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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  <a href="finset_1.html#V1">finite</a> <a href="trees_2.html#NM6">DecoratedTree</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</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>2</sub></font>;<br/>





<b>assume </b><a NAME="E1:27"/><i><font color="Green">E26</font></i>: 
(  <a href="trees_2.html#K1">succ</a> <span class="p1">(<span class="default"><a href="modal_1.html#K1" target="_self">Root</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K6">}</a></span> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font> is <a href="finset_1.html#V1">finite</a> )
 ;<br/>

<a NAME="E2:27"/><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> <span class="p2">(<span class="default"><a href="modal_1.html#K1" target="_self">Root</a> <span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="card_1.html#T79">CARD_1:79</a></i>;<br/>
<a NAME="E3:27"/><b>then </b>
 <a href="trees_2.html#K10">branchdeg</a> <span class="p1">(<span class="default"><a href="modal_1.html#K1" target="_self">Root</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="trees_2.html#D13">TREES_2:def 13</a></i>;<br/>
<a NAME="E4:27"/><b>then </b>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="txt" href="modal_1.html#E1:27"><i><font color="Green">E26</font></i></a>, <a class="ref" href="modal_1.html#T23" target="_self">Th23</a></i>;<br/>
<a NAME="E5:27"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E6:27"/><b>then </b><i><font color="Green">E27</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">0</span><a href="finseq_1.html#K12">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="trees_1.html#K2">elementary_tree</a> 1;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1">(<span class="default"><a href="trees_1.html#K2">elementary_tree</a> 1</span>)</span> <a href="trees_2.html#K9">--&gt;</a> <span class="p1">(<span class="default"><a href="modal_1.html#K2" target="_self">Root</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/>
<a NAME="E7:27"/><i><font color="Green">E28</font></i>: 
<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="trees_1.html#K2">elementary_tree</a> 1
 
<b>by </b><i><a class="ref" href="modal_1.html#T9" target="_self">Th9</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E8:27"/><i><font color="Green">E29</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> 1</span>)</span> <a href="trees_2.html#K9">--&gt;</a> <span class="p2">(<span class="default"><a href="modal_1.html#K2" target="_self">Root</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="trees_1.html#K2">elementary_tree</a> 1 &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="trees_1.html#K2">elementary_tree</a> 1</span>)</span> <a href="trees_2.html#K9">--&gt;</a> <span class="p2">(<span class="default"><a href="modal_1.html#K2" target="_self">Root</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="modal_1.html#K2" target="_self">Root</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span> )
 
<b>by </b><i><a class="ref" href="funcop_1.html#T14">FUNCOP_1:14</a>, <a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/>
<a NAME="E9:27"/><i><font color="Green">E30</font></i>: 
<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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="trees_1.html#K2">elementary_tree</a> 1</span>)</span> <a href="trees_2.html#K9">--&gt;</a> <span class="p2">(<span class="default"><a href="modal_1.html#K2" target="_self">Root</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="modal_1.html#E7:27"><i><font color="Green">E28</font></i></a>, <a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/>
<a NAME="E10:27"/><i><font color="Green">E31</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="trees_2.html#K7">|</a> <font color="Maroon">c<sub>3</sub></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">c<sub>2</sub></font></span>)</span> <a href="trees_1.html#K4">|</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="trees_2.html#D11">TREES_2:def 11</a></i>;<br/>
<a NAME="E11:27"/><b>then </b>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="trees_1.html#K2">elementary_tree</a> 1</span>)</span> <a href="trees_2.html#K9">--&gt;</a> <span class="p3">(<span class="default"><a href="modal_1.html#K2" target="_self">Root</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="trees_2.html#K8">with-replacement</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="trees_2.html#K7">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="trees_1.html#K2">elementary_tree</a> 1</span>)</span> <a href="trees_1.html#K5">with-replacement</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>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="trees_1.html#K4">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="modal_1.html#E7:27"><i><font color="Green">E28</font></i></a>, <a class="txt" href="modal_1.html#E8:27"><i><font color="Green">E29</font></i></a>, <a class="ref" href="trees_2.html#D12">TREES_2:def 12</a></i>;<br/>
<a NAME="E12:27"/><b>then </b><i><font color="Green">E32</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="trees_1.html#K2">elementary_tree</a> 1</span>)</span> <a href="trees_2.html#K9">--&gt;</a> <span class="p3">(<span class="default"><a href="modal_1.html#K2" target="_self">Root</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="trees_2.html#K8">with-replacement</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="trees_2.html#K7">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="modal_1.html#E1:27"><i><font color="Green">E26</font></i></a>, <a class="ref" href="modal_1.html#T27" target="_self">Th27</a></i>;<br/>
<a NAME="E13:27"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="trees_1.html#K2">elementary_tree</a> 1</span>)</span> <a href="trees_2.html#K9">--&gt;</a> <span class="p3">(<span class="default"><a href="modal_1.html#K2" target="_self">Root</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="trees_2.html#K8">with-replacement</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="trees_2.html#K7">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="trees_1.html#K2">elementary_tree</a> 1</span>)</span> <a href="trees_2.html#K9">--&gt;</a> <span class="p3">(<span class="default"><a href="modal_1.html#K2" target="_self">Root</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="trees_2.html#K8">with-replacement</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="trees_2.html#K7">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1"><b>proof </b></a><div class="add">

