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

<b>let </b><font color="Maroon">N</font> be  <a href="setfam_1.html#V1">with_non-empty_elements</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">S</font> be  non <a href="struct_0.html#V3">empty</a> non <a href="ami_1.html#V2">void</a> <a href="ami_1.html#V4">halting</a> <a href="ami_1.html#V5">IC-Ins-separated</a> <a href="ami_1.html#V8">definite</a> <a href="ami_1.html#V10">realistic</a> <a href="amistd_1.html#V4">standard</a>  <a href="ami_1.html#L1">AMI-Struct</a> of <font color="Maroon">N</font>;<br/>



<b>set </b><font color="Maroon">D</font> =  <a href="amistd_3.html#K1" target="_self">TrivialInfiniteTree</a> ;<br/>
<b>set </b><font color="Maroon">M</font> =  <a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font>;<br/>
<b>set </b><font color="Maroon">E</font> =  <a href="amistd_3.html#K5" target="_self">ExecTree</a> <span class="p1">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span>;<br/>
<a NAME="E1:54"/><i><font color="Green">E35</font></i>: 
 <a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="amistd_1.html#K5">il.</a> <font color="Maroon">S</font>,0</span>)</span> <a href="ami_3.html#K14">.--&gt;</a> <span class="p1">(<span class="default"><a href="ami_1.html#K5">halt</a> <font color="Maroon">S</font></span>)</span>
 
<b>by </b><i><a class="ref" href="amistd_2.html#D13">AMISTD_2:def 13</a></i>;<br/>
<a NAME="E2:54"/><b>then </b><i><font color="Green">E36</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="amistd_1.html#K5">il.</a> <font color="Maroon">S</font>,0</span>)</span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>;<br/>
<a NAME="E3:54"/><i><font color="Green">E38</font></i>: 
<span class="p1">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="amistd_1.html#K5">il.</a> <font color="Maroon">S</font>,0</span>)</span> <a href="hidden.html#R1">=</a>  <a href="ami_1.html#K5">halt</a> <font color="Maroon">S</font>
 
<b>by </b><i>, <a class="ref" href="cqc_lang.html#T6">CQC_LANG:6</a></i>;<br/>
<a NAME="E4:54"/><i><font color="Green">E40</font></i>: 
<span class="p1">(<span class="default"><a href="amistd_3.html#K5" target="_self">ExecTree</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a>  <a href="amistd_3.html#K2" target="_self">FirstLoc</a> <span class="p1">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E5:54"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">t</font> being   <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="amistd_3.html#K5" target="_self">ExecTree</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span></span>)</span> holds   <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="amistd_1.html#K1">NIC</a> <span class="p2">(<span class="default"><a href="ami_1.html#K5">halt</a> <font color="Maroon">S</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="amistd_3.html#K5" target="_self">ExecTree</a> <span class="p4">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Olive">t</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">t</font> be    <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="amistd_3.html#K5" target="_self">ExecTree</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span></span>)</span>;<br/>

<a NAME="E1:54_1"/>
 <a href="amistd_1.html#K1">NIC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K5">halt</a> <font color="Maroon">S</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="amistd_3.html#K5" target="_self">ExecTree</a> <span class="p3">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">t</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="amistd_3.html#K5" target="_self">ExecTree</a> <span class="p4">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">t</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="amistd_1.html#T15">AMISTD_1:15</a></i>;<br/>
<b>hence </b><a NAME="E2:54_1"/>
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="amistd_1.html#K1">NIC</a> <span class="p2">(<span class="default"><a href="ami_1.html#K5">halt</a> <font color="Maroon">S</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="amistd_3.html#K5" target="_self">ExecTree</a> <span class="p4">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">t</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="card_1.html#T50">CARD_1:50</a>, <a class="ref" href="card_5.html#T1">CARD_5:1</a></i>;<br/>


</div><b>end;</b></div>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b><span class="p1">(<span class="default"><a href="amistd_3.html#K5" target="_self">ExecTree</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span>;<br/>
<a NAME="E6:54"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="finseq_1.html#K6">&lt;*&gt;</a> <a href="numbers.html#K5">NAT</a> ]
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:54"/><i><font color="Green">E69</font></i>: 
for <font color="Olive">f</font> being   <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="amistd_3.html#K5" target="_self">ExecTree</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span></span>)</span>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">f</font>] holds <br/>for <font color="Olive">a</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">f</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Olive">a</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="amistd_3.html#K5" target="_self">ExecTree</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span></span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">f</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Olive">a</font></span><a href="finseq_1.html#K12">*&gt;</a></span>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">f</font> be    <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="amistd_3.html#K5" target="_self">ExecTree</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span></span>)</span>;<br/>

