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

<b>let </b><font color="Maroon">T1</font> be   <a href="trees_2.html#NM6">DecoratedTree</a> of the <a href="ami_1.html#U2">Instruction-Locations</a> of <font color="Maroon">S</font>, <font color="Maroon">T2</font> be   <a href="trees_2.html#NM6">DecoratedTree</a> of the <a href="ami_1.html#U2">Instruction-Locations</a> of <font color="Maroon">S</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:53_1_2"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">T1</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a>  <a href="amistd_3.html#K2" target="_self">FirstLoc</a> <font color="Maroon">M</font>
 <b>and </b><br/><a NAME="E2:53_1_2"/><i><font color="Green">E77</font></i>: 
for <font color="Olive">t</font> being   <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">T1</font> holds <br/> (  <a href="trees_2.html#K1">succ</a> <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">t</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#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="amistd_1.html#K1">NIC</a> <span class="p2">(<span class="default"><a href="ami_5.html#K5">pi</a> <font color="Maroon">M</font>,<span class="p3">(<span class="default"><font color="Maroon">T1</font> <a href="trees_2.html#K3">.</a> <font color="Olive">t</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">T1</font> <a href="trees_2.html#K3">.</a> <font color="Olive">t</font></span>)</span></span>)</span> </span>}</span>  &amp; ( for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</font> <a href="hidden.html#R2">in</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="amistd_1.html#K1">NIC</a> <span class="p2">(<span class="default"><a href="ami_5.html#K5">pi</a> <font color="Maroon">M</font>,<span class="p3">(<span class="default"><font color="Maroon">T1</font> <a href="trees_2.html#K3">.</a> <font color="Olive">t</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">T1</font> <a href="trees_2.html#K3">.</a> <font color="Olive">t</font></span>)</span></span>)</span> holds <br/><font color="Maroon">T1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">t</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Olive">m</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="amistd_3.html#K4" target="_self">LocSeq</a> <span class="p2">(<span class="default"><a href="amistd_1.html#K1">NIC</a> <span class="p3">(<span class="default"><a href="ami_5.html#K5">pi</a> <font color="Maroon">M</font>,<span class="p4">(<span class="default"><font color="Maroon">T1</font> <a href="trees_2.html#K3">.</a> <font color="Olive">t</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">T1</font> <a href="trees_2.html#K3">.</a> <font color="Olive">t</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">m</font> ) )
 <b>and </b><br/><a NAME="E3:53_1_2"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">T2</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a>  <a href="amistd_3.html#K2" target="_self">FirstLoc</a> <font color="Maroon">M</font>
 <b>and </b><br/><a NAME="E4:53_1_2"/><i><font color="Green">E79</font></i>: 
for <font color="Olive">t</font> being   <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">T2</font> holds <br/> (  <a href="trees_2.html#K1">succ</a> <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">t</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#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="amistd_1.html#K1">NIC</a> <span class="p2">(<span class="default"><a href="ami_5.html#K5">pi</a> <font color="Maroon">M</font>,<span class="p3">(<span class="default"><font color="Maroon">T2</font> <a href="trees_2.html#K3">.</a> <font color="Olive">t</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">T2</font> <a href="trees_2.html#K3">.</a> <font color="Olive">t</font></span>)</span></span>)</span> </span>}</span>  &amp; ( for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</font> <a href="hidden.html#R2">in</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="amistd_1.html#K1">NIC</a> <span class="p2">(<span class="default"><a href="ami_5.html#K5">pi</a> <font color="Maroon">M</font>,<span class="p3">(<span class="default"><font color="Maroon">T2</font> <a href="trees_2.html#K3">.</a> <font color="Olive">t</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">T2</font> <a href="trees_2.html#K3">.</a> <font color="Olive">t</font></span>)</span></span>)</span> holds <br/><font color="Maroon">T2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">t</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Olive">m</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="amistd_3.html#K4" target="_self">LocSeq</a> <span class="p2">(<span class="default"><a href="amistd_1.html#K1">NIC</a> <span class="p3">(<span class="default"><a href="ami_5.html#K5">pi</a> <font color="Maroon">M</font>,<span class="p4">(<span class="default"><font color="Maroon">T2</font> <a href="trees_2.html#K3">.</a> <font color="Olive">t</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">T2</font> <a href="trees_2.html#K3">.</a> <font color="Olive">t</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">m</font> ) )
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b><span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">T1</font></span>)</span> <a href="trees_2.html#K2">-level</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">T2</font></span>)</span> <a href="trees_2.html#K2">-level</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E5:53_1_2"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_2_1"><b>proof </b></a><div class="add">

<b>thus </b><span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">T1</font></span>)</span> <a href="trees_2.html#K2">-level</a> 0 = 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="qc_lang4.html#T17">QC_LANG4:17</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">T2</font></span>)</span> <a href="trees_2.html#K2">-level</a> 0
<b>by </b><i><a class="ref" href="qc_lang4.html#T17">QC_LANG4:17</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E6:53_1_2"/><i><font color="Green">E81</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:53_1_2_2"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>]
 ;<br/>

