font color="Olive">v</font> <a href="xxreal_0.html#R1">&lt;=</a> 2 &amp; (  not  <a href="modal_1.html#K5" target="_self">branchdeg</a> <font color="Olive">v</font> <a href="hidden.html#R1">=</a> 0 or <font color="Olive">y</font> <a href="trees_2.html#K3">.</a> <font color="Olive">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="Olive">y</font> <a href="trees_2.html#K3">.</a> <font color="Olive">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> ) &amp; (  not  <a href="modal_1.html#K5" target="_self">branchdeg</a> <font color="Olive">v</font> <a href="hidden.html#R1">=</a> 1 or <font color="Olive">y</font> <a href="trees_2.html#K3">.</a> <font color="Olive">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="Olive">y</font> <a href="trees_2.html#K3">.</a> <font color="Olive">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> ) &amp; (  <a href="modal_1.html#K5" target="_self">branchdeg</a> <font color="Olive">v</font> <a href="hidden.html#R1">=</a> 2 implies <font color="Olive">y</font> <a href="trees_2.html#K3">.</a> <font color="Olive">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="40_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be  <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>;<br/>

<b>assume </b><a NAME="E1:40_1_1_3"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">T</font>
 ;<br/>

<b>thus </b><a NAME="E2:40_1_1_3"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">y</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  <a href="relat_1.html#K1">dom</a> <font color="Maroon">y</font>;<br/>

<a NAME="E3:40_1_1_3"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="trees_1.html#T56">TREES_1:56</a></i>;<br/>
<a NAME="E4:40_1_1_3"/><i><font color="Green">E64</font></i>: 
 <a href="trees_2.html#K1">succ</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1_3_1"><b>proof </b></a><div class="add">

<a NAME="E1:40_1_1_3_1"/><i><font color="Green">E69</font></i>: 
 <a href="trees_2.html#K1">succ</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Olive">n</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Maroon">v</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Olive">n</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">y</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="trees_2.html#D5">TREES_2:def 5</a></i>;<br/>
<b>assume </b><a NAME="E2:40_1_1_3_1"/><i><font color="Green">E70</font></i>: 
not  <a href="trees_2.html#K1">succ</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="trees_2.html#K1">succ</a> <font color="Maroon">v</font>;<br/>
<a NAME="E4:40_1_1_3_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="trees_2.html#K1">succ</a> <font color="Maroon">v</font>
 
<b>by </b><i><a class="txt" href="modal_1.html#E3:40_1_1"><i><font color="Green">E38</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">n</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="E6:40_1_1_3_1"/><i><font color="Green">E71</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">n</font></span><a href="finseq_1.html#K12">*&gt;</a></span> &amp; <font color="Maroon">v</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">n</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="txt" href="modal_1.html#E2:40_1_1"><i><font color="Green">E37</font></i></a></i>;<br/>
<a NAME="E7:40_1_1_3_1"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, , , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="trees_1.html#T56">TREES_1:56</a></i>;<br/>
<a NAME="E8:40_1_1_3_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">n</font></span><a href="finseq_1.html#K12">*&gt;</a></span>
 
<b>by </b><i>, , <a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a></i>;<br/>
<b>hence </b><a NAME="E9:40_1_1_3_1"/>
contradiction
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E5:40_1_1_3"/><b>then </b>
 <a href="modal_1.html#K5" target="_self">branchdeg</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="card_1.html#T78">CARD_1:78</a>, <a class="ref" href="trees_2.html#D13">TREES_2:def 13</a></i>;<br/>
<b>hence </b><a NAME="E6:40_1_1_3"/>
 <a href="modal_1.html#K5" target="_self">branchdeg</a> <font color="Maroon">v</font> <a href="xxreal_0.html#R1">&lt;=</a> 2
 ;<br/>

<b>thus </b><a NAME="E7:40_1_1_3"/>
( (  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">y</font> <a href="trees_2.html#K3">.</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">y</font> <a href="trees_2.html#K3">.</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> ) &amp; (  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">y</font> <a href="trees_2.html#K3">.</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">y</font> <a href="trees_2.html#K3">.</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> ) &amp; (  <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">y</font> <a href="trees_2.html#K3">.</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="modal_1.html#T3" target="_self">Th3</a>, <a class="ref" href="card_1.html#T78">CARD_1:78</a>, <a class="ref" href="trees_2.html#D13">TREES_2:def 13</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">Y</font> = <font color="Maroon">Y</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i>, <a class="txt" href="modal_1.html#E1"><i><font color="Green">Lemma33</font></i></a></i>;<br/>
<a NAME="E11:40_1_1"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> holds <br/><font color="Olive">x</font> is   <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/>;<br/>
<b>then reconsider </b><font color="Maroon">Y</font> = <font color="Maroon">Y</font> as    <a href="modal_1.html#M1" target="_self">DOMAIN_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/>;<br/>
<b>take </b>
<font color="Maroon">Y</font>
;<br/>

<b>thus </b><a NAME="E13:40_1_1"/>
( ( for <font color="Olive">x</font> being  <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>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> holds <br/><font color="Olive">x</font> is <a href="finset_1.html#V1">finite</a> ) &amp; ( for <font color="Olive">x</font> being <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> holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> iff for <font color="Olive">v</font> being   <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Olive">x</font> holds <br/> (  <a href="modal_1.html#K5" target="_self">branchdeg</a> <font color="Olive">v</font> <a href="xxreal_0.html#R1">&lt;=</a> 2 &amp; (  not  <a href="modal_1.html#K5" target="_self">branchdeg</a> <font color="Olive">v</font> <a href="hidden.html#R1">=</a> 0 or <font color="Olive">x</font> <a href="trees_2.html#K3">.</a> <font color="Olive">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="Olive">x</font> <a href="trees_2.html#K3">.</a> <font color="Olive">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> ) &amp; (  not  <a href="modal_1.html#K5" target="_self">branchdeg</a> <font color="Olive">v</font> <a href="hidden.html#R1">=</a> 1 or <font color="Olive">x</font> <a href="trees_2.html#K3">.</a> <font color="Olive">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="Olive">x</font> <a href="trees_2.html#K3">.</a> <font color="Olive">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> ) &amp; (  <a href="modal_1.html#K5" target="_self">branchdeg</a> <font color="Olive">v</font> <a href="hidden.html#R1">=</a> 2 implies <font color="Olive">x</font> <a href="trees_2.html#K3">.</a> <font color="Olive">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>, </i>;<br/>


</div>
