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

<b>let </b><font color="Maroon">T1</font> be   <a href="trees_1.html#NM1">Tree</a>, <font color="Maroon">T2</font> be   <a href="trees_1.html#NM1">Tree</a>;<br/><b>let </b><font color="Maroon">p</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><br/>



<a NAME="E1:125"/><i><font color="Green">E39</font></i>: 
( <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">T1</font>,<font color="Maroon">T2</font></span><a href="finseq_1.html#K10">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">T1</font> &amp; <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">T1</font>,<font color="Maroon">T2</font></span><a href="finseq_1.html#K10">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">T2</font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">T1</font>,<font color="Maroon">T2</font></span><a href="finseq_1.html#K10">*&gt;</a></span> <a href="hidden.html#R1">=</a> 2 &amp; 1 <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> 2 &amp; 0 <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> 1 &amp;  <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">T1</font> &amp;  <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">T2</font> &amp; 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2 &amp; 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2 &amp; 0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a>, <a class="ref" href="trees_1.html#T47">TREES_1:47</a></i>;<br/>
<a NAME="E2:125"/><i><font color="Green">E40</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 &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> <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; 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> <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="enumset1.html#D1">ENUMSET1:def 1</a>, <a class="ref" href="modal_1.html#T7">MODAL_1:7</a>, <a class="ref" href="modal_1.html#T10">MODAL_1:10</a></i>;<br/>
<a NAME="E3:125"/><b>then </b><i><font color="Green">E47</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>,<font color="Maroon">T1</font> &amp; <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> <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>,<font color="Maroon">T1</font> )
 
<b>by </b><i><a class="ref" href="trees_1.html#D12">TREES_1:def 12</a></i>;<br/>
<b>thus </b><a NAME="E4:125"/>
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="trees_3.html#K15" target="_self">tree</a> <font color="Maroon">T1</font>,<font color="Maroon">T2</font> implies <font color="Maroon">p</font> <a href="hidden.html#R2">in</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>,<font color="Maroon">T1</font></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>,<font color="Maroon">T2</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="125_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:125_1"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="trees_3.html#K15" target="_self">tree</a> <font color="Maroon">T1</font>,<font color="Maroon">T2</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="125_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:125_1_1"/>
<font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><b>then consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">q</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E3:125_1_1"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">T1</font>,<font color="Maroon">T2</font></span><a href="finseq_1.html#K10">*&gt;</a></span> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">T1</font>,<font color="Maroon">T2</font></span><a href="finseq_1.html#K10">*&gt;</a></span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">n</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">q</font> )
 <b>by </b><i><a class="ref" href="trees_3.html#T18" target="_self">Th18</a>, <a class="ref" href="trees_3.html#D5" target="_self">Def5</a></i>;<br/><b>reconsider </b><font color="Maroon">q</font> = <font color="Maroon">q</font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i>, <a class="ref" href="finseq_1.html#T50">FINSEQ_1:50</a></i>;<br/><a NAME="E5:125_1_1"/><i><font color="Green">E68</font></i>: 
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="tarski.html#R1">is_a_prefix_of</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="Maroon">q</font>
 <b>by </b><i><a class="ref" href="modal_1.html#T6">MODAL_1:6</a></i>;<br/><div><i><font color="Green">E107</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="125_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:125_1_1_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:125_1_1_1"/><b>then </b>
( 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> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">n</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q</font> &amp; <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">n</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q</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>,<font color="Maroon">T1</font> )
 <b>by </b><i><a class="ref" href="trees_3.html#T15" target="_self">Th15</a>, <a class="ref" href="trees_3.html#T16" target="_self">Th16</a>, , <a class="ref" href="trees_3.html#D6" target="_self">Def6</a>, <a class="ref" href="trees_1.html#D12">TREES_1:def 12</a>, <a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/><b>hence </b><a NAME="E3:125_1_1_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</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>,<font color="Maroon">T1</font></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>,<font color="Maroon">T2</font>
 <b>by </b><i><a class="ref" href="trees_3.html#T17" target="_self">Th17</a>, , <a class="ref" href="trees_1.html#D12">TREES_1:def 12</a></i>;<br/></div><b>end;</b></div><a NAME="E7:125_1_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> 0 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="trees_3.html#T15" target="_self">Th15</a>, , <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E8:125_1_1"/><b>then </b>
( ( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1 &amp; ( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1 implies <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">n</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</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>,<font color="Maroon">T1</font></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>,<font color="Maroon">T2</font> ) ) or ( <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> 0 ) )
 <b>by </b><i><a class="ref" href="trees_3.html#T15" target="_self">Th15</a>, <a class="ref" href="trees_3.html#T17" target="_self">Th17</a>, , <a class="ref" href="nat_1.html#T26">NAT_1:26</a>, <a class="ref" href="trees_1.html#D12">TREES_1:def 12</a></i>;<br/><b>hence </b><a NAME="E9:125_1_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</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>,<font color="Maroon">T1</font></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>,<font color="Maroon">T2</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:125_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</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>,<font color="Maroon">T1</font></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>,<font color="Maroon">T2</font>
 <b>by </b><i><a class="ref" href="trees_1.html#T47">TREES_1:47</a></i>;<br/>