<b>set </b><font color="Maroon">U1</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  <a href="relat_1.html#K1">dom</a> <font color="Maroon">T1</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>set </b><font color="Maroon">U2</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  <a href="relat_1.html#K1">dom</a> <font color="Maroon">T2</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/>
<a NAME="E2:53_1_2_2"/><i><font color="Green">E111</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">T1</font></span>)</span> <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="hidden.html#R1">=</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  <a href="relat_1.html#K1">dom</a> <font color="Maroon">T1</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#T18">QC_LANG4:18</a></i>;<br/>
<a NAME="E3:53_1_2_2"/><i><font color="Green">E112</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">T1</font></span>)</span> <a href="trees_2.html#K2">-level</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">v</font> where <font color="Olive">v</font> is    <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">T1</font> :  <a href="finseq_1.html#K3">len</a> <font color="Olive">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="trees_2.html#D6">TREES_2:def 6</a></i>;<br/>
<a NAME="E4:53_1_2_2"/><i><font color="Green">E114</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">T2</font></span>)</span> <a href="trees_2.html#K2">-level</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">v</font> where <font color="Olive">v</font> is    <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">T2</font> :  <a href="finseq_1.html#K3">len</a> <font color="Olive">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="trees_2.html#D6">TREES_2:def 6</a></i>;<br/>
<a NAME="E5:53_1_2_2"/>
 <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  <a href="relat_1.html#K1">dom</a> <font color="Maroon">T1</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="hidden.html#R1">=</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  <a href="relat_1.html#K1">dom</a> <font color="Maroon">T2</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> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_2_2_1"><b>proof </b></a><div class="add">

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a>,<a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a>
<div class="add"><b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:53_1_2_2_1_1"/>
<font color="Maroon">a</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  <a href="relat_1.html#K1">dom</a> <font color="Maroon">T1</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">A</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:53_1_2_2_1_1"/><i><font color="Green">E115</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E4:53_1_2_2_1_1"/><i><font color="Green">E116</font></i>: 
<font color="Maroon">A</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  <a href="relat_1.html#K1">dom</a> <font color="Maroon">T1</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  <a href="relat_1.html#K1">dom</a> <font color="Maroon">T1</font><b> such that </b><br/><a NAME="E6:53_1_2_2_1_1"/><i><font color="Green">E117</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="trees_2.html#K1">succ</a> <font color="Maroon">w</font>
 <b>and </b><br/><a NAME="E7:53_1_2_2_1_1"/><i><font color="Green">E118</font></i>: 
 <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/><a NAME="E8:53_1_2_2_1_1"/>
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">T1</font></span>)</span> <a href="trees_2.html#K2">-level</a> <font color="Maroon">n</font>
 <b>by </b><i>, </i>;<br/><b>then consider </b><font color="Maroon">v</font> being    <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">T2</font><b> such that </b><br/><a NAME="E10:53_1_2_2_1_1"/><i><font color="Green">E119</font></i>: 
<font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font>
 <b>and </b><br/><a NAME="E11:53_1_2_2_1_1"/><i><font color="Green">E120</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 <b>by </b><i>, </i>;<br/><a NAME="E12:53_1_2_2_1_1"/><i><font color="Green">E121</font></i>: 
 <a href="trees_2.html#K1">succ</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">v</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#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="amistd_1.html#K1">NIC</a> <span class="p2">(<span class="default"><a href="ami_5.html#K5">pi</a> <font color="Maroon">M</font>,<span class="p3">(<span class="default"><font color="Maroon">T2</font> <a href="trees_2.html#K3">.</a> <font color="Maroon">v</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">T2</font> <a href="trees_2.html#K3">.</a> <font color="Maroon">v</font></span>)</span></span>)</span> </span>}</span> 
 <b>by </b><i/>;<br/><a NAME="E13:53_1_2_2_1_1"/><i><font color="Green">E122</font></i>: 
 <a href="trees_2.html#K1">succ</a> <font color="Maroon">w</font> <a href="hidden.html#R1">=</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#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="amistd_1.html#K1">NIC</a> <span class="p2">(<span class="default"><a href="ami_5.html#K5">pi</a> <font color="Maroon">M</font>,<span class="p3">(<span class="default"><font color="Maroon">T1</font> <a href="trees_2.html#K3">.</a> <font color="Maroon">w</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">T1</font> <a href="trees_2.html#K3">.</a> <font color="Maroon">w</font></span>)</span></span>)</span> </span>}</span> 
 <b>by </b><i/>;<br/><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">w</font> &amp; <font color="Maroon">w</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">T1</font> &amp; <font color="Maroon">v</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">T2</font> implies <font color="Maroon">T1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">T2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span> );<br/><a NAME="E14:53_1_2_2_1_1"/><i><font color="Green">E123</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_2_2_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:53_1_2_2_1_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">w</font>
 ;<br/>

<b>assume </b><a NAME="E2:53_1_2_2_1_1_1"/>
<font color="Maroon">w</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> 0</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">T1</font>
 ;<br/>

<b>assume </b><a NAME="E3:53_1_2_2_1_1_1"/>
<font color="Maroon">v</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> 0</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">T2</font>
 ;<br/>

<a NAME="E4:53_1_2_2_1_1_1"/>
<font color="Maroon">w</font> <a href="partfun1.html#K2">|</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="relat_1.html#T110">RELAT_1:110</a></i>;<br/>
<b>hence </b><a NAME="E5:53_1_2_2_1_1_1"/>
<font color="Maroon">T1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> 0</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">T2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> 0</span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a></i>;<br/>


