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

<b>let </b><font color="Maroon" title="c4">T1</font>, <font color="Maroon" title="c5">T2</font> be    <a href="ami_1.html#M7" title="AMI_1:mode.7">IL-DecoratedTree</a> of <font color="Maroon" title="c2">S</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:51_1_2"/><i><font color="Green" title="E25">A16</font></i>: 
<font color="Maroon" title="c4">T1</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="amistd_3.html#K2" title="AMISTD_3:func.2">FirstLoc</a> <font color="Maroon" title="c3">M</font>
 <b>and </b><br/><a NAME="E2:51_1_2"/><i><font color="Green" title="E26">A17</font></i>: 
for <font color="Olive" title="b1">t</font> being   <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">T1</font> holds <br/> (  <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">t</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Olive" title="b2">k</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> where <font color="Olive" title="b2">k</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Olive" title="b2">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> <span class="p2">(<span class="default"><a href="ami_1.html#K31" title="AMI_1:func.31">pi</a> <font color="Maroon" title="c3">M</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">T1</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Olive" title="b1">t</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">T1</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Olive" title="b1">t</font></span>)</span></span>)</span> </span>}</span>  &amp; ( for <font color="Olive" title="b2">m</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> <span class="p2">(<span class="default"><a href="ami_1.html#K31" title="AMI_1:func.31">pi</a> <font color="Maroon" title="c3">M</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">T1</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Olive" title="b1">t</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">T1</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Olive" title="b1">t</font></span>)</span></span>)</span> holds <br/><font color="Maroon" title="c4">T1</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">t</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Olive" title="b2">m</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="amistd_3.html#K4" title="AMISTD_3:func.4">LocSeq</a> <span class="p2">(<span class="default"><a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> <span class="p3">(<span class="default"><a href="ami_1.html#K31" title="AMI_1:func.31">pi</a> <font color="Maroon" title="c3">M</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">T1</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Olive" title="b1">t</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">T1</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Olive" title="b1">t</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">m</font> ) )
 <b>and </b><br/><a NAME="E3:51_1_2"/><i><font color="Green" title="E27">A18</font></i>: 
<font color="Maroon" title="c5">T2</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="amistd_3.html#K2" title="AMISTD_3:func.2">FirstLoc</a> <font color="Maroon" title="c3">M</font>
 <b>and </b><br/><a NAME="E4:51_1_2"/><i><font color="Green" title="E28">A19</font></i>: 
for <font color="Olive" title="b1">t</font> being   <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">T2</font> holds <br/> (  <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">t</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Olive" title="b2">k</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> where <font color="Olive" title="b2">k</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Olive" title="b2">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> <span class="p2">(<span class="default"><a href="ami_1.html#K31" title="AMI_1:func.31">pi</a> <font color="Maroon" title="c3">M</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">T2</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Olive" title="b1">t</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">T2</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Olive" title="b1">t</font></span>)</span></span>)</span> </span>}</span>  &amp; ( for <font color="Olive" title="b2">m</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> <span class="p2">(<span class="default"><a href="ami_1.html#K31" title="AMI_1:func.31">pi</a> <font color="Maroon" title="c3">M</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">T2</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Olive" title="b1">t</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">T2</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Olive" title="b1">t</font></span>)</span></span>)</span> holds <br/><font color="Maroon" title="c5">T2</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">t</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Olive" title="b2">m</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="amistd_3.html#K4" title="AMISTD_3:func.4">LocSeq</a> <span class="p2">(<span class="default"><a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> <span class="p3">(<span class="default"><a href="ami_1.html#K31" title="AMI_1:func.31">pi</a> <font color="Maroon" title="c3">M</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c5">T2</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Olive" title="b1">t</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">T2</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Olive" title="b1">t</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">m</font> ) )
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b><span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">T1</font></span>)</span> <a href="trees_2.html#K2" title="TREES_2:func.2">-level</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">T2</font></span>)</span> <a href="trees_2.html#K2" title="TREES_2:func.2">-level</a> $1;<br/>
<a NAME="E5:51_1_2"/><i><font color="Green" title="E29">A20</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_2_1"><b>proof </b></a><div class="add">

<b>thus </b><span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">T1</font></span>)</span> <a href="trees_2.html#K2" title="TREES_2:func.2">-level</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  = 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<b>by </b><i><a class="ref" href="qc_lang4.html#T17" title="QC_LANG4:th.17">QC_LANG4:17</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">T2</font></span>)</span> <a href="trees_2.html#K2" title="TREES_2:func.2">-level</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="ref" href="qc_lang4.html#T17" title="QC_LANG4:th.17">QC_LANG4:17</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E6:51_1_2"/><i><font color="Green" title="E30">A21</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:51_1_2_2"/><i><font color="Green" title="E31">A22</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">n</font>]
 ;<br/>

