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

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">T</font> be   <a href="trees_1.html#NM1">Tree</a>;<br/>



<b>thus </b><a NAME="E1:16"/>
<font color="Maroon">T</font> <a href="trees_2.html#K2">-level</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="trees_2.html#K1">succ</a> <font color="Olive">w</font></span>)</span> where <font color="Olive">w</font> is    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">T</font> :  <a href="finseq_1.html#K3">len</a> <font color="Olive">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> </span>}</span> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:16_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font> <a href="trees_2.html#K2">-level</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">t</font> = <font color="Maroon">x</font> as    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">T</font> ;<br/>
<a NAME="E3:16_1"/>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">w'</font> where <font color="Olive">w'</font> is    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">T</font> :  <a href="finseq_1.html#K3">len</a> <font color="Olive">w'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">w'</font> being    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E5:16_1"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w'</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">w'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/>
<a NAME="E6:16_1"/>
<font color="Maroon">t</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span> is    <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#T23">FINSEQ_1:23</a></i>;<br/>
<b>then consider </b><font color="Maroon">s</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="E8:16_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span>
 ;<br/>
<a NAME="E9:16_1"/>
<font color="Maroon">s</font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Maroon">t</font>
 
<b>by </b><i>, <a class="ref" href="trees_1.html#D1">TREES_1:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">s</font> = <font color="Maroon">s</font> as    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">T</font> <b>by </b><i><a class="ref" href="trees_1.html#T45">TREES_1:45</a></i>;<br/>
<a NAME="E11:16_1"/>
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E12:16_1"/><b>then </b><i><font color="Green">E32</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 
<b>by </b><i>, , <a class="ref" href="finseq_1.html#T21">FINSEQ_1:21</a></i>;<br/>
<a NAME="E13:16_1"/>
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">t</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E14:16_1"/><b>then </b>
( <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">t</font> &amp; <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="relat_1.html#T98">RELAT_1:98</a></i>;<br/>
<b>then consider </b><font color="Maroon">m</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E16:16_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</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">m</font></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i>, </i>;<br/>
<a NAME="E17:16_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="trees_2.html#K1">succ</a> <font color="Maroon">s</font>
 
<b>by </b><i/>;<br/>
<a NAME="E18:16_1"/>
 <a href="trees_2.html#K1">succ</a> <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="trees_2.html#K1">succ</a> <font color="Olive">w</font></span>)</span> where <font color="Olive">w</font> is    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">T</font> :  <a href="finseq_1.html#K3">len</a> <font color="Olive">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="qc_lang4.html#T4" target="_self">Th4</a></i>;<br/>
<b>hence </b><a NAME="E19:16_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="trees_2.html#K1">succ</a> <font color="Olive">w</font></span>)</span> where <font color="Olive">w</font> is    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">T</font> :  <a href="finseq_1.html#K3">len</a> <font color="Olive">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> </span>}</span> 
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


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

<b>thus </b><a NAME="E2:16"/>
 <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="trees_2.html#K1">succ</a> <font color="Olive">w</font></span>)</span> where <font color="Olive">w</font> is    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">T</font> :  <a href="finseq_1.html#K3">len</a> <font color="Olive">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">T</font> <a href="trees_2.html#K2">-level</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>set </b><font color="Maroon">X</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="trees_2.html#K1">succ</a> <font color="Olive">w</font></span>)</span> where <font color="Olive">w</font> is    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">T</font> :  <a href="finseq_1.html#K3">len</a> <font color="Olive">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> </span>}</span> ;<br/>
<b>assume </b><a NAME="E1:16_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="trees_2.html#K1">succ</a> <font color="Olive">w</font></span>)</span> where <font color="Olive">w</font> is    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">T</font> :  <a href="finseq_1.html#K3">len</a> <font color="Olive">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">Y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:16_2"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="trees_2.html#K1">succ</a> <font color="Olive">w</font></span>)</span> where <font color="Olive">w</font> is    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">T</font> :  <a href="finseq_1.html#K3">len</a> <font color="Olive">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> </span>}</span>  )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">w</font> being    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E5:16_2"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a>  <a href="trees_2.html#K1">succ</a> <font color="Maroon">w</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> )
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">t</font> = <font color="Maroon">x</font> as    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">T</font> <b>by </b><i>, </i>;<br/>
<a NAME="E7:16_2"/>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">w</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Olive">k</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> where <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Maroon">w</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Olive">k</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font> </span>}</span> 
 
<b>by </b><i>, </i>;<br/>
<b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E9:16_2"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w</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">k</font></span><a href="finseq_1.html#K12">*&gt;</a></span> &amp; <font color="Maroon">w</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">k</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font> )
 ;<br/>
<a NAME="E10:16_2"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">k</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/>
<a NAME="E11:16_2"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i>, , <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/>
<a NAME="E12:16_2"/><b>then </b>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">s</font> where <font color="Olive">s</font> is    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">T</font> :  <a href="finseq_1.html#K3">len</a> <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 </span>}</span> 
 
;<br/>
<b>hence </b><a NAME="E13:16_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font> <a href="trees_2.html#K2">-level</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/>


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


</div>