</div><b>end;</b></div><a NAME="E15:53_1_2_2_1_1"/><i><font color="Green">E124</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_2_2_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:53_1_2_2_1_1_2"/><i><font color="Green">E125</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">n</font>]
 <b>and </b><br/><a NAME="E2:53_1_2_2_1_1_2"/><i><font color="Green">E126</font></i>: 
<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">w</font>
 <b>and </b><br/><a NAME="E3:53_1_2_2_1_1_2"/><i><font color="Green">E127</font></i>: 
<font color="Maroon">w</font> <a href="partfun1.html#K2">|</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> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">T1</font>
 <b>and </b><br/><a NAME="E4:53_1_2_2_1_1_2"/><i><font color="Green">E128</font></i>: 
<font color="Maroon">v</font> <a href="partfun1.html#K2">|</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> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">T2</font>
 ;<br/>

<b>set </b><font color="Maroon">t1</font> = <font color="Maroon">w</font> <a href="partfun1.html#K2">|</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="E5:53_1_2_2_1_1_2"/><i><font color="Green">E130</font></i>: 
<font color="Maroon">n</font> <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="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E6:53_1_2_2_1_1_2"/><b>then </b><i><font color="Green">E131</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> <a href="tarski.html#R1">c=</a>  <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>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T7">FINSEQ_1:7</a></i>;<br/>
<a NAME="E7:53_1_2_2_1_1_2"/><b>then </b>
<font color="Maroon">w</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T103">RELAT_1:103</a></i>;<br/>
<a NAME="E8:53_1_2_2_1_1_2"/><b>then </b><i><font color="Green">E132</font></i>: 
<font color="Maroon">w</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Maroon">w</font> <a href="partfun1.html#K2">|</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="trees_1.html#D1">TREES_1:def 1</a></i>;<br/>
<a NAME="E9:53_1_2_2_1_1_2"/>
<font color="Maroon">v</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="amistd_3.html#T9" target="_self">Th9</a>, <a class="ref" href="relat_1.html#T103">RELAT_1:103</a></i>;<br/>
<a NAME="E10:53_1_2_2_1_1_2"/><b>then </b><i><font color="Green">E133</font></i>: 
<font color="Maroon">v</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Maroon">v</font> <a href="partfun1.html#K2">|</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="trees_1.html#D1">TREES_1:def 1</a></i>;<br/>
<a NAME="E11:53_1_2_2_1_1_2"/><i><font color="Green">E134</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">w</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E12:53_1_2_2_1_1_2"/><i><font color="Green">E135</font></i>: 
1 <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="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E13:53_1_2_2_1_1_2"/><b>then </b><i><font color="Green">E136</font></i>: 
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">w</font>
 
<b>by </b><i><a class="ref" href="amistd_3.html#T6" target="_self">Th6</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E14:53_1_2_2_1_1_2"/><i><font color="Green">E137</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <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="amistd_3.html#T6" target="_self">Th6</a>, <a class="ref" href="finseq_1.html#T21">FINSEQ_1:21</a></i>;<br/>
<a NAME="E15:53_1_2_2_1_1_2"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <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>
 
<b>by </b><i><a class="ref" href="amistd_3.html#T14" target="_self">Th14</a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E16:53_1_2_2_1_1_2"/><b>then </b>
<font color="Maroon">w</font> <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> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">w</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>
<a NAME="E17:53_1_2_2_1_1_2"/><b>then </b><i><font color="Green">E138</font></i>: 
<font color="Maroon">w</font> <a href="partfun1.html#K2">|</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> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">w</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><font color="Maroon">w</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p0">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="amistd_3.html#T15" target="_self">Th15</a>, <a class="ref" href="finseq_5.html#T11">FINSEQ_5:11</a></i>;<br/>
<b>reconsider </b><font color="Maroon">t1</font> = <font color="Maroon">w</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span> as    <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">T1</font> <b>by </b><i><a class="ref" href="amistd_3.html#T7" target="_self">Th7</a>, <a class="ref" href="amistd_3.html#T10" target="_self">Th10</a>, <a class="ref" href="trees_1.html#T45">TREES_1:45</a></i>;<br/>
<b>reconsider </b><font color="Maroon">t2</font> = <font color="Maroon">t1</font> as    <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">T2</font> <b>by </b><i>, , <a class="txt" href="amistd_3.html#E13"><i><font color="Green">Lemma54</font></i></a>, <a class="ref" href="trees_1.html#T45">TREES_1:45</a></i>;<br/>
<a NAME="E20:53_1_2_2_1_1_2"/><i><font color="Green">E140</font></i>: 
 <a href="trees_2.html#K1">succ</a> <font color="Maroon">t1</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">t1</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#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="amistd_1.html#K1">NIC</a> <span class="p2">(<span class="default"><a href="ami_5.html#K5">pi</a> <font color="Maroon">M</font>,<span class="p3">(<span class="default"><font color="Maroon">T1</font> <a href="trees_2.html#K3">.</a> <font color="Maroon">t1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">T1</font> <a href="trees_2.html#K3">.</a> <font color="Maroon">t1</font></span>)</span></span>)</span> </span>}</span> 
 
<b>by </b><i/>;<br/>
<a NAME="E21:53_1_2_2_1_1_2"/>
<font color="Maroon">t1</font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">w</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><font color="Maroon">w</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p0">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="hidden.html#R2">in</a>  <a href="trees_2.html#K1">succ</a> <font color="Maroon">t1</font>
 