<b>set </b><font color="Maroon" title="c7">U1</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Olive" title="b1">w</font></span>)</span> where <font color="Olive" title="b1">w</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">T1</font> :  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">n</font> </span>}</span> ;<br/>
<b>set </b><font color="Maroon" title="c8">U2</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Olive" title="b1">w</font></span>)</span> where <font color="Olive" title="b1">w</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">T2</font> :  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">n</font> </span>}</span> ;<br/>
<a NAME="E2:51_1_2_2"/><i><font color="Green" title="E32">A23</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">T1</font></span>)</span> <a href="trees_2.html#K2" title="TREES_2:func.2">-level</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Olive" title="b1">w</font></span>)</span> where <font color="Olive" title="b1">w</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">T1</font> :  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">n</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="qc_lang4.html#T18" title="QC_LANG4:th.18">QC_LANG4:18</a></i>;<br/>
<a NAME="E3:51_1_2_2"/><i><font color="Green" title="E33">A24</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">T1</font></span>)</span> <a href="trees_2.html#K2" title="TREES_2:func.2">-level</a> <font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">v</font> where <font color="Olive" title="b1">v</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">T1</font> :  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">n</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="trees_2.html#D6" title="TREES_2:def.6">TREES_2:def 6</a></i>;<br/>
<a NAME="E4:51_1_2_2"/><i><font color="Green" title="E34">A25</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">T2</font></span>)</span> <a href="trees_2.html#K2" title="TREES_2:func.2">-level</a> <font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">v</font> where <font color="Olive" title="b1">v</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">T2</font> :  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">n</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="trees_2.html#D6" title="TREES_2:def.6">TREES_2:def 6</a></i>;<br/>
<a NAME="E5:51_1_2_2"/>
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Olive" title="b1">w</font></span>)</span> where <font color="Olive" title="b1">w</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">T1</font> :  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">n</font> </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Olive" title="b1">w</font></span>)</span> where <font color="Olive" title="b1">w</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">T2</font> :  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">n</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_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" title="TARSKI:def.3">TARSKI:def 3</a>,<a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a>
<div class="add"><b>let </b><font color="Maroon" title="c9">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:51_1_2_2_1_1"/>
<font color="Maroon" title="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Olive" title="b1">w</font></span>)</span> where <font color="Olive" title="b1">w</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">T1</font> :  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">n</font> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon" title="c10">A</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:51_1_2_2_1_1"/><i><font color="Green" title="E35">A26</font></i>: 
<font color="Maroon" title="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A</font>
 <b>and </b><br/><a NAME="E4:51_1_2_2_1_1"/><i><font color="Green" title="E36">A27</font></i>: 
<font color="Maroon" title="c10">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Olive" title="b1">w</font></span>)</span> where <font color="Olive" title="b1">w</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">T1</font> :  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">n</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon" title="c11">w</font> being    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">T1</font><b> such that </b><br/><a NAME="E6:51_1_2_2_1_1"/><i><font color="Green" title="E37">A28</font></i>: 
<font color="Maroon" title="c10">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c11">w</font>
 <b>and </b><br/><a NAME="E7:51_1_2_2_1_1"/><i><font color="Green" title="E38">A29</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">n</font>
 <b>by </b><i><a class="txt" href="amistd_3.html#E4:51_1_2_2_1_1"><i><font color="Green" title="E36">A27</font></i></a></i>;<br/><a NAME="E8:51_1_2_2_1_1"/>
<font color="Maroon" title="c11">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">T1</font></span>)</span> <a href="trees_2.html#K2" title="TREES_2:func.2">-level</a> <font color="Maroon" title="c6">n</font>
 <b>by </b><i><a class="txt" href="amistd_3.html#E3:51_1_2_2"><i><font color="Green" title="E33">A24</font></i></a>, <a class="txt" href="amistd_3.html#E7:51_1_2_2_1_1"><i><font color="Green" title="E38">A29</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c12">v</font> being    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">T2</font><b> such that </b><br/><a NAME="E10:51_1_2_2_1_1"/><i><font color="Green" title="E39">A30</font></i>: 
<font color="Maroon" title="c11">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">v</font>
 <b>and </b><br/><a NAME="E11:51_1_2_2_1_1"/><i><font color="Green" title="E40">A31</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">n</font>
 <b>by </b><i><a class="txt" href="amistd_3.html#E1:51_1_2_2"><i><font color="Green" title="E31">A22</font></i></a>, <a class="txt" href="amistd_3.html#E4:51_1_2_2"><i><font color="Green" title="E34">A25</font></i></a></i>;<br/><a NAME="E12:51_1_2_2_1_1"/><i><font color="Green" title="E41">A32</font></i>: 
 <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c12">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c12">v</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Olive" title="b1">k</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> where <font color="Olive" title="b1">k</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> <span class="p2">(<span class="default"><a href="ami_1.html#K31" title="AMI_1:func.31">pi</a> <font color="Maroon" title="c3">M</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">T2</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Maroon" title="c12">v</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">T2</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Maroon" title="c12">v</font></span>)</span></span>)</span> </span>}</span> 
 <b>by </b><i><a class="txt" href="amistd_3.html#E4:51_1_2"><i><font color="Green" title="E28">A19</font></i></a></i>;<br/><a NAME="E13:51_1_2_2_1_1"/><i><font color="Green" title="E42">A33</font></i>: 
 <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c11">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c11">w</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Olive" title="b1">k</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> where <font color="Olive" title="b1">k</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> <span class="p2">(<span class="default"><a href="ami_1.html#K31" title="AMI_1:func.31">pi</a> <font color="Maroon" title="c3">M</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">T1</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Maroon" title="c11">w</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">T1</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Maroon" title="c11">w</font></span>)</span></span>)</span> </span>}</span> 
 <b>by </b><i><a class="txt" href="amistd_3.html#E2:51_1_2"><i><font color="Green" title="E26">A17</font></i></a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b>( $1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">w</font> &amp; <font color="Maroon" title="c11">w</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> $1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">T1</font> &amp; <font color="Maroon" title="c12">v</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> $1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">T2</font> implies <font color="Maroon" title="c4">T1</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">w</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> $1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">T2</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">w</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> $1</span>)</span></span>)</span> );<br/><a NAME="E14:51_1_2_2_1_1"/><i><font color="Green" title="E43">A34</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_2_2_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:51_1_2_2_1_1_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">w</font>
 ;<br/>

<b>assume </b><a NAME="E2:51_1_2_2_1_1_1"/>
<font color="Maroon" title="c11">w</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">T1</font>
 ;<br/>

<b>assume </b><a NAME="E3:51_1_2_2_1_1_1"/>
<font color="Maroon" title="c12">v</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">T2</font>
 ;<br/>

<a NAME="E4:51_1_2_2_1_1_1"/>
<font color="Maroon" title="c11">w</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="relat_1.html#T110" title="RELAT_1:th.110">RELAT_1:110</a></i>;<br/>
<b>hence </b><a NAME="E5:51_1_2_2_1_1_1"/>
<font color="Maroon" title="c4">T1</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">w</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">T2</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">w</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="amistd_3.html#E1:51_1_2"><i><font color="Green" title="E25">A16</font></i></a>, <a class="txt" href="amistd_3.html#E3:51_1_2"><i><font color="Green" title="E27">A18</font></i></a>, <a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a></i>;<br/>


</div><b>end;</b></div><a NAME="E15:51_1_2_2_1_1"/><i><font color="Green" title="E44">A35</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_2_2_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:51_1_2_2_1_1_2"/><i><font color="Green" title="E45">A36</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c13">n</font>]
 <b>and </b><br/><a NAME="E2:51_1_2_2_1_1_2"/><i><font color="Green" title="E46">A37</font></i>: 
<font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">w</font>
 <b>and </b><br/><a NAME="E3:51_1_2_2_1_1_2"/><i><font color="Green" title="E47">A38</font></i>: 
<font color="Maroon" title="c11">w</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">T1</font>
 <b>and </b><br/><a NAME="E4:51_1_2_2_1_1_2"/><i><font color="Green" title="E48">A39</font></i>: 
<font color="Maroon" title="c12">v</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">T2</font>
 ;<br/>

<b>set </b><font color="Maroon" title="c14">t1</font> = <font color="Maroon" title="c11">w</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c13">n</font></span>)</span>;<br/>
<a NAME="E5:51_1_2_2_1_1_2"/><i><font color="Green" title="E49">A40</font></i>: 
<font color="Maroon" title="c13">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<a NAME="E6:51_1_2_2_1_1_2"/><b>then </b><i><font color="Green" title="E50">A41</font></i>: 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c13">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T7" title="FINSEQ_1:th.7">FINSEQ_1:7</a></i>;<br/>
<a NAME="E7:51_1_2_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c11">w</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">w</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c13">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T103" title="RELAT_1:th.103">RELAT_1:103</a></i>;<br/>
<a NAME="E8:51_1_2_2_1_1_2"/><b>then </b><i><font color="Green" title="E51">A42</font></i>: 
<font color="Maroon" title="c11">w</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">is_a_prefix_of</a> <font color="Maroon" title="c11">w</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="trees_1.html#D1" title="TREES_1:def.1">TREES_1:def 1</a></i>;<br/>
<a NAME="E9:51_1_2_2_1_1_2"/>
<font color="Maroon" title="c12">v</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">v</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c13">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="amistd_3.html#E6:51_1_2_2_1_1_2"><i><font color="Green" title="E50">A41</font></i></a>, <a class="ref" href="relat_1.html#T103" title="RELAT_1:th.103">RELAT_1:103</a></i>;<br/>
<a NAME="E10:51_1_2_2_1_1_2"/><b>then </b><i><font color="Green" title="E52">A43</font></i>: 
<font color="Maroon" title="c12">v</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">is_a_prefix_of</a> <font color="Maroon" title="c12">v</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="trees_1.html#D1" title="TREES_1:def.1">TREES_1:def 1</a></i>;<br/>
<a NAME="E11:51_1_2_2_1_1_2"/><i><font color="Green" title="E53">A44</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">w</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">w</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E12:51_1_2_2_1_1_2"/><i><font color="Green" title="E54">A45</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<a NAME="E13:51_1_2_2_1_1_2"/><b>then </b><i><font color="Green" title="E55">A46</font></i>: 
<font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">w</font>
 