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

<b>assume </b><a NAME="E5:125"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</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>,<font color="Maroon">T1</font></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>,<font color="Maroon">T2</font>
 ;<br/>

<a NAME="E6:125"/><b>then </b><i><font color="Green">E108</font></i>: 
( ( <font color="Maroon">p</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>,<font color="Maroon">T1</font> &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">p</font> ) or  ex <font color="Olive">r</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T2</font> &amp; <font color="Maroon">p</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">r</font> ) )
 
<b>by </b><i><a class="ref" href="trees_3.html#T17" target="_self">Th17</a>, <a class="ref" href="trees_1.html#D12">TREES_1:def 12</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="125_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:125_2"/>
<font color="Maroon">p</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>,<font color="Maroon">T1</font>
 ;<br/><a NAME="E2:125_2"/><b>then </b><i><font color="Green">E109</font></i>: 
( ( <font color="Maroon">p</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">p</font> ) or  ex <font color="Olive">r</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T1</font> &amp; <font color="Maroon">p</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">r</font> ) )
 <b>by </b><i><a class="ref" href="trees_3.html#T16" target="_self">Th16</a>, <a class="ref" href="trees_1.html#D12">TREES_1:def 12</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="125_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:125_2_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="trees_1.html#K2">elementary_tree</a> 2
 ;<br/><a NAME="E2:125_2_1"/><b>then </b>
( ( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">p</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">p</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; <font color="Maroon">p</font> <a href="finseq_1.html#K7">^</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> )
 <b>by </b><i><a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a>, <a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a>, <a class="ref" href="modal_1.html#T10">MODAL_1:10</a></i>;<br/><b>hence </b><a NAME="E3:125_2_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="trees_3.html#K15" target="_self">tree</a> <font color="Maroon">T1</font>,<font color="Maroon">T2</font>
 <b>by </b><i><a class="ref" href="trees_3.html#T15" target="_self">Th15</a>, <a class="ref" href="trees_3.html#D5" target="_self">Def5</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:125_2"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="trees_3.html#K15" target="_self">tree</a> <font color="Maroon">T1</font>,<font color="Maroon">T2</font>
 <b>by </b><i><a class="ref" href="trees_3.html#T15" target="_self">Th15</a>, , <a class="ref" href="trees_3.html#D5" target="_self">Def5</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:125"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="trees_3.html#K15" target="_self">tree</a> <font color="Maroon">T1</font>,<font color="Maroon">T2</font>
 <b>by </b><i><a class="ref" href="trees_3.html#T15" target="_self">Th15</a>, <a class="ref" href="trees_3.html#T18" target="_self">Th18</a>, <a class="ref" href="trees_3.html#D5" target="_self">Def5</a></i>;<br/>


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