<b>by </b><i><a class="ref" href="amistd_3.html#T7" target="_self">Th7</a>, <a class="ref" href="amistd_3.html#T13" target="_self">Th13</a>, , <a class="ref" href="trees_2.html#T14">TREES_2:14</a></i>;<br/>
<b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E23:53_1_2_2_1_1_2"/><i><font color="Green">E141</font></i>: 
<font color="Maroon">t1</font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">w</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><font color="Maroon">w</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p0">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">t1</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>
 <b>and </b><br/><a NAME="E24:53_1_2_2_1_1_2"/><i><font color="Green">E142</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="amistd_1.html#K1">NIC</a> <span class="p2">(<span class="default"><a href="ami_5.html#K5">pi</a> <font color="Maroon">M</font>,<span class="p3">(<span class="default"><font color="Maroon">T1</font> <a href="trees_2.html#K3">.</a> <font color="Maroon">t1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">T1</font> <a href="trees_2.html#K3">.</a> <font color="Maroon">t1</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/>
<a NAME="E25:53_1_2_2_1_1_2"/><i><font color="Green">E143</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">w</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="amistd_3.html#T16" target="_self">Th16</a>, <a class="ref" href="finseq_2.html#T20">FINSEQ_2:20</a></i>;<br/>
<a NAME="E26:53_1_2_2_1_1_2"/><i><font color="Green">E144</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">w</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="amistd_1.html#K1">NIC</a> <span class="p2">(<span class="default"><a href="ami_5.html#K5">pi</a> <font color="Maroon">M</font>,<span class="p3">(<span class="default"><font color="Maroon">T2</font> <a href="trees_2.html#K3">.</a> <font color="Maroon">t2</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">T2</font> <a href="trees_2.html#K3">.</a> <font color="Maroon">t2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="amistd_3.html#T5" target="_self">Th5</a>, <a class="ref" href="amistd_3.html#T6" target="_self">Th6</a>, , , <a class="txt" href="amistd_3.html#E13"><i><font color="Green">Lemma54</font></i></a>, <a class="ref" href="amistd_3.html#T16" target="_self">Th16</a>, <a class="ref" href="amistd_3.html#T17" target="_self">Th17</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="ref" href="finseq_2.html#T20">FINSEQ_2:20</a>, <a class="ref" href="trees_1.html#T45">TREES_1:45</a></i>;<br/>
<b>thus </b><font color="Maroon">T1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="amistd_3.html#K4" target="_self">LocSeq</a> <span class="p2">(<span class="default"><a href="amistd_1.html#K1">NIC</a> <span class="p3">(<span class="default"><a href="ami_5.html#K5">pi</a> <font color="Maroon">M</font>,<span class="p4">(<span class="default"><font color="Maroon">T1</font> <a href="trees_2.html#K3">.</a> <font color="Maroon">t1</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">T1</font> <a href="trees_2.html#K3">.</a> <font color="Maroon">t1</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">w</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">w</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="amistd_3.html#T17" target="_self">Th17</a>, </i>
<br/>.= 
<font color="Maroon">T2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="amistd_3.html#T5" target="_self">Th5</a>, <a class="ref" href="amistd_3.html#T6" target="_self">Th6</a>, , , <a class="txt" href="amistd_3.html#E13"><i><font color="Green">Lemma54</font></i></a>, <a class="ref" href="amistd_3.html#T13" target="_self">Th13</a>, , , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="ref" href="trees_1.html#T45">TREES_1:45</a></i>
;<br/>


</div><b>end;</b></div><a NAME="E16:53_1_2_2_1_1"/><i><font color="Green">E145</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">n</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i><a NAME="E17:53_1_2_2_1_1"/>
<font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">w</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_3.html#T55">FINSEQ_3:55</a></i>;<br/><a NAME="E18:53_1_2_2_1_1"/><b>then </b><i><font color="Green">E146</font></i>: 
<font color="Maroon">T1</font> <a href="trees_2.html#K3">.</a> <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">T2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">w</font>
 <b>by </b><i>, <a class="ref" href="amistd_3.html#T5" target="_self">Th5</a></i>;<br/><a NAME="E19:53_1_2_2_1_1"/>
 <a href="trees_2.html#K1">succ</a> <font color="Maroon">v</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  <a href="relat_1.html#K1">dom</a> <font color="Maroon">T2</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/>;<br/><b>hence </b><a NAME="E20:53_1_2_2_1_1"/>
<font color="Maroon">a</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  <a href="relat_1.html#K1">dom</a> <font color="Maroon">T2</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="amistd_3.html#T2" target="_self">Th2</a>, <a class="ref" href="amistd_3.html#T3" target="_self">Th3</a>, <a class="ref" href="amistd_3.html#T6" target="_self">Th6</a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">a</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="E2:53_1_2_2_1"/>
<font color="Maroon">a</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  <a href="relat_1.html#K1">dom</a> <font color="Maroon">T2</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">A</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:53_1_2_2_1"/><i><font color="Green">E147</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E5:53_1_2_2_1"/><i><font color="Green">E148</font></i>: 
<font color="Maroon">A</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  <a href="relat_1.html#K1">dom</a> <font color="Maroon">T2</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  <a href="relat_1.html#K1">dom</a> <font color="Maroon">T2</font><b> such that </b><br/><a NAME="E7:53_1_2_2_1"/><i><font color="Green">E149</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="trees_2.html#K1">succ</a> <font color="Maroon">w</font>
 <b>and </b><br/><a NAME="E8:53_1_2_2_1"/><i><font color="Green">E150</font></i>: 
 <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/>
<a NAME="E9:53_1_2_2_1"/>
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">T2</font></span>)</span> <a href="trees_2.html#K2">-level</a> <font color="Maroon">n</font>
 