<b>by </b><i><a class="txt" href="amistd_3.html#E2:51_1_2_2_1_1_2"><i><font color="Green" title="E46">A37</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E14:51_1_2_2_1_1_2"/><i><font color="Green" title="E56">A47</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">w</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="amistd_3.html#E2:51_1_2_2_1_1_2"><i><font color="Green" title="E46">A37</font></i></a>, <a class="ref" href="finseq_1.html#T21" title="FINSEQ_1:th.21">FINSEQ_1:21</a></i>;<br/>
<a NAME="E15:51_1_2_2_1_1_2"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">w</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="amistd_3.html#E12:51_1_2_2_1_1_2"><i><font color="Green" title="E54">A45</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E16:51_1_2_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c11">w</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">w</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">w</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="amistd_3.html#E14:51_1_2_2_1_1_2"><i><font color="Green" title="E56">A47</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/>
<a NAME="E17:51_1_2_2_1_1_2"/><b>then </b><i><font color="Green" title="E57">A48</font></i>: 
<font color="Maroon" title="c11">w</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">w</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c13">n</font></span>)</span></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">w</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p5">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c11">w</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p0">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="amistd_3.html#E13:51_1_2_2_1_1_2"><i><font color="Green" title="E55">A46</font></i></a>, <a class="ref" href="finseq_5.html#T11" title="FINSEQ_5:th.11">FINSEQ_5:11</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c15">t1</font> = <font color="Maroon" title="c11">w</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c13">n</font></span>)</span> as    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">T1</font> <b>by </b><i><a class="txt" href="amistd_3.html#E3:51_1_2_2_1_1_2"><i><font color="Green" title="E47">A38</font></i></a>, <a class="txt" href="amistd_3.html#E8:51_1_2_2_1_1_2"><i><font color="Green" title="E51">A42</font></i></a>, <a class="ref" href="trees_1.html#T45" title="TREES_1:th.45">TREES_1:45</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c16">t2</font> = <font color="Maroon" title="c15">t1</font> as    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">T2</font> <b>by </b><i><a class="txt" href="amistd_3.html#E10:51_1_2_2_1_1"><i><font color="Green" title="E39">A30</font></i></a>, <a class="txt" href="amistd_3.html#E4:51_1_2_2_1_1_2"><i><font color="Green" title="E48">A39</font></i></a>, <a class="txt" href="amistd_3.html#E10:51_1_2_2_1_1_2"><i><font color="Green" title="E52">A43</font></i></a>, <a class="ref" href="trees_1.html#T45" title="TREES_1:th.45">TREES_1:45</a></i>;<br/>
<a NAME="E20:51_1_2_2_1_1_2"/><i><font color="Green" title="E58">A49</font></i>: 
 <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c15">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c15">t1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Olive" title="b1">k</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> where <font color="Olive" title="b1">k</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> <span class="p2">(<span class="default"><a href="ami_1.html#K31" title="AMI_1:func.31">pi</a> <font color="Maroon" title="c3">M</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">T1</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Maroon" title="c15">t1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">T1</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Maroon" title="c15">t1</font></span>)</span></span>)</span> </span>}</span> 
 
<b>by </b><i><a class="txt" href="amistd_3.html#E2:51_1_2"><i><font color="Green" title="E26">A17</font></i></a></i>;<br/>
<a NAME="E21:51_1_2_2_1_1_2"/>
<font color="Maroon" title="c15">t1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">w</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p5">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c11">w</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p0">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c15">t1</font>
 
<b>by </b><i><a class="txt" href="amistd_3.html#E3:51_1_2_2_1_1_2"><i><font color="Green" title="E47">A38</font></i></a>, <a class="txt" href="amistd_3.html#E11:51_1_2_2_1_1_2"><i><font color="Green" title="E53">A44</font></i></a>, <a class="txt" href="amistd_3.html#E17:51_1_2_2_1_1_2"><i><font color="Green" title="E57">A48</font></i></a>, <a class="ref" href="trees_2.html#T14" title="TREES_2:th.14">TREES_2:14</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c17">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E23:51_1_2_2_1_1_2"/><i><font color="Green" title="E59">A50</font></i>: 
<font color="Maroon" title="c15">t1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">w</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p5">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c11">w</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p0">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">t1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c17">k</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <b>and </b><br/><a NAME="E24:51_1_2_2_1_1_2"/><i><font color="Green" title="E60">A51</font></i>: 
<font color="Maroon" title="c17">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> <span class="p2">(<span class="default"><a href="ami_1.html#K31" title="AMI_1:func.31">pi</a> <font color="Maroon" title="c3">M</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">T1</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Maroon" title="c15">t1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">T1</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Maroon" title="c15">t1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="amistd_3.html#E20:51_1_2_2_1_1_2"><i><font color="Green" title="E58">A49</font></i></a></i>;<br/>
<a NAME="E25:51_1_2_2_1_1_2"/><i><font color="Green" title="E61">A52</font></i>: 
<font color="Maroon" title="c17">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">w</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">w</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="amistd_3.html#E23:51_1_2_2_1_1_2"><i><font color="Green" title="E59">A50</font></i></a>, <a class="ref" href="finseq_2.html#T20" title="FINSEQ_2:th.20">FINSEQ_2:20</a></i>;<br/>
<a NAME="E26:51_1_2_2_1_1_2"/><i><font color="Green" title="E62">A53</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">w</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">w</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> <span class="p2">(<span class="default"><a href="ami_1.html#K31" title="AMI_1:func.31">pi</a> <font color="Maroon" title="c3">M</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">T2</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Maroon" title="c16">t2</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">T2</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Maroon" title="c16">t2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="amistd_3.html#E1:51_1_2_2_1_1_2"><i><font color="Green" title="E45">A36</font></i></a>, <a class="txt" href="amistd_3.html#E2:51_1_2_2_1_1_2"><i><font color="Green" title="E46">A37</font></i></a>, <a class="txt" href="amistd_3.html#E4:51_1_2_2_1_1_2"><i><font color="Green" title="E48">A39</font></i></a>, <a class="txt" href="amistd_3.html#E5:51_1_2_2_1_1_2"><i><font color="Green" title="E49">A40</font></i></a>, <a class="txt" href="amistd_3.html#E10:51_1_2_2_1_1_2"><i><font color="Green" title="E52">A43</font></i></a>, <a class="txt" href="amistd_3.html#E23:51_1_2_2_1_1_2"><i><font color="Green" title="E59">A50</font></i></a>, <a class="txt" href="amistd_3.html#E24:51_1_2_2_1_1_2"><i><font color="Green" title="E60">A51</font></i></a>, <a class="ref" href="finseq_2.html#T20" title="FINSEQ_2:th.20">FINSEQ_2:20</a>, <a class="ref" href="trees_1.html#T45" title="TREES_1:th.45">TREES_1:45</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c4">T1</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">w</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="amistd_3.html#K4" title="AMISTD_3:func.4">LocSeq</a> <span class="p2">(<span class="default"><a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> <span class="p3">(<span class="default"><a href="ami_1.html#K31" title="AMI_1:func.31">pi</a> <font color="Maroon" title="c3">M</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">T1</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Maroon" title="c15">t1</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">T1</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Maroon" title="c15">t1</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">w</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">w</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p5">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="amistd_3.html#E2:51_1_2"><i><font color="Green" title="E26">A17</font></i></a>, <a class="txt" href="amistd_3.html#E17:51_1_2_2_1_1_2"><i><font color="Green" title="E57">A48</font></i></a>, <a class="txt" href="amistd_3.html#E24:51_1_2_2_1_1_2"><i><font color="Green" title="E60">A51</font></i></a>, <a class="txt" href="amistd_3.html#E25:51_1_2_2_1_1_2"><i><font color="Green" title="E61">A52</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c5">T2</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">w</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="amistd_3.html#E4:51_1_2"><i><font color="Green" title="E28">A19</font></i></a>, <a class="txt" href="amistd_3.html#E1:51_1_2_2_1_1_2"><i><font color="Green" title="E45">A36</font></i></a>, <a class="txt" href="amistd_3.html#E2:51_1_2_2_1_1_2"><i><font color="Green" title="E46">A37</font></i></a>, <a class="txt" href="amistd_3.html#E4:51_1_2_2_1_1_2"><i><font color="Green" title="E48">A39</font></i></a>, <a class="txt" href="amistd_3.html#E5:51_1_2_2_1_1_2"><i><font color="Green" title="E49">A40</font></i></a>, <a class="txt" href="amistd_3.html#E10:51_1_2_2_1_1_2"><i><font color="Green" title="E52">A43</font></i></a>, <a class="txt" href="amistd_3.html#E11:51_1_2_2_1_1_2"><i><font color="Green" title="E53">A44</font></i></a>, <a class="txt" href="amistd_3.html#E17:51_1_2_2_1_1_2"><i><font color="Green" title="E57">A48</font></i></a>, <a class="txt" href="amistd_3.html#E26:51_1_2_2_1_1_2"><i><font color="Green" title="E62">A53</font></i></a>, <a class="ref" href="trees_1.html#T45" title="TREES_1:th.45">TREES_1:45</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>
;<br/>


