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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="trees_1.html#NM1">Tree</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:77_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="trees_1.html#K3">Leaves</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><a NAME="E2:77_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <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>
 <b>by </b><i><a class="ref" href="trees_1.html#T30">TREES_1:30</a></i>;<br/><b>hence </b><a NAME="E3:77_1"/>
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> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="modal_1.html#E1:77_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="trees_1.html#D8">TREES_1:def 8</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:77"/><i><font color="Green">E58</font></i>: 
( 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> <font color="Maroon">c<sub>1</sub></font> &amp; not <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="trees_1.html#K3">Leaves</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>3</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:77"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="ref" href="trees_1.html#D8">TREES_1:def 8</a></i>;<br/>
<a NAME="E5:77"/>
<font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="modal_1.html#E4:77"><i><font color="Green">E59</font></i></a>, <a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E7:77"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="trees_1.html#T8">TREES_1:8</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="modal_1.html#E7:77"><i><font color="Green">E60</font></i></a>, <a class="ref" href="finseq_1.html#T50">FINSEQ_1:50</a></i>;<br/>
<a NAME="E9:77"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="modal_1.html#E7:77"><i><font color="Green">E60</font></i></a>, <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/>
<a NAME="E10:77"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="modal_1.html#E4:77"><i><font color="Green">E59</font></i></a>, <a class="ref" href="trees_1.html#T24">TREES_1:24</a></i>;<br/>
<a NAME="E11:77"/><b>then </b>
<font color="Maroon">c<sub>5</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="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">c<sub>7</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E13:77"/><i><font color="Green">E61</font></i>: 
<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>7</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="modal_1.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E14:77"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="modal_1.html#E7:77"><i><font color="Green">E60</font></i></a>, <a class="txt" href="modal_1.html#E13:77"><i><font color="Green">E61</font></i></a>, <a class="ref" href="finseq_1.html#T45">FINSEQ_1:45</a></i>;<br/>
<a NAME="E15:77"/><b>then </b>
( <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>7</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> )
 
<b>by </b><i><a class="txt" href="modal_1.html#E4:77"><i><font color="Green">E59</font></i></a>, <a class="ref" href="nat_1.html#T18">NAT_1:18</a>, <a class="ref" href="trees_1.html#T46">TREES_1:46</a></i>;<br/>
<b>hence </b><a NAME="E16:77"/>
contradiction
 <b>by </b><i><a class="txt" href="modal_1.html#E2:77"><i><font color="Green">E58</font></i></a>, <a class="ref" href="trees_1.html#D5">TREES_1:def 5</a></i>;<br/>


</div>