<b>assume </b><a NAME="E1:54_2"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">f</font>]
 ;<br/>

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

<b>assume </b><a NAME="E2:54_2"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">f</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">a</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="amistd_3.html#K5" target="_self">ExecTree</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E3:54_2"/><i><font color="Green">E73</font></i>: 
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="amistd_1.html#K1">NIC</a> <span class="p2">(<span class="default"><a href="ami_1.html#K5">halt</a> <font color="Maroon">S</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="amistd_3.html#K5" target="_self">ExecTree</a> <span class="p4">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E4:54_2"/><i><font color="Green">E74</font></i>: 
 <a href="trees_2.html#K1">succ</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Olive">k</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="amistd_1.html#K1">NIC</a> <span class="p2">(<span class="default"><a href="ami_5.html#K5">pi</a> <span class="p3">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="amistd_3.html#K5" target="_self">ExecTree</a> <span class="p5">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="amistd_3.html#K5" target="_self">ExecTree</a> <span class="p4">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">f</font></span>)</span></span>)</span> </span>}</span> 
 
<b>by </b><i/>;<br/>
<a NAME="E5:54_2"/><i><font color="Green">E75</font></i>: 
<span class="p1">(<span class="default"><a href="amistd_3.html#K5" target="_self">ExecTree</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="amistd_1.html#K5">il.</a> <font color="Maroon">S</font>,0
 
<b>by </b><i>, <a class="txt" href="amistd_3.html#E1:54"><i><font color="Green">E35</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E6:54_2"/><b>then </b><i><font color="Green">E76</font></i>: 
 <a href="amistd_1.html#K7">locnum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="amistd_3.html#K5" target="_self">ExecTree</a> <span class="p3">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="amistd_1.html#D13">AMISTD_1:def 13</a></i>;<br/>
<a NAME="E7:54_2"/><i><font color="Green">E77</font></i>: 
 <a href="ami_5.html#K5">pi</a> <span class="p1">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="amistd_3.html#K5" target="_self">ExecTree</a> <span class="p3">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="amistd_3.html#K5" target="_self">ExecTree</a> <span class="p3">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i><a class="txt" href="amistd_3.html#E1:54"><i><font color="Green">E35</font></i></a>, <a class="ref" href="ami_5.html#D5">AMI_5:def 5</a></i>;<br/>
<a NAME="E8:54_2"/><b>then </b><i><font color="Green">E78</font></i>: 
0 <a href="hidden.html#R2">in</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="amistd_1.html#K1">NIC</a> <span class="p2">(<span class="default"><a href="ami_5.html#K5">pi</a> <span class="p3">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="amistd_3.html#K5" target="_self">ExecTree</a> <span class="p5">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="amistd_3.html#K5" target="_self">ExecTree</a> <span class="p4">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 
<b>by </b><i>, , , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E9:54_2"/><i><font color="Green">E79</font></i>: 
 <a href="trees_2.html#K1">succ</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_2_1"><b>proof </b></a><div class="add">

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a>,<a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a>
<div class="add"><b>let </b><font color="Maroon">s</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:54_2_1_1"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="trees_2.html#K1">succ</a> <font color="Maroon">f</font>
 ;<br/><b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:54_2_1_1"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">k</font></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>and </b><br/><a NAME="E4:54_2_1_1"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="amistd_1.html#K1">NIC</a> <span class="p2">(<span class="default"><a href="ami_5.html#K5">pi</a> <span class="p3">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="amistd_3.html#K5" target="_self">ExecTree</a> <span class="p5">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="amistd_3.html#K5" target="_self">ExecTree</a> <span class="p4">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="amistd_3.html#E3:54"><i><font color="Green">E38</font></i></a></i>;<br/><a NAME="E5:54_2_1_1"/>
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, , , , , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E6:54_2_1_1"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="amistd_3.html#T3" target="_self">Th3</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div>
<b>end;</b></div>