</div><b>end;</b></div><a NAME="E16:51_1_2_2_1_1"/><i><font color="Green" title="E45">A54</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">n</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="amistd_3.html#E14:51_1_2_2_1_1"><i><font color="Green" title="E43">A34</font></i></a>, <a class="txt" href="amistd_3.html#E15:51_1_2_2_1_1"><i><font color="Green" title="E44">A35</font></i></a>);<br/></i><a NAME="E17:51_1_2_2_1_1"/>
<font color="Maroon" title="c11">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">w</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">w</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_3.html#T55" title="FINSEQ_3:th.55">FINSEQ_3:55</a></i>;<br/><a NAME="E18:51_1_2_2_1_1"/><b>then </b><i><font color="Green" title="E46">A55</font></i>: 
<font color="Maroon" title="c4">T1</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Maroon" title="c11">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">T2</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Maroon" title="c11">w</font>
 <b>by </b><i><a class="txt" href="amistd_3.html#E10:51_1_2_2_1_1"><i><font color="Green" title="E39">A30</font></i></a>, <a class="txt" href="amistd_3.html#E16:51_1_2_2_1_1"><i><font color="Green" title="E45">A54</font></i></a></i>;<br/><a NAME="E19:51_1_2_2_1_1"/>
 <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c12">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Olive" title="b1">w</font></span>)</span> where <font color="Olive" title="b1">w</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">T2</font> :  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">n</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="amistd_3.html#E11:51_1_2_2_1_1"><i><font color="Green" title="E40">A31</font></i></a></i>;<br/><b>hence </b><a NAME="E20:51_1_2_2_1_1"/>
<font color="Maroon" title="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Olive" title="b1">w</font></span>)</span> where <font color="Olive" title="b1">w</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">T2</font> :  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">n</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="amistd_3.html#E3:51_1_2_2_1_1"><i><font color="Green" title="E35">A26</font></i></a>, <a class="txt" href="amistd_3.html#E6:51_1_2_2_1_1"><i><font color="Green" title="E37">A28</font></i></a>, <a class="txt" href="amistd_3.html#E10:51_1_2_2_1_1"><i><font color="Green" title="E39">A30</font></i></a>, <a class="txt" href="amistd_3.html#E12:51_1_2_2_1_1"><i><font color="Green" title="E41">A32</font></i></a>, <a class="txt" href="amistd_3.html#E13:51_1_2_2_1_1"><i><font color="Green" title="E42">A33</font></i></a>, <a class="txt" href="amistd_3.html#E18:51_1_2_2_1_1"><i><font color="Green" title="E46">A55</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon" title="c9">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:51_1_2_2_1"/>
<font color="Maroon" title="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Olive" title="b1">w</font></span>)</span> where <font color="Olive" title="b1">w</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">T2</font> :  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">n</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c10">A</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:51_1_2_2_1"/><i><font color="Green" title="E35">A56</font></i>: 
<font color="Maroon" title="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A</font>
 <b>and </b><br/><a NAME="E5:51_1_2_2_1"/><i><font color="Green" title="E36">A57</font></i>: 
<font color="Maroon" title="c10">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Olive" title="b1">w</font></span>)</span> where <font color="Olive" title="b1">w</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">T2</font> :  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">n</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c11">w</font> being    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">T2</font><b> such that </b><br/><a NAME="E7:51_1_2_2_1"/><i><font color="Green" title="E37">A58</font></i>: 
<font color="Maroon" title="c10">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c11">w</font>
 <b>and </b><br/><a NAME="E8:51_1_2_2_1"/><i><font color="Green" title="E38">A59</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">n</font>
 <b>by </b><i><a class="txt" href="amistd_3.html#E5:51_1_2_2_1"><i><font color="Green" title="E36">A57</font></i></a></i>;<br/>
<a NAME="E9:51_1_2_2_1"/>
<font color="Maroon" title="c11">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">T2</font></span>)</span> <a href="trees_2.html#K2" title="TREES_2:func.2">-level</a> <font color="Maroon" title="c6">n</font>
 
<b>by </b><i><a class="txt" href="amistd_3.html#E4:51_1_2_2"><i><font color="Green" title="E34">A25</font></i></a>, <a class="txt" href="amistd_3.html#E8:51_1_2_2_1"><i><font color="Green" title="E38">A59</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c12">v</font> being    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">T1</font><b> such that </b><br/><a NAME="E11:51_1_2_2_1"/><i><font color="Green" title="E39">A60</font></i>: 
<font color="Maroon" title="c11">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">v</font>
 <b>and </b><br/><a NAME="E12:51_1_2_2_1"/><i><font color="Green" title="E40">A61</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">n</font>
 <b>by </b><i><a class="txt" href="amistd_3.html#E1:51_1_2_2"><i><font color="Green" title="E31">A22</font></i></a>, <a class="txt" href="amistd_3.html#E3:51_1_2_2"><i><font color="Green" title="E33">A24</font></i></a></i>;<br/>
<a NAME="E13:51_1_2_2_1"/><i><font color="Green" title="E41">A62</font></i>: 
 <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c12">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c12">v</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Olive" title="b1">k</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> where <font color="Olive" title="b1">k</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> <span class="p2">(<span class="default"><a href="ami_1.html#K31" title="AMI_1:func.31">pi</a> <font color="Maroon" title="c3">M</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">T1</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Maroon" title="c12">v</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">T1</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Maroon" title="c12">v</font></span>)</span></span>)</span> </span>}</span> 
 
<b>by </b><i><a class="txt" href="amistd_3.html#E2:51_1_2"><i><font color="Green" title="E26">A17</font></i></a></i>;<br/>
<a NAME="E14:51_1_2_2_1"/><i><font color="Green" title="E42">A63</font></i>: 
 <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c11">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c11">w</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Olive" title="b1">k</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> where <font color="Olive" title="b1">k</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> <span class="p2">(<span class="default"><a href="ami_1.html#K31" title="AMI_1:func.31">pi</a> <font color="Maroon" title="c3">M</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">T2</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Maroon" title="c11">w</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">T2</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Maroon" title="c11">w</font></span>)</span></span>)</span> </span>}</span> 
 