<a NAME="E1:27_1"/><i><font color="Green">E33</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="trees_1.html#K2">elementary_tree</a> 1</span>)</span> <a href="trees_2.html#K9">--&gt;</a> <span class="p3">(<span class="default"><a href="modal_1.html#K2" target="_self">Root</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="trees_2.html#K8">with-replacement</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="trees_2.html#K7">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="trees_1.html#K2">elementary_tree</a> 1</span>)</span> <a href="trees_2.html#K9">--&gt;</a> <span class="p3">(<span class="default"><a href="modal_1.html#K2" target="_self">Root</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="trees_1.html#K5">with-replacement</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>,<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="trees_2.html#K7">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="modal_1.html#E9:27"><i><font color="Green">E30</font></i></a>, <a class="ref" href="trees_2.html#D12">TREES_2:def 12</a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E2:27_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="trees_1.html#K2">elementary_tree</a> 1</span>)</span> <a href="trees_2.html#K9">--&gt;</a> <span class="p3">(<span class="default"><a href="modal_1.html#K2" target="_self">Root</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="trees_2.html#K8">with-replacement</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="trees_2.html#K7">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E3:27_1"/><b>then </b><i><font color="Green">E35</font></i>: 
( ( not <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="tarski.html#R1">is_a_prefix_of</a> <font color="Maroon">c<sub>6</sub></font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="trees_1.html#K2">elementary_tree</a> 1</span>)</span> <a href="trees_2.html#K9">--&gt;</a> <span class="p3">(<span class="default"><a href="modal_1.html#K2" target="_self">Root</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="trees_2.html#K8">with-replacement</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="trees_2.html#K7">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="trees_1.html#K2">elementary_tree</a> 1</span>)</span> <a href="trees_2.html#K9">--&gt;</a> <span class="p2">(<span class="default"><a href="modal_1.html#K2" target="_self">Root</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> ) or ex <font color="Olive">b<sub>1</sub></font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="trees_2.html#K7">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</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="finseq_1.html#K8">^</a> <font color="Olive">b<sub>1</sub></font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="trees_1.html#K2">elementary_tree</a> 1</span>)</span> <a href="trees_2.html#K9">--&gt;</a> <span class="p3">(<span class="default"><a href="modal_1.html#K2" target="_self">Root</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="trees_2.html#K8">with-replacement</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="trees_2.html#K7">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="trees_2.html#K7">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> ) )
 
<b>by </b><i><a class="txt" href="modal_1.html#E9:27"><i><font color="Green">E30</font></i></a>, <a class="txt" href="modal_1.html#E1:27_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="trees_2.html#D12">TREES_2:def 12</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:27_1_1_1"/>
( ( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <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> 1</span>)</span> <a href="trees_2.html#K9">--&gt;</a> <span class="p2">(<span class="default"><a href="modal_1.html#K2" target="_self">Root</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> &amp; not <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="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Maroon">c<sub>6</sub></font> ) or ex <font color="Olive">b<sub>1</sub></font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="trees_2.html#K7">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</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="finseq_1.html#K8">^</a> <font color="Olive">b<sub>1</sub></font> ) )
 </span><b>by </b><i><a class="txt" href="modal_1.html#E9:27"><i><font color="Green">E30</font></i></a>, <a class="txt" href="modal_1.html#E1:27_1"><i><font color="Green">E33</font></i></a>, <a class="txt" href="modal_1.html#E2:27_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="trees_1.html#D12">TREES_1:def 12</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_1_1"><b>suppose </b></a><a NAME="E1:27_1_1_1_1"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <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> 1</span>)</span> <a href="trees_2.html#K9">--&gt;</a> <span class="p2">(<span class="default"><a href="modal_1.html#K2" target="_self">Root</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> &amp; not <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="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_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="E1:27_1_1_1_1_1_1"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</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> )
 </span><b>by </b><i><a class="txt" href="modal_1.html#E8:27"><i><font color="Green">E29</font></i></a>, <a class="txt" href="modal_1.html#E1:27_1_1_1_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="modal_1.html#T9" target="_self">Th9</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:27_1_1_1_1_1_1_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:27_1_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="trees_1.html#K2">elementary_tree</a> 1
 <b>by </b><i><a class="ref" href="trees_1.html#T47">TREES_1:47</a></i>;<br/><a NAME="E3:27_1_1_1_1_1_1_1"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="trees_2.html#K7">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> or  not <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</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="finseq_1.html#K8">^</a> <font color="Olive">b<sub>1</sub></font> or  not <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="trees_1.html#K2">elementary_tree</a> 1</span>)</span> <a href="trees_2.html#K9">--&gt;</a> <span class="p3">(<span class="default"><a href="modal_1.html#K2" target="_self">Root</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="trees_2.html#K8">with-replacement</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="trees_2.html#K7">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="trees_2.html#K7">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="modal_1.html#E1:27_1_1_1_1_1_1_1"><i><font color="Green">E37</font></i></a></i>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="trees_1.html#K2">elementary_tree</a> 1</span>)</span> <a href="trees_2.html#K9">--&gt;</a> <span class="p2">(<span class="default"><a href="modal_1.html#K2" target="_self">Root</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> = 
 <a href="modal_1.html#K2" target="_self">Root</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="modal_1.html#E2:27_1_1_1_1_1_1_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="trees_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="modal_1.html#K1" target="_self">Root</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>