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

<b>assume </b><a NAME="E2:54_2_1"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<a NAME="E3:54_2_1"/><b>then </b>
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:54_2_1"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="trees_2.html#K1">succ</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="txt" href="amistd_3.html#E3:54"><i><font color="Green">E38</font></i></a>, <a class="txt" href="amistd_3.html#E5:54"><i><font color="Green">E42</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:54_2"/>
<font color="Maroon">f</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">a</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R2">in</a>  <a href="trees_2.html#K1">succ</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="txt" href="amistd_3.html#E2:54"><i><font color="Green">E36</font></i></a>, <a class="ref" href="trees_2.html#T14">TREES_2:14</a></i>;<br/>
<a NAME="E11:54_2"/><b>then </b><i><font color="Green">E108</font></i>: 
<font color="Maroon">f</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">a</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="amistd_3.html#T2" target="_self">Th2</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E12:54_2"/>
 <a href="amistd_3.html#K3" target="_self">LocNums</a> <span class="p1">(<span class="default"><a href="amistd_1.html#K1">NIC</a> <span class="p2">(<span class="default"><a href="ami_1.html#K5">halt</a> <font color="Maroon">S</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="amistd_3.html#K5" target="_self">ExecTree</a> <span class="p4">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="txt" href="amistd_3.html#E4:54"><i><font color="Green">E40</font></i></a>, </i>;<br/>
<a NAME="E13:54_2"/><b>then </b>
 <a href="wellord1.html#K3">canonical_isomorphism_of</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <span class="p2">(<span class="default"><a href="wellord2.html#K2">order_type_of</a> <span class="p3">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <span class="p4">(<span class="default"><a href="amistd_3.html#K3" target="_self">LocNums</a> <span class="p5">(<span class="default"><a href="amistd_1.html#K1">NIC</a> <span class="p0">(<span class="default"><a href="ami_5.html#K5">pi</a> <span class="p1">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="amistd_3.html#K5" target="_self">ExecTree</a> <span class="p3">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="amistd_3.html#K5" target="_self">ExecTree</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <span class="p2">(<span class="default"><a href="amistd_3.html#K3" target="_self">LocNums</a> <span class="p3">(<span class="default"><a href="amistd_1.html#K1">NIC</a> <span class="p4">(<span class="default"><a href="ami_5.html#K5">pi</a> <span class="p5">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="amistd_3.html#K5" target="_self">ExecTree</a> <span class="p1">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="amistd_3.html#K5" target="_self">ExecTree</a> <span class="p0">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0 <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p1">(<span class="default"><a href="amistd_1.html#K7">locnum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="amistd_3.html#K5" target="_self">ExecTree</a> <span class="p4">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 
<b>by </b><i>, , <a class="txt" href="amistd_3.html#E4:54"><i><font color="Green">E40</font></i></a>, , </i>;<br/>
<b>then </b><i><font color="Green">E111</font></i>: <span class="p1">(<span class="default"><a href="wellord1.html#K3">canonical_isomorphism_of</a> <span class="p2">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <span class="p3">(<span class="default"><a href="wellord2.html#K2">order_type_of</a> <span class="p4">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <span class="p5">(<span class="default"><a href="amistd_3.html#K3" target="_self">LocNums</a> <span class="p0">(<span class="default"><a href="amistd_1.html#K1">NIC</a> <span class="p1">(<span class="default"><a href="ami_5.html#K5">pi</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="amistd_3.html#K5" target="_self">ExecTree</a> <span class="p4">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="amistd_3.html#K5" target="_self">ExecTree</a> <span class="p3">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <span class="p3">(<span class="default"><a href="amistd_3.html#K3" target="_self">LocNums</a> <span class="p4">(<span class="default"><a href="amistd_1.html#K1">NIC</a> <span class="p5">(<span class="default"><a href="ami_5.html#K5">pi</a> <span class="p0">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="amistd_3.html#K5" target="_self">ExecTree</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="amistd_3.html#K5" target="_self">ExecTree</a> <span class="p1">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> 0 = 
 <a href="amistd_1.html#K7">locnum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="amistd_3.html#K5" target="_self">ExecTree</a> <span class="p3">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">f</font></span>)</span>
<b>by </b><i><a class="ref" href="cqc_lang.html#T6">CQC_LANG:6</a></i>
<br/>.= 
0
<b>by </b><i>, <a class="ref" href="amistd_1.html#D13">AMISTD_1:def 13</a></i>
;<br/>
<span class="p1">(<span class="default"><a href="amistd_3.html#K5" target="_self">ExecTree</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">a</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> = 
<span class="p1">(<span class="default"><a href="amistd_3.html#K4" target="_self">LocSeq</a> <span class="p2">(<span class="default"><a href="amistd_1.html#K1">NIC</a> <span class="p3">(<span class="default"><a href="ami_5.html#K5">pi</a> <span class="p4">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="amistd_3.html#K5" target="_self">ExecTree</a> <span class="p0">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="amistd_3.html#K5" target="_self">ExecTree</a> <span class="p5">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> 0
<b>by </b><i><a class="txt" href="amistd_3.html#E5:54"><i><font color="Green">E42</font></i></a>, <a class="ref" href="amistd_3.html#T3" target="_self">Th3</a>, </i>
<br/>.= 
 <a href="amistd_1.html#K5">il.</a> <font color="Maroon">S</font>,0
<b>by </b><i><a class="txt" href="amistd_3.html#E5:54"><i><font color="Green">E42</font></i></a>, , </i>
;<br/>
<b>hence </b><a NAME="E16:54_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">f</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">a</font></span><a href="finseq_1.html#K12">*&gt;</a></span>]
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:54"/><i><font color="Green">E112</font></i>: 
for <font color="Olive">f</font> being   <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="amistd_3.html#K5" target="_self">ExecTree</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span></span>)</span> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">f</font>]
 