<b>by </b><i><a class="txt" href="amistd_3.html#E4:51_1_2"><i><font color="Green" title="E28">A19</font></i></a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b>( $1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">w</font> &amp; <font color="Maroon" title="c11">w</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> $1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">T1</font> &amp; <font color="Maroon" title="c12">v</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> $1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">T2</font> implies <font color="Maroon" title="c4">T1</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">w</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> $1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">T2</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">w</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> $1</span>)</span></span>)</span> );<br/>
<a NAME="E15:51_1_2_2_1"/><i><font color="Green" title="E43">A64</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_2_2_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:51_1_2_2_1_2"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">w</font>
 ;<br/>

<b>assume </b><a NAME="E2:51_1_2_2_1_2"/>
<font color="Maroon" title="c11">w</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">T1</font>
 ;<br/>

<b>assume </b><a NAME="E3:51_1_2_2_1_2"/>
<font color="Maroon" title="c12">v</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">T2</font>
 ;<br/>

<a NAME="E4:51_1_2_2_1_2"/>
<font color="Maroon" title="c11">w</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="relat_1.html#T110" title="RELAT_1:th.110">RELAT_1:110</a></i>;<br/>
<b>hence </b><a NAME="E5:51_1_2_2_1_2"/>
<font color="Maroon" title="c4">T1</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">w</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">T2</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">w</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="amistd_3.html#E1:51_1_2"><i><font color="Green" title="E25">A16</font></i></a>, <a class="txt" href="amistd_3.html#E3:51_1_2"><i><font color="Green" title="E27">A18</font></i></a>, <a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E16:51_1_2_2_1"/><i><font color="Green" title="E44">A65</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_2_2_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:51_1_2_2_1_3"/><i><font color="Green" title="E45">A66</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c13">n</font>]
 <b>and </b><br/><a NAME="E2:51_1_2_2_1_3"/><i><font color="Green" title="E46">A67</font></i>: 
<font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">w</font>
 <b>and </b><br/><a NAME="E3:51_1_2_2_1_3"/><i><font color="Green" title="E47">A68</font></i>: 
<font color="Maroon" title="c11">w</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">T1</font>
 <b>and </b><br/><a NAME="E4:51_1_2_2_1_3"/><i><font color="Green" title="E48">A69</font></i>: 
<font color="Maroon" title="c12">v</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">T2</font>
 ;<br/>

<b>set </b><font color="Maroon" title="c14">t1</font> = <font color="Maroon" title="c11">w</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c13">n</font></span>)</span>;<br/>
<a NAME="E5:51_1_2_2_1_3"/><i><font color="Green" title="E49">A70</font></i>: 
<font color="Maroon" title="c13">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<a NAME="E6:51_1_2_2_1_3"/><b>then </b><i><font color="Green" title="E50">A71</font></i>: 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c13">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T7" title="FINSEQ_1:th.7">FINSEQ_1:7</a></i>;<br/>
<a NAME="E7:51_1_2_2_1_3"/><b>then </b>
<font color="Maroon" title="c11">w</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">w</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c13">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T103" title="RELAT_1:th.103">RELAT_1:103</a></i>;<br/>
<a NAME="E8:51_1_2_2_1_3"/><b>then </b><i><font color="Green" title="E51">A72</font></i>: 
<font color="Maroon" title="c11">w</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">is_a_prefix_of</a> <font color="Maroon" title="c11">w</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="trees_1.html#D1" title="TREES_1:def.1">TREES_1:def 1</a></i>;<br/>
<a NAME="E9:51_1_2_2_1_3"/>
<font color="Maroon" title="c12">v</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">v</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c13">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="amistd_3.html#E6:51_1_2_2_1_3"><i><font color="Green" title="E50">A71</font></i></a>, <a class="ref" href="relat_1.html#T103" title="RELAT_1:th.103">RELAT_1:103</a></i>;<br/>
<a NAME="E10:51_1_2_2_1_3"/><b>then </b><i><font color="Green" title="E52">A73</font></i>: 
<font color="Maroon" title="c12">v</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">is_a_prefix_of</a> <font color="Maroon" title="c12">v</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="trees_1.html#D1" title="TREES_1:def.1">TREES_1:def 1</a></i>;<br/>
<a NAME="E11:51_1_2_2_1_3"/><i><font color="Green" title="E53">A74</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">w</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">w</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E12:51_1_2_2_1_3"/><i><font color="Green" title="E54">A75</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<a NAME="E13:51_1_2_2_1_3"/><b>then </b><i><font color="Green" title="E55">A76</font></i>: 
<font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">w</font>
 
<b>by </b><i><a class="txt" href="amistd_3.html#E2:51_1_2_2_1_3"><i><font color="Green" title="E46">A67</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E14:51_1_2_2_1_3"/><i><font color="Green" title="E56">A77</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">w</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="amistd_3.html#E2:51_1_2_2_1_3"><i><font color="Green" title="E46">A67</font></i></a>, <a class="ref" href="finseq_1.html#T21" title="FINSEQ_1:th.21">FINSEQ_1:21</a></i>;<br/>
<a NAME="E15:51_1_2_2_1_3"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">w</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="amistd_3.html#E12:51_1_2_2_1_3"><i><font color="Green" title="E54">A75</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E16:51_1_2_2_1_3"/><b>then </b>
<font color="Maroon" title="c11">w</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">w</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">w</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="amistd_3.html#E14:51_1_2_2_1_3"><i><font color="Green" title="E56">A77</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/>
<a NAME="E17:51_1_2_2_1_3"/><b>then </b><i><font color="Green" title="E57">A78</font></i>: 
<font color="Maroon" title="c11">w</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">w</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c13">n</font></span>)</span></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">w</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p5">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c11">w</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p0">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="amistd_3.html#E13:51_1_2_2_1_3"><i><font color="Green" title="E55">A76</font></i></a>, <a class="ref" href="finseq_5.html#T11" title="FINSEQ_5:th.11">FINSEQ_5:11</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c15">t1</font> = <font color="Maroon" title="c11">w</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c13">n</font></span>)</span> as    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">T1</font> <b>by </b><i><a class="txt" href="amistd_3.html#E3:51_1_2_2_1_3"><i><font color="Green" title="E47">A68</font></i></a>, <a class="txt" href="amistd_3.html#E8:51_1_2_2_1_3"><i><font color="Green" title="E51">A72</font></i></a>, <a class="ref" href="trees_1.html#T45" title="TREES_1:th.45">TREES_1:45</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c16">t2</font> = <font color="Maroon" title="c15">t1</font> as    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">T2</font> <b>by </b><i><a class="txt" href="amistd_3.html#E11:51_1_2_2_1"><i><font color="Green" title="E39">A60</font></i></a>, <a class="txt" href="amistd_3.html#E4:51_1_2_2_1_3"><i><font color="Green" title="E48">A69</font></i></a>, <a class="txt" href="amistd_3.html#E10:51_1_2_2_1_3"><i><font color="Green" title="E52">A73</font></i></a>, <a class="ref" href="trees_1.html#T45" title="TREES_1:th.45">TREES_1:45</a></i>;<br/>
<a NAME="E20:51_1_2_2_1_3"/><i><font color="Green" title="E58">A79</font></i>: 
 <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c15">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c15">t1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Olive" title="b1">k</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> where <font color="Olive" title="b1">k</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> <span class="p2">(<span class="default"><a href="ami_1.html#K31" title="AMI_1:func.31">pi</a> <font color="Maroon" title="c3">M</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">T1</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Maroon" title="c15">t1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">T1</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Maroon" title="c15">t1</font></span>)</span></span>)</span> </span>}</span> 
 