<b>by </b><i>, </i>;<br/>
<b>then consider </b><font color="Maroon">v</font> being    <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">T1</font><b> such that </b><br/><a NAME="E11:53_1_2_2_1"/><i><font color="Green">E151</font></i>: 
<font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font>
 <b>and </b><br/><a NAME="E12:53_1_2_2_1"/><i><font color="Green">E152</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 <b>by </b><i>, </i>;<br/>
<a NAME="E13:53_1_2_2_1"/><i><font color="Green">E153</font></i>: 
 <a href="trees_2.html#K1">succ</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">v</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#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="amistd_1.html#K1">NIC</a> <span class="p2">(<span class="default"><a href="ami_5.html#K5">pi</a> <font color="Maroon">M</font>,<span class="p3">(<span class="default"><font color="Maroon">T1</font> <a href="trees_2.html#K3">.</a> <font color="Maroon">v</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">T1</font> <a href="trees_2.html#K3">.</a> <font color="Maroon">v</font></span>)</span></span>)</span> </span>}</span> 
 
<b>by </b><i/>;<br/>
<a NAME="E14:53_1_2_2_1"/><i><font color="Green">E154</font></i>: 
 <a href="trees_2.html#K1">succ</a> <font color="Maroon">w</font> <a href="hidden.html#R1">=</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#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="amistd_1.html#K1">NIC</a> <span class="p2">(<span class="default"><a href="ami_5.html#K5">pi</a> <font color="Maroon">M</font>,<span class="p3">(<span class="default"><font color="Maroon">T2</font> <a href="trees_2.html#K3">.</a> <font color="Maroon">w</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">T2</font> <a href="trees_2.html#K3">.</a> <font color="Maroon">w</font></span>)</span></span>)</span> </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">w</font> &amp; <font color="Maroon">w</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">T1</font> &amp; <font color="Maroon">v</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">T2</font> implies <font color="Maroon">T1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">T2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span> );<br/>
<a NAME="E15:53_1_2_2_1"/><i><font color="Green">E155</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_2_2_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:53_1_2_2_1_2"/>
0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">w</font>
 ;<br/>

<b>assume </b><a NAME="E2:53_1_2_2_1_2"/>
<font color="Maroon">w</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> 0</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">T1</font>
 ;<br/>

<b>assume </b><a NAME="E3:53_1_2_2_1_2"/>
<font color="Maroon">v</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> 0</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">T2</font>
 ;<br/>

<a NAME="E4:53_1_2_2_1_2"/>
<font color="Maroon">w</font> <a href="partfun1.html#K2">|</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="relat_1.html#T110">RELAT_1:110</a></i>;<br/>
<b>hence </b><a NAME="E5:53_1_2_2_1_2"/>
<font color="Maroon">T1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> 0</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">T2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> 0</span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E16:53_1_2_2_1"/><i><font color="Green">E156</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_2_2_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:53_1_2_2_1_3"/><i><font color="Green">E157</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">n</font>]
 <b>and </b><br/><a NAME="E2:53_1_2_2_1_3"/><i><font color="Green">E158</font></i>: 
<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">w</font>
 <b>and </b><br/><a NAME="E3:53_1_2_2_1_3"/><i><font color="Green">E159</font></i>: 
<font color="Maroon">w</font> <a href="partfun1.html#K2">|</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> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">T1</font>
 <b>and </b><br/><a NAME="E4:53_1_2_2_1_3"/><i><font color="Green">E160</font></i>: 
<font color="Maroon">v</font> <a href="partfun1.html#K2">|</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> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">T2</font>
 ;<br/>

<b>set </b><font color="Maroon">t1</font> = <font color="Maroon">w</font> <a href="partfun1.html#K2">|</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="E5:53_1_2_2_1_3"/><i><font color="Green">E161</font></i>: 
<font color="Maroon">n</font> <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="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E6:53_1_2_2_1_3"/><b>then </b><i><font color="Green">E162</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> <a href="tarski.html#R1">c=</a>  <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>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T7">FINSEQ_1:7</a></i>;<br/>
<a NAME="E7:53_1_2_2_1_3"/><b>then </b>
<font color="Maroon">w</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T103">RELAT_1:103</a></i>;<br/>
<a NAME="E8:53_1_2_2_1_3"/><b>then </b><i><font color="Green">E163</font></i>: 
<font color="Maroon">w</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Maroon">w</font> <a href="partfun1.html#K2">|</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="trees_1.html#D1">TREES_1:def 1</a></i>;<br/>
<a NAME="E9:53_1_2_2_1_3"/>
<font color="Maroon">v</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="amistd_3.html#T9" target="_self">Th9</a>, <a class="ref" href="relat_1.html#T103">RELAT_1:103</a></i>;<br/>
<a NAME="E10:53_1_2_2_1_3"/><b>then </b><i><font color="Green">E164</font></i>: 
<font color="Maroon">v</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Maroon">v</font> <a href="partfun1.html#K2">|</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="trees_1.html#D1">TREES_1:def 1</a></i>;<br/>
<a NAME="E11:53_1_2_2_1_3"/><i><font color="Green">E165</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">w</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E12:53_1_2_2_1_3"/><i><font color="Green">E166</font></i>: 
1 <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="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E13:53_1_2_2_1_3"/><b>then </b><i><font color="Green">E167</font></i>: 
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">w</font>
 