<b>from </b><i><a class="ref" href="hilbert2.html#S1">HILBERT2:sch 1</a>(, <a class="ref" href="amistd_3.html#T1" target="_self">Th1</a>);<br/></i>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b><span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><a href="amistd_3.html#K5" target="_self">ExecTree</a> <span class="p3">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span></span>)</span></span>)</span> <a href="trees_2.html#K2">-level</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <a href="amistd_3.html#K1" target="_self">TrivialInfiniteTree</a>  <a href="trees_2.html#K2">-level</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E9:54"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_3"><b>proof </b></a><div class="add">

<b>thus </b><span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><a href="amistd_3.html#K5" target="_self">ExecTree</a> <span class="p3">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span></span>)</span></span>)</span> <a href="trees_2.html#K2">-level</a> 0 = 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="qc_lang4.html#T17">QC_LANG4:17</a></i>
<br/>.= 
<a href="amistd_3.html#K1" target="_self">TrivialInfiniteTree</a>  <a href="trees_2.html#K2">-level</a> 0
<b>by </b><i><a class="ref" href="qc_lang4.html#T17">QC_LANG4:17</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E10:54"/><i><font color="Green">E115</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_4"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:54_4"/><i><font color="Green">E116</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">n</font>]
 ;<br/>

<b>set </b><font color="Maroon">f0</font> = <font color="Maroon">n</font> <a href="finseq_2.html#K2">|-&gt;</a> 0;<br/>
<b>set </b><font color="Maroon">f1</font> = <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="finseq_2.html#K2">|-&gt;</a> 0;<br/>
<a NAME="E2:54_4"/><i><font color="Green">E117</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><a href="amistd_3.html#K5" target="_self">ExecTree</a> <span class="p3">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span></span>)</span></span>)</span> <a href="trees_2.html#K2">-level</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">w</font> where <font color="Olive">w</font> is    <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="amistd_3.html#K5" target="_self">ExecTree</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span></span>)</span> :  <a href="finseq_1.html#K3">len</a> <font color="Olive">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 </span>}</span> 
 
<b>by </b><i><a class="ref" href="trees_2.html#D6">TREES_2:def 6</a></i>;<br/>
<a NAME="E3:54_4"/><i><font color="Green">E118</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="finseq_2.html#K2">|-&gt;</a> 0</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="finseq_2.html#T69">FINSEQ_2:69</a></i>;<br/>
<a NAME="E4:54_4"/>
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><a href="amistd_3.html#K5" target="_self">ExecTree</a> <span class="p3">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span></span>)</span></span>)</span> <a href="trees_2.html#K2">-level</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="finseq_2.html#K2">|-&gt;</a> 0</span>)</span></span><a href="tarski.html#K1">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_4_1"><b>proof </b></a><div class="add">