<b>by </b><i><a class="txt" href="amistd_3.html#E2:51_1_2"><i><font color="Green" title="E26">A17</font></i></a></i>;<br/>
<a NAME="E21:51_1_2_2_1_3"/>
<font color="Maroon" title="c15">t1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">w</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p5">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c11">w</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p0">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c15">t1</font>
 
<b>by </b><i><a class="txt" href="amistd_3.html#E3:51_1_2_2_1_3"><i><font color="Green" title="E47">A68</font></i></a>, <a class="txt" href="amistd_3.html#E11:51_1_2_2_1_3"><i><font color="Green" title="E53">A74</font></i></a>, <a class="txt" href="amistd_3.html#E17:51_1_2_2_1_3"><i><font color="Green" title="E57">A78</font></i></a>, <a class="ref" href="trees_2.html#T14" title="TREES_2:th.14">TREES_2:14</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c17">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E23:51_1_2_2_1_3"/><i><font color="Green" title="E59">A80</font></i>: 
<font color="Maroon" title="c15">t1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">w</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p5">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c11">w</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p0">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">t1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c17">k</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <b>and </b><br/><a NAME="E24:51_1_2_2_1_3"/><i><font color="Green" title="E60">A81</font></i>: 
<font color="Maroon" title="c17">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> <span class="p2">(<span class="default"><a href="ami_1.html#K31" title="AMI_1:func.31">pi</a> <font color="Maroon" title="c3">M</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">T1</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Maroon" title="c15">t1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">T1</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Maroon" title="c15">t1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="amistd_3.html#E20:51_1_2_2_1_3"><i><font color="Green" title="E58">A79</font></i></a></i>;<br/>
<a NAME="E25:51_1_2_2_1_3"/><i><font color="Green" title="E61">A82</font></i>: 
<font color="Maroon" title="c17">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">w</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">w</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="amistd_3.html#E23:51_1_2_2_1_3"><i><font color="Green" title="E59">A80</font></i></a>, <a class="ref" href="finseq_2.html#T20" title="FINSEQ_2:th.20">FINSEQ_2:20</a></i>;<br/>
<a NAME="E26:51_1_2_2_1_3"/><i><font color="Green" title="E62">A83</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">w</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">w</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> <span class="p2">(<span class="default"><a href="ami_1.html#K31" title="AMI_1:func.31">pi</a> <font color="Maroon" title="c3">M</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">T2</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Maroon" title="c16">t2</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">T2</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Maroon" title="c16">t2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="amistd_3.html#E1:51_1_2_2_1_3"><i><font color="Green" title="E45">A66</font></i></a>, <a class="txt" href="amistd_3.html#E2:51_1_2_2_1_3"><i><font color="Green" title="E46">A67</font></i></a>, <a class="txt" href="amistd_3.html#E4:51_1_2_2_1_3"><i><font color="Green" title="E48">A69</font></i></a>, <a class="txt" href="amistd_3.html#E5:51_1_2_2_1_3"><i><font color="Green" title="E49">A70</font></i></a>, <a class="txt" href="amistd_3.html#E10:51_1_2_2_1_3"><i><font color="Green" title="E52">A73</font></i></a>, <a class="txt" href="amistd_3.html#E23:51_1_2_2_1_3"><i><font color="Green" title="E59">A80</font></i></a>, <a class="txt" href="amistd_3.html#E24:51_1_2_2_1_3"><i><font color="Green" title="E60">A81</font></i></a>, <a class="ref" href="finseq_2.html#T20" title="FINSEQ_2:th.20">FINSEQ_2:20</a>, <a class="ref" href="trees_1.html#T45" title="TREES_1:th.45">TREES_1:45</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c4">T1</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">w</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="amistd_3.html#K4" title="AMISTD_3:func.4">LocSeq</a> <span class="p2">(<span class="default"><a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> <span class="p3">(<span class="default"><a href="ami_1.html#K31" title="AMI_1:func.31">pi</a> <font color="Maroon" title="c3">M</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">T1</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Maroon" title="c15">t1</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">T1</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Maroon" title="c15">t1</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">w</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">w</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p5">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="amistd_3.html#E2:51_1_2"><i><font color="Green" title="E26">A17</font></i></a>, <a class="txt" href="amistd_3.html#E17:51_1_2_2_1_3"><i><font color="Green" title="E57">A78</font></i></a>, <a class="txt" href="amistd_3.html#E24:51_1_2_2_1_3"><i><font color="Green" title="E60">A81</font></i></a>, <a class="txt" href="amistd_3.html#E25:51_1_2_2_1_3"><i><font color="Green" title="E61">A82</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c5">T2</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">w</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="amistd_3.html#E4:51_1_2"><i><font color="Green" title="E28">A19</font></i></a>, <a class="txt" href="amistd_3.html#E1:51_1_2_2_1_3"><i><font color="Green" title="E45">A66</font></i></a>, <a class="txt" href="amistd_3.html#E2:51_1_2_2_1_3"><i><font color="Green" title="E46">A67</font></i></a>, <a class="txt" href="amistd_3.html#E4:51_1_2_2_1_3"><i><font color="Green" title="E48">A69</font></i></a>, <a class="txt" href="amistd_3.html#E5:51_1_2_2_1_3"><i><font color="Green" title="E49">A70</font></i></a>, <a class="txt" href="amistd_3.html#E10:51_1_2_2_1_3"><i><font color="Green" title="E52">A73</font></i></a>, <a class="txt" href="amistd_3.html#E11:51_1_2_2_1_3"><i><font color="Green" title="E53">A74</font></i></a>, <a class="txt" href="amistd_3.html#E17:51_1_2_2_1_3"><i><font color="Green" title="E57">A78</font></i></a>, <a class="txt" href="amistd_3.html#E26:51_1_2_2_1_3"><i><font color="Green" title="E62">A83</font></i></a>, <a class="ref" href="trees_1.html#T45" title="TREES_1:th.45">TREES_1:45</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E17:51_1_2_2_1"/><i><font color="Green" title="E45">A84</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="amistd_3.html#E15:51_1_2_2_1"><i><font color="Green" title="E43">A64</font></i></a>, <a class="txt" href="amistd_3.html#E16:51_1_2_2_1"><i><font color="Green" title="E44">A65</font></i></a>);<br/></i>
<a NAME="E18:51_1_2_2_1"/>
<font color="Maroon" title="c11">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">w</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">w</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T55" title="FINSEQ_3:th.55">FINSEQ_3:55</a></i>;<br/>
<a NAME="E19:51_1_2_2_1"/><b>then </b><i><font color="Green" title="E46">A85</font></i>: 
<font color="Maroon" title="c4">T1</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Maroon" title="c11">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">T2</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Maroon" title="c11">w</font>
 