<b>by </b><i><a class="ref" href="amistd_3.html#T6" target="_self">Th6</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E14:53_1_2_2_1_3"/><i><font color="Green">E168</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <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="amistd_3.html#T6" target="_self">Th6</a>, <a class="ref" href="finseq_1.html#T21">FINSEQ_1:21</a></i>;<br/>
<a NAME="E15:53_1_2_2_1_3"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <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>
 
<b>by </b><i><a class="ref" href="amistd_3.html#T14" target="_self">Th14</a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E16:53_1_2_2_1_3"/><b>then </b>
<font color="Maroon">w</font> <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> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">w</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>
<a NAME="E17:53_1_2_2_1_3"/><b>then </b><i><font color="Green">E169</font></i>: 
<font color="Maroon">w</font> <a href="partfun1.html#K2">|</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> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">w</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><font color="Maroon">w</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p0">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="amistd_3.html#T15" target="_self">Th15</a>, <a class="ref" href="finseq_5.html#T11">FINSEQ_5:11</a></i>;<br/>
<b>reconsider </b><font color="Maroon">t1</font> = <font color="Maroon">w</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span> as    <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">T1</font> <b>by </b><i><a class="ref" href="amistd_3.html#T7" target="_self">Th7</a>, <a class="ref" href="amistd_3.html#T10" target="_self">Th10</a>, <a class="ref" href="trees_1.html#T45">TREES_1:45</a></i>;<br/>
<b>reconsider </b><font color="Maroon">t2</font> = <font color="Maroon">t1</font> as    <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">T2</font> <b>by </b><i>, , <a class="txt" href="amistd_3.html#E13"><i><font color="Green">Lemma54</font></i></a>, <a class="ref" href="trees_1.html#T45">TREES_1:45</a></i>;<br/>
<a NAME="E20:53_1_2_2_1_3"/><i><font color="Green">E170</font></i>: 
 <a href="trees_2.html#K1">succ</a> <font color="Maroon">t1</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">t1</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#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="amistd_1.html#K1">NIC</a> <span class="p2">(<span class="default"><a href="ami_5.html#K5">pi</a> <font color="Maroon">M</font>,<span class="p3">(<span class="default"><font color="Maroon">T1</font> <a href="trees_2.html#K3">.</a> <font color="Maroon">t1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">T1</font> <a href="trees_2.html#K3">.</a> <font color="Maroon">t1</font></span>)</span></span>)</span> </span>}</span> 
 
<b>by </b><i/>;<br/>
<a NAME="E21:53_1_2_2_1_3"/>
<font color="Maroon">t1</font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">w</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><font color="Maroon">w</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p0">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="hidden.html#R2">in</a>  <a href="trees_2.html#K1">succ</a> <font color="Maroon">t1</font>
 
<b>by </b><i><a class="ref" href="amistd_3.html#T7" target="_self">Th7</a>, <a class="ref" href="amistd_3.html#T13" target="_self">Th13</a>, , <a class="ref" href="trees_2.html#T14">TREES_2:14</a></i>;<br/>
<b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E23:53_1_2_2_1_3"/><i><font color="Green">E171</font></i>: 
<font color="Maroon">t1</font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">w</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><font color="Maroon">w</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p0">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">t1</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>
 <b>and </b><br/><a NAME="E24:53_1_2_2_1_3"/><i><font color="Green">E172</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="amistd_1.html#K1">NIC</a> <span class="p2">(<span class="default"><a href="ami_5.html#K5">pi</a> <font color="Maroon">M</font>,<span class="p3">(<span class="default"><font color="Maroon">T1</font> <a href="trees_2.html#K3">.</a> <font color="Maroon">t1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">T1</font> <a href="trees_2.html#K3">.</a> <font color="Maroon">t1</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/>
<a NAME="E25:53_1_2_2_1_3"/><i><font color="Green">E173</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">w</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="amistd_3.html#T16" target="_self">Th16</a>, <a class="ref" href="finseq_2.html#T20">FINSEQ_2:20</a></i>;<br/>
<a NAME="E26:53_1_2_2_1_3"/><i><font color="Green">E174</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">w</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="amistd_1.html#K1">NIC</a> <span class="p2">(<span class="default"><a href="ami_5.html#K5">pi</a> <font color="Maroon">M</font>,<span class="p3">(<span class="default"><font color="Maroon">T2</font> <a href="trees_2.html#K3">.</a> <font color="Maroon">t2</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">T2</font> <a href="trees_2.html#K3">.</a> <font color="Maroon">t2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="amistd_3.html#T5" target="_self">Th5</a>, <a class="ref" href="amistd_3.html#T6" target="_self">Th6</a>, , , <a class="txt" href="amistd_3.html#E13"><i><font color="Green">Lemma54</font></i></a>, <a class="ref" href="amistd_3.html#T16" target="_self">Th16</a>, <a class="ref" href="amistd_3.html#T17" target="_self">Th17</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="ref" href="finseq_2.html#T20">FINSEQ_2:20</a>, <a class="ref" href="trees_1.html#T45">TREES_1:45</a></i>;<br/>
<b>thus </b><font color="Maroon">T1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="amistd_3.html#K4" target="_self">LocSeq</a> <span class="p2">(<span class="default"><a href="amistd_1.html#K1">NIC</a> <span class="p3">(<span class="default"><a href="ami_5.html#K5">pi</a> <font color="Maroon">M</font>,<span class="p4">(<span class="default"><font color="Maroon">T1</font> <a href="trees_2.html#K3">.</a> <font color="Maroon">t1</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">T1</font> <a href="trees_2.html#K3">.</a> <font color="Maroon">t1</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">w</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">w</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="amistd_3.html#T17" target="_self">Th17</a>, </i>
<br/>.= 
<font color="Maroon">T2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="amistd_3.html#T5" target="_self">Th5</a>, <a class="ref" href="amistd_3.html#T6" target="_self">Th6</a>, , , <a class="txt" href="amistd_3.html#E13"><i><font color="Green">Lemma54</font></i></a>, <a class="ref" href="amistd_3.html#T13" target="_self">Th13</a>, , , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="ref" href="trees_1.html#T45">TREES_1:45</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E17:53_1_2_2_1"/><i><font color="Green">E175</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<a NAME="E18:53_1_2_2_1"/>
<font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">w</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T55">FINSEQ_3:55</a></i>;<br/>
<a NAME="E19:53_1_2_2_1"/><b>then </b><i><font color="Green">E176</font></i>: 
<font color="Maroon">T1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">T2</font> <a href="trees_2.html#K3">.</a> <font color="Maroon">w</font>
 
