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

<b>set </b><font color="Maroon">c<sub>1</sub></font> =  <a href="trees_1.html#K2">elementary_tree</a> 2;<br/>
<a NAME="E1:28"/><i><font color="Green">E27</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> 2 &amp; <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">1</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> 2 )
 
<b>by </b><i><a class="ref" href="modal_1.html#T10" target="_self">Th10</a>, <a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="trees_1.html#NM1">Tree</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E2:28"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="finset_1.html#V1">finite</a>
 <b>and </b><br/><a NAME="E3:28"/><i><font color="Green">E28</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> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12">*&gt;</a></span> &amp;  <a href="trees_2.html#K1">succ</a> <span class="p1">(<span class="default"><a href="modal_1.html#K1" target="_self">Root</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K7">}</a></span> )
 ;<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> 2</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"><font color="Maroon">c<sub>2</sub></font> <a href="trees_1.html#K4">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>;<br/>
<a NAME="E4:28"/><i><font color="Green">E29</font></i>: 
<span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="trees_1.html#K2">elementary_tree</a> 2</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"><font color="Maroon">c<sub>2</sub></font> <a href="trees_1.html#K4">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:28_1_1"/>
<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> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12">*&gt;</a></span>
 ;<br/><a NAME="E2:28_1_1"/><b>then </b>
<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> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/><b>hence </b><a NAME="E3:28_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="trees_1.html#T16">TREES_1:16</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:28_1"/>
<span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="trees_1.html#K2">elementary_tree</a> 2</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"><font color="Maroon">c<sub>2</sub></font> <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#E1:28"><i><font color="Green">E27</font></i></a>, <a class="ref" href="trees_1.html#D12">TREES_1:def 12</a></i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2"><b>now </b></a><div class="add"><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>thus </b><a NAME="E1:28_2"/>
(  not <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> or ( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="trees_1.html#K2">elementary_tree</a> 2</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"><font color="Maroon">c<sub>2</sub></font> <a href="trees_1.html#K4">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp; not <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">1</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> <font color="Maroon">c<sub>2</sub></font> <a href="trees_1.html#K4">|</a> <font color="Maroon">c<sub>4</sub></font> &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">1</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> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:28_2_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:28_2_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#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_2_1_1_1"><b>suppose </b></a><a NAME="E1:28_2_1_1_1"/>
<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"><b>hence </b><a NAME="E2:28_2_1_1_1"/>
( ( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="trees_1.html#K2">elementary_tree</a> 2</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"><font color="Maroon">c<sub>2</sub></font> <a href="trees_1.html#K4">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp; not <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">1</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> <font color="Maroon">c<sub>2</sub></font> <a href="trees_1.html#K4">|</a> <font color="Maroon">c<sub>4</sub></font> &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">1</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> ) )
 <b>by </b><i><a class="ref" href="trees_1.html#T36">TREES_1:36</a>, <a class="ref" href="trees_1.html#T39">TREES_1:39</a>, <a class="ref" href="trees_1.html#T47">TREES_1:47</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_2_1_1_2"><b>suppose </b></a><a NAME="E1:28_2_1_1_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">c<sub>8</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:28_2_1_1_2"/><i><font color="Green">E31</font></i>: 
<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"><font color="Maroon">c<sub>8</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="modal_1.html#T4" target="_self">Th4</a></i>;<br/><a NAME="E4:28_2_1_1_2"/>
<span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>8</sub></font></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>
 <b>by </b><i><a class="txt" href="modal_1.html#E3:28_2_1_1_2"><i><font color="Green">E31</font></i></a>, <a class="ref" href="trees_1.html#T8">TREES_1:8</a></i>;<br/><a NAME="E5:28_2_1_1_2"/><b>then </b><i><font color="Green">E32</font></i>: 
<span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>8</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="modal_1.html#E1:28_2_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="trees_1.html#T45">TREES_1:45</a></i>;<br/><span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>8</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> = 