<b>by </b><i><a class="txt" href="amistd_3.html#E11:51_1_2_2_1"><i><font color="Green" title="E39">A60</font></i></a>, <a class="txt" href="amistd_3.html#E17:51_1_2_2_1"><i><font color="Green" title="E45">A84</font></i></a></i>;<br/>
<a NAME="E20:51_1_2_2_1"/>
 <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c12">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Olive" title="b1">w</font></span>)</span> where <font color="Olive" title="b1">w</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">T1</font> :  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">n</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="amistd_3.html#E12:51_1_2_2_1"><i><font color="Green" title="E40">A61</font></i></a></i>;<br/>
<b>hence </b><a NAME="E21:51_1_2_2_1"/>
<font color="Maroon" title="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Olive" title="b1">w</font></span>)</span> where <font color="Olive" title="b1">w</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">T1</font> :  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">n</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="amistd_3.html#E4:51_1_2_2_1"><i><font color="Green" title="E35">A56</font></i></a>, <a class="txt" href="amistd_3.html#E7:51_1_2_2_1"><i><font color="Green" title="E37">A58</font></i></a>, <a class="txt" href="amistd_3.html#E11:51_1_2_2_1"><i><font color="Green" title="E39">A60</font></i></a>, <a class="txt" href="amistd_3.html#E13:51_1_2_2_1"><i><font color="Green" title="E41">A62</font></i></a>, <a class="txt" href="amistd_3.html#E14:51_1_2_2_1"><i><font color="Green" title="E42">A63</font></i></a>, <a class="txt" href="amistd_3.html#E19:51_1_2_2_1"><i><font color="Green" title="E46">A85</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:51_1_2_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <b>by </b><i><a class="txt" href="amistd_3.html#E2:51_1_2_2"><i><font color="Green" title="E32">A23</font></i></a>, <a class="ref" href="qc_lang4.html#T18" title="QC_LANG4:th.18">QC_LANG4:18</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:51_1_2"/><i><font color="Green" title="E31">A86</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="amistd_3.html#E5:51_1_2"><i><font color="Green" title="E29">A20</font></i></a>, <a class="txt" href="amistd_3.html#E6:51_1_2"><i><font color="Green" title="E30">A21</font></i></a>);<br/></i>
<a NAME="E8:51_1_2"/><b>then </b><i><font color="Green" title="E32">A87</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">T1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">T2</font>
 
<b>by </b><i><a class="ref" href="amistd_3.html#T18" target="_self" title="AMISTD_3:th.18">Th18</a></i>;<br/>
<a NAME="E9:51_1_2"/>
for <font color="Olive" title="b1">p</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">T1</font> holds <br/><font color="Maroon" title="c4">T1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">T2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">p</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">p</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:51_1_2_3"/><i><font color="Green" title="E33">A88</font></i>: 
<font color="Maroon" title="c6">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">T1</font>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b>( $1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">p</font> &amp; <font color="Maroon" title="c6">p</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> $1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">T1</font> implies <font color="Maroon" title="c4">T1</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> $1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">T2</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> $1</span>)</span></span>)</span> );<br/>
<a NAME="E2:51_1_2_3"/><i><font color="Green" title="E34">A89</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_2_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:51_1_2_3_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">p</font>
 ;<br/>

<b>assume </b><a NAME="E2:51_1_2_3_1"/>
<font color="Maroon" title="c6">p</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">T1</font>
 ;<br/>

<a NAME="E3:51_1_2_3_1"/>
<font color="Maroon" title="c6">p</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="relat_1.html#T110" title="RELAT_1:th.110">RELAT_1:110</a></i>;<br/>
<b>hence </b><a NAME="E4:51_1_2_3_1"/>
<font color="Maroon" title="c4">T1</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">T2</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="amistd_3.html#E1:51_1_2"><i><font color="Green" title="E25">A16</font></i></a>, <a class="txt" href="amistd_3.html#E3:51_1_2"><i><font color="Green" title="E27">A18</font></i></a>, <a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:51_1_2_3"/><i><font color="Green" title="E35">A90</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_2_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:51_1_2_3_2"/><i><font color="Green" title="E36">A91</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c7">n</font>]
 <b>and </b><br/><a NAME="E2:51_1_2_3_2"/><i><font color="Green" title="E37">A92</font></i>: 
<font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">p</font>
 <b>and </b><br/><a NAME="E3:51_1_2_3_2"/><i><font color="Green" title="E38">A93</font></i>: 
<font color="Maroon" title="c6">p</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">T1</font>
 ;<br/>

<b>set </b><font color="Maroon" title="c8">t1</font> = <font color="Maroon" title="c6">p</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c7">n</font></span>)</span>;<br/>
<a NAME="E4:51_1_2_3_2"/><i><font color="Green" title="E39">A94</font></i>: 
<font color="Maroon" title="c7">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<a NAME="E5:51_1_2_3_2"/><b>then </b>
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c7">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T7" title="FINSEQ_1:th.7">FINSEQ_1:7</a></i>;<br/>
<a NAME="E6:51_1_2_3_2"/><b>then </b>
<font color="Maroon" title="c6">p</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c7">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T103" title="RELAT_1:th.103">RELAT_1:103</a></i>;<br/>
<a NAME="E7:51_1_2_3_2"/><b>then </b><i><font color="Green" title="E40">A95</font></i>: 
<font color="Maroon" title="c6">p</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">is_a_prefix_of</a> <font color="Maroon" title="c6">p</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="trees_1.html#D1" title="TREES_1:def.1">TREES_1:def 1</a></i>;<br/>
<a NAME="E8:51_1_2_3_2"/><i><font color="Green" title="E41">A96</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E9:51_1_2_3_2"/><i><font color="Green" title="E42">A97</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<a NAME="E10:51_1_2_3_2"/><b>then </b><i><font color="Green" title="E43">A98</font></i>: 
<font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">p</font>
 
<b>by </b><i><a class="txt" href="amistd_3.html#E2:51_1_2_3_2"><i><font color="Green" title="E37">A92</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E11:51_1_2_3_2"/><i><font color="Green" title="E44">A99</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="amistd_3.html#E2:51_1_2_3_2"><i><font color="Green" title="E37">A92</font></i></a>, <a class="ref" href="finseq_1.html#T21" title="FINSEQ_1:th.21">FINSEQ_1:21</a></i>;<br/>
<a NAME="E12:51_1_2_3_2"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="amistd_3.html#E9:51_1_2_3_2"><i><font color="Green" title="E42">A97</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E13:51_1_2_3_2"/><b>then </b>
<font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="amistd_3.html#E11:51_1_2_3_2"><i><font color="Green" title="E44">A99</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/>
<a NAME="E14:51_1_2_3_2"/><b>then </b><i><font color="Green" title="E45">A100</font></i>: 
<font color="Maroon" title="c6">p</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p5">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p0">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="amistd_3.html#E10:51_1_2_3_2"><i><font color="Green" title="E43">A98</font></i></a>, <a class="ref" href="finseq_5.html#T11" title="FINSEQ_5:th.11">FINSEQ_5:11</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">t1</font> = <font color="Maroon" title="c6">p</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c7">n</font></span>)</span> as    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">T1</font> <b>by </b><i><a class="txt" href="amistd_3.html#E3:51_1_2_3_2"><i><font color="Green" title="E38">A93</font></i></a>, <a class="txt" href="amistd_3.html#E7:51_1_2_3_2"><i><font color="Green" title="E40">A95</font></i></a>, <a class="ref" href="trees_1.html#T45" title="TREES_1:th.45">TREES_1:45</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">t2</font> = <font color="Maroon" title="c9">t1</font> as    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">T2</font> <b>by </b><i><a class="txt" href="amistd_3.html#E7:51_1_2"><i><font color="Green" title="E31">A86</font></i></a>, <a class="ref" href="amistd_3.html#T18" target="_self" title="AMISTD_3:th.18">Th18</a></i>;<br/>
<a NAME="E17:51_1_2_3_2"/><i><font color="Green" title="E46">A101</font></i>: 
 <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c9">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c9">t1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Olive" title="b1">k</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> where <font color="Olive" title="b1">k</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> <span class="p2">(<span class="default"><a href="ami_1.html#K31" title="AMI_1:func.31">pi</a> <font color="Maroon" title="c3">M</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">T1</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Maroon" title="c9">t1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">T1</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Maroon" title="c9">t1</font></span>)</span></span>)</span> </span>}</span> 
 
