<?xml version="1.0"?>
<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>set </b><font color="Maroon">c<sub>2</sub></font> = <span class="p1">(<span class="default"><a href="trees_1.html#K2">elementary_tree</a> 0</span>)</span> <a href="trees_2.html#K9">--&gt;</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default">3,<font color="Maroon">c<sub>1</sub></font></span><a href="domain_1.html#K1">]</a></span>;<br/>
<a NAME="E1:51"/><i><font color="Green">E36</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="trees_1.html#K2">elementary_tree</a> 0</span>)</span> <a href="trees_2.html#K9">--&gt;</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default">3,<font color="Maroon">c<sub>1</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a>, <a class="ref" href="trees_1.html#T56">TREES_1:56</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>3</sub></font> = <span class="p1">(<span class="default"><a href="trees_1.html#K2">elementary_tree</a> 0</span>)</span> <a href="trees_2.html#K9">--&gt;</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default">3,<font color="Maroon">c<sub>1</sub></font></span><a href="domain_1.html#K1">]</a></span> 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="txt" href="modal_1.html#E31"><i><font color="Green">Lemma24</font></i></a></i>;<br/>
<a NAME="E3:51"/><i><font color="Green">E37</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="trees_1.html#K2">elementary_tree</a> 0
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/>
<a NAME="E4:51"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font> holds <br/>(  <a href="modal_1.html#K5" target="_self">branchdeg</a> <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>3</sub></font> <a href="trees_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></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">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>3</sub></font> <a href="trees_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default">3,<font color="Olive">b<sub>2</sub></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">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>3</sub></font> <a href="trees_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></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">c<sub>3</sub></font> <a href="trees_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 2 implies <font color="Maroon">c<sub>3</sub></font> <a href="trees_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></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="51_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font>;<br/>

<a NAME="E1:51_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>4</sub></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#E3:51"><i><font color="Green">E37</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/>
<a NAME="E2:51_1"/><i><font color="Green">E39</font></i>: 
 <a href="trees_2.html#K1">succ</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:51_1_1"/><i><font color="Green">E40</font></i>: 
not  <a href="trees_2.html#K1">succ</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="trees_2.html#K1">succ</a> <font color="Maroon">c<sub>4</sub></font>;<br/>
<a NAME="E3:51_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="trees_2.html#K1">succ</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="modal_1.html#E1:51_1_1"><i><font color="Green">E40</font></i></a></i>;<br/>
<a NAME="E4:51_1_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></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">b<sub>1</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Maroon">c<sub>4</sub></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">b<sub>1</sub></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">c<sub>3</sub></font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="trees_2.html#D5">TREES_2:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E6:51_1_1"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></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">c<sub>6</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> &amp; <font color="Maroon">c<sub>4</sub></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">c<sub>6</sub></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">c<sub>3</sub></font> )
 ;<br/>
<a NAME="E7:51_1_1"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> &amp; <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>6</sub></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">c<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="modal_1.html#E1:51_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="modal_1.html#E6:51_1_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a></i>;<br/>
<a NAME="E8:51_1_1"/><b>then </b>
<span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="modal_1.html#E1:51"><i><font color="Green">E36</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E9:51_1_1"/>
contradiction
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:51_1"/><b>then </b>
0 <a href="hidden.html#R1">=</a>  <a href="modal_1.html#K5" target="_self">branchdeg</a> <font color="Maroon">c<sub>4</sub></font>
 
<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="E4:51_1"/>
 <a href="modal_1.html#K5" target="_self">branchdeg</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 2
 ;<br/>

<b>thus </b><a NAME="E5:51_1"/>
(  not  <a href="modal_1.html#K5" target="_self">branchdeg</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>3</sub></font> <a href="trees_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></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">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>3</sub></font> <a href="trees_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default">3,<font color="Olive">b<sub>1</sub></font></span><a href="domain_1.html#K1">]</a></span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:51_1_2"/>
 <a href="modal_1.html#K5" target="_self">branchdeg</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<a NAME="E2:51_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="trees_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default">3,<font color="Maroon">c<sub>1</sub></font></span><a href="domain_1.html#K1">]</a></span>
 
<b>by </b><i><a class="txt" href="modal_1.html#E3:51"><i><font color="Green">E37</font></i></a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>
<b>hence </b><a NAME="E3:51_1_2"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="trees_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></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">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>3</sub></font> <a href="trees_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default">3,<font color="Olive">b<sub>1</sub></font></span><a href="domain_1.html#K1">]</a></span> )
 ;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E6:51_1"/>
( (  not  <a href="modal_1.html#K5" target="_self">branchdeg</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>3</sub></font> <a href="trees_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></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">c<sub>3</sub></font> <a href="trees_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></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">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 2 implies <font color="Maroon">c<sub>3</sub></font> <a href="trees_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></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#E2:51_1"><i><font color="Green">E39</font></i></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>hence </b><a NAME="E5:51"/>
<span class="p1">(<span class="default"><a href="trees_1.html#K2">elementary_tree</a> 0</span>)</span> <a href="trees_2.html#K9">--&gt;</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default">3,<font color="Maroon">c<sub>1</sub></font></span><a href="domain_1.html#K1">]</a></span> is   <a href="modal_1.html#NM5">MP-wff</a>
 <b>by </b><i><a class="ref" href="modal_1.html#D6" target="_self">Def6</a></i>;<br/>


</div>