<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="txt" href="modal_1.html#E1:27_1_1_1_1_1_1_1"><i><font color="Green">E37</font></i></a></i>
;<br/><b>hence </b><a NAME="E5:27_1_1_1_1_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="trees_1.html#K2">elementary_tree</a> 1</span>)</span> <a href="trees_2.html#K9">--&gt;</a> <span class="p3">(<span class="default"><a href="modal_1.html#K2" target="_self">Root</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="trees_2.html#K8">with-replacement</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="trees_2.html#K7">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="modal_1.html#E9:27"><i><font color="Green">E30</font></i></a>, <a class="txt" href="modal_1.html#E1:27_1"><i><font color="Green">E33</font></i></a>, <a class="txt" href="modal_1.html#E2:27_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="modal_1.html#E3:27_1_1_1_1_1_1_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="trees_2.html#D12">TREES_2:def 12</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:27_1_1_1_1_1_1_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</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>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:27_1_1_1_1_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="trees_1.html#K2">elementary_tree</a> 1</span>)</span> <a href="trees_2.html#K9">--&gt;</a> <span class="p3">(<span class="default"><a href="modal_1.html#K2" target="_self">Root</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="trees_2.html#K8">with-replacement</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="trees_2.html#K7">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="modal_1.html#E6:27"><i><font color="Green">E27</font></i></a>, <a class="txt" href="modal_1.html#E10:27"><i><font color="Green">E31</font></i></a>, <a class="txt" href="modal_1.html#E3:27_1"><i><font color="Green">E35</font></i></a>, <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>hence </b><a NAME="E3:27_1_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="trees_1.html#K2">elementary_tree</a> 1</span>)</span> <a href="trees_2.html#K9">--&gt;</a> <span class="p3">(<span class="default"><a href="modal_1.html#K2" target="_self">Root</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="trees_2.html#K8">with-replacement</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="trees_2.html#K7">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_1_2"><b>suppose </b></a><a NAME="E1:27_1_1_1_2"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="trees_2.html#K7">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</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="finseq_1.html#K8">^</a> <font color="Olive">b<sub>1</sub></font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:27_1_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="trees_1.html#K2">elementary_tree</a> 1</span>)</span> <a href="trees_2.html#K9">--&gt;</a> <span class="p3">(<span class="default"><a href="modal_1.html#K2" target="_self">Root</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="trees_2.html#K8">with-replacement</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="trees_2.html#K7">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="modal_1.html#E6:27"><i><font color="Green">E27</font></i></a>, <a class="txt" href="modal_1.html#E10:27"><i><font color="Green">E31</font></i></a>, <a class="txt" href="modal_1.html#E3:27_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="trees_1.html#T8">TREES_1:8</a>, <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>hence </b><a NAME="E5:27_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="trees_1.html#K2">elementary_tree</a> 1</span>)</span> <a href="trees_2.html#K9">--&gt;</a> <span class="p3">(<span class="default"><a href="modal_1.html#K2" target="_self">Root</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="trees_2.html#K8">with-replacement</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="trees_2.html#K7">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E14:27"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="trees_1.html#K2">elementary_tree</a> 1</span>)</span> <a href="trees_2.html#K9">--&gt;</a> <span class="p2">(<span class="default"><a href="modal_1.html#K2" target="_self">Root</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="trees_2.html#K8">with-replacement</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>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="trees_2.html#K7">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="modal_1.html#E12:27"><i><font color="Green">E32</font></i></a>, <a class="ref" href="trees_2.html#T33">TREES_2:33</a></i>;<br/>


</div>