<b>by </b><i><a class="txt" href="amistd_3.html#E2:51_1_2"><i><font color="Green" title="E26">A17</font></i></a></i>;<br/>
<a NAME="E18:51_1_2_3_2"/>
<font color="Maroon" title="c9">t1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p5">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p0">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c9">t1</font>
 
<b>by </b><i><a class="txt" href="amistd_3.html#E3:51_1_2_3_2"><i><font color="Green" title="E38">A93</font></i></a>, <a class="txt" href="amistd_3.html#E8:51_1_2_3_2"><i><font color="Green" title="E41">A96</font></i></a>, <a class="txt" href="amistd_3.html#E14:51_1_2_3_2"><i><font color="Green" title="E45">A100</font></i></a>, <a class="ref" href="trees_2.html#T14" title="TREES_2:th.14">TREES_2:14</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c11">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E20:51_1_2_3_2"/><i><font color="Green" title="E47">A102</font></i>: 
<font color="Maroon" title="c9">t1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p5">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p0">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">t1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c11">k</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <b>and </b><br/><a NAME="E21:51_1_2_3_2"/><i><font color="Green" title="E48">A103</font></i>: 
<font color="Maroon" title="c11">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> <span class="p2">(<span class="default"><a href="ami_1.html#K31" title="AMI_1:func.31">pi</a> <font color="Maroon" title="c3">M</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">T1</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Maroon" title="c9">t1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">T1</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Maroon" title="c9">t1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="amistd_3.html#E17:51_1_2_3_2"><i><font color="Green" title="E46">A101</font></i></a></i>;<br/>
<a NAME="E22:51_1_2_3_2"/><i><font color="Green" title="E49">A104</font></i>: 
<font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="amistd_3.html#E20:51_1_2_3_2"><i><font color="Green" title="E47">A102</font></i></a>, <a class="ref" href="finseq_2.html#T20" title="FINSEQ_2:th.20">FINSEQ_2:20</a></i>;<br/>
<a NAME="E23:51_1_2_3_2"/><i><font color="Green" title="E50">A105</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> <span class="p2">(<span class="default"><a href="ami_1.html#K31" title="AMI_1:func.31">pi</a> <font color="Maroon" title="c3">M</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">T2</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Maroon" title="c10">t2</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">T2</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Maroon" title="c10">t2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="amistd_3.html#E1:51_1_2_3_2"><i><font color="Green" title="E36">A91</font></i></a>, <a class="txt" href="amistd_3.html#E2:51_1_2_3_2"><i><font color="Green" title="E37">A92</font></i></a>, <a class="txt" href="amistd_3.html#E4:51_1_2_3_2"><i><font color="Green" title="E39">A94</font></i></a>, <a class="txt" href="amistd_3.html#E20:51_1_2_3_2"><i><font color="Green" title="E47">A102</font></i></a>, <a class="txt" href="amistd_3.html#E21:51_1_2_3_2"><i><font color="Green" title="E48">A103</font></i></a>, <a class="ref" href="finseq_2.html#T20" title="FINSEQ_2:th.20">FINSEQ_2:20</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c4">T1</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="amistd_3.html#K4" title="AMISTD_3:func.4">LocSeq</a> <span class="p2">(<span class="default"><a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> <span class="p3">(<span class="default"><a href="ami_1.html#K31" title="AMI_1:func.31">pi</a> <font color="Maroon" title="c3">M</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">T1</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Maroon" title="c9">t1</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">T1</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Maroon" title="c9">t1</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p5">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="amistd_3.html#E2:51_1_2"><i><font color="Green" title="E26">A17</font></i></a>, <a class="txt" href="amistd_3.html#E14:51_1_2_3_2"><i><font color="Green" title="E45">A100</font></i></a>, <a class="txt" href="amistd_3.html#E21:51_1_2_3_2"><i><font color="Green" title="E48">A103</font></i></a>, <a class="txt" href="amistd_3.html#E22:51_1_2_3_2"><i><font color="Green" title="E49">A104</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c5">T2</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="amistd_3.html#E4:51_1_2"><i><font color="Green" title="E28">A19</font></i></a>, <a class="txt" href="amistd_3.html#E1:51_1_2_3_2"><i><font color="Green" title="E36">A91</font></i></a>, <a class="txt" href="amistd_3.html#E2:51_1_2_3_2"><i><font color="Green" title="E37">A92</font></i></a>, <a class="txt" href="amistd_3.html#E4:51_1_2_3_2"><i><font color="Green" title="E39">A94</font></i></a>, <a class="txt" href="amistd_3.html#E8:51_1_2_3_2"><i><font color="Green" title="E41">A96</font></i></a>, <a class="txt" href="amistd_3.html#E14:51_1_2_3_2"><i><font color="Green" title="E45">A100</font></i></a>, <a class="txt" href="amistd_3.html#E23:51_1_2_3_2"><i><font color="Green" title="E50">A105</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E4:51_1_2_3"/><i><font color="Green" title="E36">A106</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="amistd_3.html#E2:51_1_2_3"><i><font color="Green" title="E34">A89</font></i></a>, <a class="txt" href="amistd_3.html#E3:51_1_2_3"><i><font color="Green" title="E35">A90</font></i></a>);<br/></i>
<a NAME="E5:51_1_2_3"/><i><font color="Green" title="E37">A107</font></i>: 
<font color="Maroon" title="c6">p</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">p</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T55" title="FINSEQ_3:th.55">FINSEQ_3:55</a></i>;<br/>
<a NAME="E6:51_1_2_3"/>
( <font color="Maroon" title="c4">T1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">T1</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Maroon" title="c6">p</font> &amp; <font color="Maroon" title="c5">T2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">T2</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Maroon" title="c6">p</font> )
 
<b>by </b><i><a class="txt" href="amistd_3.html#E8:51_1_2"><i><font color="Green" title="E32">A87</font></i></a>, <a class="txt" href="amistd_3.html#E1:51_1_2_3"><i><font color="Green" title="E33">A88</font></i></a>, <a class="ref" href="ami_1.html#D38" title="AMI_1:def.38">AMI_1:def 38</a></i>;<br/>
<b>hence </b><a NAME="E7:51_1_2_3"/>
<font color="Maroon" title="c4">T1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">T2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">p</font>
 <b>by </b><i><a class="txt" href="amistd_3.html#E1:51_1_2_3"><i><font color="Green" title="E33">A88</font></i></a>, <a class="txt" href="amistd_3.html#E4:51_1_2_3"><i><font color="Green" title="E36">A106</font></i></a>, <a class="txt" href="amistd_3.html#E5:51_1_2_3"><i><font color="Green" title="E37">A107</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E10:51_1_2"/>
<font color="Maroon" title="c4">T1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">T2</font>
 <b>by </b><i><a class="txt" href="amistd_3.html#E8:51_1_2"><i><font color="Green" title="E32">A87</font></i></a>, <a class="ref" href="trees_2.html#T33" title="TREES_2:th.33">TREES_2:33</a></i>;<br/>


</div>