<a href="xboole_0.html#K1">{}</a>  <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>8</sub></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/>.= 
<span class="p1">(<span class="default"><a href="modal_1.html#K1" target="_self">Root</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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>8</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span>

;<br/><a NAME="E7:28_2_1_1_2"/><b>then </b><i><font color="Green">E33</font></i>: 
<span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>8</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R2">in</a>  <a href="trees_2.html#K1">succ</a> <span class="p1">(<span class="default"><a href="modal_1.html#K1" target="_self">Root</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="modal_1.html#E5:28_2_1_1_2"><i><font color="Green">E32</font></i></a>, <a class="ref" href="trees_2.html#T14">TREES_2:14</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2_1_1_2_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_2_1_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:28_2_1_1_2_2_1"/>
( <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>8</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">0</span><a href="finseq_1.html#K12">*&gt;</a></span> or <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>8</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">1</span><a href="finseq_1.html#K12">*&gt;</a></span> )
 </span><b>by </b><i><a class="txt" href="modal_1.html#E3:28"><i><font color="Green">E28</font></i></a>, <a class="txt" href="modal_1.html#E7:28_2_1_1_2"><i><font color="Green">E33</font></i></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="28_2_1_1_2_2_1_1"><b>suppose </b></a><a NAME="E1:28_2_1_1_2_2_1_1"/><i><font color="Green">E34</font></i>: 
<span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>8</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">0</span><a href="finseq_1.html#K12">*&gt;</a></span>
 ;<br/><div class="add"><a NAME="E2:28_2_1_1_2_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="trees_1.html#K4">|</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="modal_1.html#E3:28"><i><font color="Green">E28</font></i></a>, <a class="txt" href="modal_1.html#E1:28_2_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="modal_1.html#E3:28_2_1_1_2"><i><font color="Green">E31</font></i></a>, <a class="ref" href="trees_1.html#D9">TREES_1:def 9</a></i>;<br/><b>hence </b><a NAME="E3:28_2_1_1_2_2_1_1"/>
( ( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="trees_1.html#K2">elementary_tree</a> 2</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"><font color="Maroon">c<sub>2</sub></font> <a href="trees_1.html#K4">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp; not <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">1</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> <font color="Maroon">c<sub>2</sub></font> <a href="trees_1.html#K4">|</a> <font color="Maroon">c<sub>4</sub></font> &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">1</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> ) )
 <b>by </b><i><a class="txt" href="modal_1.html#E1:28"><i><font color="Green">E27</font></i></a>, <a class="txt" href="modal_1.html#E3:28_2_1_1_2"><i><font color="Green">E31</font></i></a>, <a class="txt" href="modal_1.html#E1:28_2_1_1_2_2_1_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="modal_1.html#T5" target="_self">Th5</a>, <a class="ref" href="trees_1.html#D12">TREES_1:def 12</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_2_1_1_2_2_1_2"><b>suppose </b></a><a NAME="E1:28_2_1_1_2_2_1_2"/><i><font color="Green">E34</font></i>: 
<span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>8</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">1</span><a href="finseq_1.html#K12">*&gt;</a></span>
 ;<br/><div class="add"><a NAME="E2:28_2_1_1_2_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="trees_1.html#K4">|</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="modal_1.html#E3:28"><i><font color="Green">E28</font></i></a>, <a class="txt" href="modal_1.html#E1:28_2_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="modal_1.html#E3:28_2_1_1_2"><i><font color="Green">E31</font></i></a>, <a class="ref" href="trees_1.html#D9">TREES_1:def 9</a></i>;<br/><b>hence </b><a NAME="E3:28_2_1_1_2_2_1_2"/>
( ( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="trees_1.html#K2">elementary_tree</a> 2</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"><font color="Maroon">c<sub>2</sub></font> <a href="trees_1.html#K4">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp; not <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">1</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> <font color="Maroon">c<sub>2</sub></font> <a href="trees_1.html#K4">|</a> <font color="Maroon">c<sub>4</sub></font> &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">1</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> ) )
 <b>by </b><i><a class="txt" href="modal_1.html#E3:28_2_1_1_2"><i><font color="Green">E31</font></i></a>, <a class="txt" href="modal_1.html#E1:28_2_1_1_2_2_1_2"><i><font color="Green">E34</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E9:28_2_1_1_2"/>
( ( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="trees_1.html#K2">elementary_tree</a> 2</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"><font color="Maroon">c<sub>2</sub></font> <a href="trees_1.html#K4">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp; not <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">1</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> <font color="Maroon">c<sub>2</sub></font> <a href="trees_1.html#K4">|</a> <font color="Maroon">c<sub>4</sub></font> &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">1</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><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>assume </b><a NAME="E2:28_2"/><i><font color="Green">E30</font></i>: 
( ( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="trees_1.html#K2">elementary_tree</a> 2</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"><font color="Maroon">c<sub>2</sub></font> <a href="trees_1.html#K4">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp; not <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">1</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> <font color="Maroon">c<sub>2</sub></font> <a href="trees_1.html#K4">|</a> <font color="Maroon">c<sub>4</sub></font> &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">1</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><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:28_2_2_1"/>
( ( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="trees_1.html#K2">elementary_tree</a> 2</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"><font color="Maroon">c<sub>2</sub></font> <a href="trees_1.html#K4">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp; not <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">1</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> <font color="Maroon">c<sub>2</sub></font> <a href="trees_1.html#K4">|</a> <font color="Maroon">c<sub>4</sub></font> &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">1</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#E2:28_2"><i><font color="Green">E30</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_2_2_1_1"><b>suppose </b></a><a NAME="E1:28_2_2_1_1"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="trees_1.html#K2">elementary_tree</a> 2</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"><font color="Maroon">c<sub>2</sub></font> <a href="trees_1.html#K4">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp; not <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">1</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="28_2_2_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_2_2_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="E1:28_2_2_1_1_1_1"/>
( ( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="trees_1.html#K2">elementary_tree</a> 2 &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> <font color="Maroon">c<sub>2</sub></font> <a href="trees_1.html#K4">|</a> <font color="Maroon">c<sub>3</sub></font> &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#E1:28"><i><font color="Green">E27</font></i></a>, <a class="txt" href="modal_1.html#E1:28_2_2_1_1"><i><font color="Green">E31</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="28_2_2_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:28_2_2_1_1_1_1_1"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="trees_1.html#K2">elementary_tree</a> 2 &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"><a NAME="E2:28_2_2_1_1_1_1_1"/><b>then </b>
( <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> 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">1</span><a href="finseq_1.html#K12">*&gt;</a></span> )
 <b>by </b><i><a class="ref" href="modal_1.html#T10" target="_self">Th10</a>, <a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/><b>hence </b><a NAME="E3:28_2_2_1_1_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="modal_1.html#E3:28"><i><font color="Green">E28</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_2_2_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:28_2_2_1_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> <font color="Maroon">c<sub>2</sub></font> <a href="trees_1.html#K4">|</a> <font color="Maroon">c<sub>3</sub></font> &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="E2:28_2_2_1_1_1_1_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="modal_1.html#E3:28"><i><font color="Green">E28</font></i></a>, <a class="ref" href="trees_1.html#D9">TREES_1:def 9</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:28_2_2_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_2_2_1_2"><b>suppose </b></a><a NAME="E1:28_2_2_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> <font color="Maroon">c<sub>2</sub></font> <a href="trees_1.html#K4">|</a> <font color="Maroon">c<sub>4</sub></font> &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">1</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="E2:28_2_2_1_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="modal_1.html#E3:28"><i><font color="Green">E28</font></i></a>, <a class="ref" href="trees_1.html#D9">TREES_1:def 9</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:28_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:28"/>
<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> 2</span>)</span> <a href="trees_1.html#K5">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_1.html#K4">|</a> <font color="Maroon">c<sub>3</sub></font></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">1</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_1.html#K4">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="modal_1.html#E4:28"><i><font color="Green">E29</font></i></a>, <a class="ref" href="trees_1.html#D12">TREES_1:def 12</a></i>;<br/>


</div>
