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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="trees_2.html#NM4">DecoratedTree</a>;<br/>

<b>reconsider </b><font color="Maroon">c<sub>2</sub></font> =  <a href="xboole_0.html#K1">{}</a>  as   <a href="trees_4.html#NM1">Node</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="trees_1.html#T47">TREES_1:47</a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:12_1"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="trees_9.html#V1" target="_self">root</a>
 ;<br/><a NAME="E2:12_1"/><b>then </b>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</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="trees_9.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E3:12_1"/><b>then </b>
not <font color="Maroon">c<sub>2</sub></font> <a href="finseq_1.html#K8">^</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> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font>
 <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/><b>hence </b><a NAME="E4:12_1"/>
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a>  <a href="trees_1.html#K3">Leaves</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>
 <b>by </b><i><a class="ref" href="modal_1.html#T53">MODAL_1:53</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E3:12"/><i><font color="Green">E10</font></i>: 
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a>  <a href="trees_1.html#K3">Leaves</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>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="trees_2.html#D1">TREES_2:def 1</a>,<a class="ref" href="trees_9.html#D1" target="_self">TREES_9:def 1</a><br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:12_2"/><i><font color="Green">E11</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font> &amp; not <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="trees_1.html#K2">elementary_tree</a> 0 )
 ;<br/><a NAME="E2:12_2"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</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/><b>then consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">c<sub>5</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:12_2"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>3</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>5</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="modal_1.html#T4">MODAL_1:4</a></i>;<br/><a NAME="E5:12_2"/>
( <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>5</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>1</sub></font> &amp; <font color="Maroon">c<sub>2</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>5</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> <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>5</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> )
 <b>by </b><i><a class="txt" href="trees_9.html#E1:12_2"><i><font color="Green">E11</font></i></a>, <a class="txt" href="trees_9.html#E4:12_2"><i><font color="Green">E12</font></i></a>, <a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a>, <a class="ref" href="trees_1.html#T46">TREES_1:46</a></i>;<br/><b>hence </b><a NAME="E6:12_2"/>
contradiction
 <b>by </b><i><a class="txt" href="trees_9.html#E3:12"><i><font color="Green">E10</font></i></a>, <a class="ref" href="modal_1.html#T54">MODAL_1:54</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E5:12"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="trees_1.html#K2">elementary_tree</a> 0
 ;<br/>

<a NAME="E6:12"/><b>then </b>
<font color="Maroon">c<sub>3</sub></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/>
<b>hence </b><a NAME="E7:12"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="trees_1.html#T47">TREES_1:47</a></i>;<br/>


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