<b>by </b><i>, <a class="ref" href="amistd_3.html#T5" target="_self">Th5</a></i>;<br/>
<a NAME="E20:53_1_2_2_1"/>
 <a href="trees_2.html#K1">succ</a> <font color="Maroon">v</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  <a href="relat_1.html#K1">dom</a> <font color="Maroon">T1</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/>;<br/>
<b>hence </b><a NAME="E21:53_1_2_2_1"/>
<font color="Maroon">a</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  <a href="relat_1.html#K1">dom</a> <font color="Maroon">T1</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="amistd_3.html#T2" target="_self">Th2</a>, <a class="ref" href="amistd_3.html#T3" target="_self">Th3</a>, <a class="ref" href="amistd_3.html#T6" target="_self">Th6</a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:53_1_2_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i><a class="ref" href="amistd_3.html#T1" target="_self">Th1</a>, <a class="ref" href="qc_lang4.html#T18">QC_LANG4:18</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:53_1_2"/><i><font color="Green">E177</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<a NAME="E8:53_1_2"/><b>then </b><i><font color="Green">E178</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">T1</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">T2</font>
 
<b>by </b><i/>;<br/>
<a NAME="E9:53_1_2"/>
for <font color="Olive">p</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">p</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">T1</font> holds <br/><font color="Maroon">T1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">T2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_2_3"><b>proof </b></a><div class="add">

<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> ;<br/>

<b>assume </b><a NAME="E1:53_1_2_3"/><i><font color="Green">E179</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">T1</font>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> &amp; <font color="Maroon">p</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">T1</font> implies <font color="Maroon">T1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">T2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span> );<br/>
<a NAME="E2:53_1_2_3"/><i><font color="Green">E180</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_2_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:53_1_2_3_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font>
 ;<br/>

<b>assume </b><a NAME="E2:53_1_2_3_1"/>
<font color="Maroon">p</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> 0</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">T1</font>
 ;<br/>

<a NAME="E3:53_1_2_3_1"/>
<font color="Maroon">p</font> <a href="partfun1.html#K2">|</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="relat_1.html#T110">RELAT_1:110</a></i>;<br/>
<b>hence </b><a NAME="E4:53_1_2_3_1"/>
<font color="Maroon">T1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> 0</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">T2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> 0</span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:53_1_2_3"/><i><font color="Green">E181</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_2_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:53_1_2_3_2"/><i><font color="Green">E182</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">n</font>]
 <b>and </b><br/><a NAME="E2:53_1_2_3_2"/><i><font color="Green">E183</font></i>: 
<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">p</font>
 <b>and </b><br/><a NAME="E3:53_1_2_3_2"/><i><font color="Green">E184</font></i>: 
<font color="Maroon">p</font> <a href="partfun1.html#K2">|</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> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">T1</font>
 ;<br/>

<b>set </b><font color="Maroon">t1</font> = <font color="Maroon">p</font> <a href="partfun1.html#K2">|</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="E4:53_1_2_3_2"/><i><font color="Green">E185</font></i>: 
<font color="Maroon">n</font> <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="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E5:53_1_2_3_2"/><b>then </b>
 <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> <a href="tarski.html#R1">c=</a>  <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>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T7">FINSEQ_1:7</a></i>;<br/>
<a NAME="E6:53_1_2_3_2"/><b>then </b>
<font color="Maroon">p</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T103">RELAT_1:103</a></i>;<br/>
<a NAME="E7:53_1_2_3_2"/><b>then </b><i><font color="Green">E186</font></i>: 
<font color="Maroon">p</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Maroon">p</font> <a href="partfun1.html#K2">|</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="trees_1.html#D1">TREES_1:def 1</a></i>;<br/>
<a NAME="E8:53_1_2_3_2"/><i><font color="Green">E187</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E9:53_1_2_3_2"/><i><font color="Green">E188</font></i>: 
1 <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="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E10:53_1_2_3_2"/><b>then </b><i><font color="Green">E189</font></i>: 
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font>
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E11:53_1_2_3_2"/><i><font color="Green">E190</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <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#T21">FINSEQ_1:21</a></i>;<br/>
<a NAME="E12:53_1_2_3_2"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <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>
 
<b>by </b><i><a class="ref" href="amistd_3.html#T3" target="_self">Th3</a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E13:53_1_2_3_2"/><b>then </b>
<font color="Maroon">p</font> <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> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>
<a NAME="E14:53_1_2_3_2"/><b>then </b><i><font color="Green">E191</font></i>: 
<font color="Maroon">p</font> <a href="partfun1.html#K2">|</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> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p0">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span>
 
<b>by </b><i>, <a class="ref" href="finseq_5.html#T11">FINSEQ_5:11</a></i>;<br/>
<b>reconsider </b><font color="Maroon">t1</font> = <font color="Maroon">p</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span> as    <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">T1</font> <b>by </b><i>, , <a class="ref" href="trees_1.html#T45">TREES_1:45</a></i>;<br/>
<b>reconsider </b><font color="Maroon">t2</font> = <font color="Maroon">t1</font> as    <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">T2</font> <b>by </b><i>, </i>;<br/>
<a NAME="E17:53_1_2_3_2"/><i><font color="Green">E192</font></i>: 
 <a href="trees_2.html#K1">succ</a> <font color="Maroon">t1</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">t1</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#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="amistd_1.html#K1">NIC</a> <span class="p2">(<span class="default"><a href="ami_5.html#K5">pi</a> <font color="Maroon">M</font>,<span class="p3">(<span class="default"><font color="Maroon">T1</font> <a href="trees_2.html#K3">.</a> <font color="Maroon">t1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">T1</font> <a href="trees_2.html#K3">.</a> <font color="Maroon">t1</font></span>)</span></span>)</span> </span>}</span> 
 
<b>by </b><i/>;<br/>
<a NAME="E18:53_1_2_3_2"/>
<font color="Maroon">t1</font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p0">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="hidden.html#R2">in</a>  <a href="trees_2.html#K1">succ</a> <font color="Maroon">t1</font>
 
<b>by </b><i>, <a class="ref" href="amistd_3.html#T2" target="_self">Th2</a>, <a class="ref" href="amistd_3.html#T5" target="_self">Th5</a>, <a class="ref" href="trees_2.html#T14">TREES_2:14</a></i>;<br/>
<b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E20:53_1_2_3_2"/><i><font color="Green">E193</font></i>: 
<font color="Maroon">t1</font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p0">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">t1</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>
 <b>and </b><br/><a NAME="E21:53_1_2_3_2"/><i><font color="Green">E194</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="amistd_1.html#K1">NIC</a> <span class="p2">(<span class="default"><a href="ami_5.html#K5">pi</a> <font color="Maroon">M</font>,<span class="p3">(<span class="default"><font color="Maroon">T1</font> <a href="trees_2.html#K3">.</a> <font color="Maroon">t1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">T1</font> <a href="trees_2.html#K3">.</a> <font color="Maroon">t1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="amistd_3.html#T6" target="_self">Th6</a></i>;<br/>
<a NAME="E22:53_1_2_3_2"/><i><font color="Green">E195</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="amistd_3.html#T7" target="_self">Th7</a>, <a class="ref" href="finseq_2.html#T20">FINSEQ_2:20</a></i>;<br/>
<a NAME="E23:53_1_2_3_2"/><i><font color="Green">E196</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="amistd_1.html#K1">NIC</a> <span class="p2">(<span class="default"><a href="ami_5.html#K5">pi</a> <font color="Maroon">M</font>,<span class="p3">(<span class="default"><font color="Maroon">T2</font> <a href="trees_2.html#K3">.</a> <font color="Maroon">t2</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">T2</font> <a href="trees_2.html#K3">.</a> <font color="Maroon">t2</font></span>)</span></span>)</span>
 
<b>by </b><i>, , , <a class="ref" href="amistd_3.html#T7" target="_self">Th7</a>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="ref" href="finseq_2.html#T20">FINSEQ_2:20</a></i>;<br/>
<b>thus </b><font color="Maroon">T1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="amistd_3.html#K4" target="_self">LocSeq</a> <span class="p2">(<span class="default"><a href="amistd_1.html#K1">NIC</a> <span class="p3">(<span class="default"><a href="ami_5.html#K5">pi</a> <font color="Maroon">M</font>,<span class="p4">(<span class="default"><font color="Maroon">T1</font> <a href="trees_2.html#K3">.</a> <font color="Maroon">t1</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">T1</font> <a href="trees_2.html#K3">.</a> <font color="Maroon">t1</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="amistd_3.html#T5" target="_self">Th5</a>, , </i>
<br/>.= 
<font color="Maroon">T2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i>, , , , <a class="ref" href="amistd_3.html#T2" target="_self">Th2</a>, <a class="ref" href="amistd_3.html#T5" target="_self">Th5</a>, <a class="ref" href="amistd_3.html#T9" target="_self">Th9</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E4:53_1_2_3"/><i><font color="Green">E197</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<a NAME="E5:53_1_2_3"/>
<font color="Maroon">p</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T55">FINSEQ_3:55</a></i>;<br/>
<b>hence </b><a NAME="E6:53_1_2_3"/>
<font color="Maroon">T1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">T2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E10:53_1_2"/>
<font color="Maroon">T1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">T2</font>
 <b>by </b><i><a class="ref" href="amistd_3.html#T1" target="_self">Th1</a>, <a class="ref" href="trees_2.html#T33">TREES_2:33</a></i>;<br/>


</div>