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a>,<a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a>
<div class="add"><b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:54_4_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><a href="amistd_3.html#K5" target="_self">ExecTree</a> <span class="p3">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span></span>)</span></span>)</span> <a href="trees_2.html#K2">-level</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">t1</font> being    <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="amistd_3.html#K5" target="_self">ExecTree</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E3:54_4_1_1"/><i><font color="Green">E119</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t1</font>
 <b>and </b><br/><a NAME="E4:54_4_1_1"/><i><font color="Green">E120</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">t1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">t0</font> = <font color="Maroon">t1</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span> as    <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="amistd_3.html#K5" target="_self">ExecTree</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span></span>)</span> <b>by </b><i/>;<br/><a NAME="E6:54_4_1_1"/><i><font color="Green">E121</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><a href="amistd_3.html#K5" target="_self">ExecTree</a> <span class="p3">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span></span>)</span></span>)</span> <a href="trees_2.html#K2">-level</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">w</font> where <font color="Olive">w</font> is    <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="amistd_3.html#K5" target="_self">ExecTree</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span></span>)</span> :  <a href="finseq_1.html#K3">len</a> <font color="Olive">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="trees_2.html#D6">TREES_2:def 6</a></i>;<br/><a NAME="E7:54_4_1_1"/><i><font color="Green">E122</font></i>: 
 <a href="trees_2.html#K1">succ</a> <font color="Maroon">t0</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">t0</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Olive">k</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="amistd_1.html#K1">NIC</a> <span class="p2">(<span class="default"><a href="ami_5.html#K5">pi</a> <span class="p3">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="amistd_3.html#K5" target="_self">ExecTree</a> <span class="p5">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">t0</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="amistd_3.html#K5" target="_self">ExecTree</a> <span class="p4">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">t0</font></span>)</span></span>)</span> </span>}</span> 
 <b>by </b><i/>;<br/><a NAME="E8:54_4_1_1"/><i><font color="Green">E123</font></i>: 
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="amistd_1.html#K1">NIC</a> <span class="p2">(<span class="default"><a href="ami_1.html#K5">halt</a> <font color="Maroon">S</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="amistd_3.html#K5" target="_self">ExecTree</a> <span class="p4">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">t0</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i/>;<br/><a NAME="E9:54_4_1_1"/><i><font color="Green">E124</font></i>: 
<span class="p1">(<span class="default"><a href="amistd_3.html#K5" target="_self">ExecTree</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">t0</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span>
 <b>by </b><i><a class="txt" href="amistd_3.html#E1:54"><i><font color="Green">E35</font></i></a></i>;<br/><a NAME="E10:54_4_1_1"/><b>then </b><i><font color="Green">E125</font></i>: 
<span class="p1">(<span class="default"><a href="amistd_3.html#K5" target="_self">ExecTree</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">t0</font> <a href="hidden.html#R1">=</a>  <a href="amistd_1.html#K5">il.</a> <font color="Maroon">S</font>,0
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E11:54_4_1_1"/><i><font color="Green">E126</font></i>: 
 <a href="ami_5.html#K5">pi</a> <span class="p1">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="amistd_3.html#K5" target="_self">ExecTree</a> <span class="p3">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">t0</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="amistd_3.html#K5" target="_self">ExecTree</a> <span class="p3">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">t0</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="ami_5.html#D5">AMI_5:def 5</a></i>;<br/><a NAME="E12:54_4_1_1"/><b>then </b><i><font color="Green">E127</font></i>: 
0 <a href="hidden.html#R2">in</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="amistd_1.html#K1">NIC</a> <span class="p2">(<span class="default"><a href="ami_5.html#K5">pi</a> <span class="p3">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="amistd_3.html#K5" target="_self">ExecTree</a> <span class="p5">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">t0</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="amistd_3.html#K5" target="_self">ExecTree</a> <span class="p4">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">t0</font></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="amistd_3.html#T5" target="_self">Th5</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E13:54_4_1_1"/><i><font color="Green">E128</font></i>: 
<font color="Maroon">t1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><a NAME="E14:54_4_1_1"/><i><font color="Green">E130</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><a href="amistd_3.html#K5" target="_self">ExecTree</a> <span class="p3">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span></span>)</span></span>)</span> <a href="trees_2.html#K2">-level</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="finseq_2.html#K2">|-&gt;</a> 0</span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="amistd_3.html#E3:54"><i><font color="Green">E38</font></i></a>, </i>;<br/><a NAME="E15:54_4_1_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E16:54_4_1_1"/><b>then </b>
 <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T7">FINSEQ_1:7</a></i>;<br/><a NAME="E17:54_4_1_1"/><b>then </b>
 <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">t1</font>
 <b>by </b><i><a class="txt" href="amistd_3.html#E5:54"><i><font color="Green">E42</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E18:54_4_1_1"/><b>then </b>
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">t0</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/><a NAME="E19:54_4_1_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">t0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E20:54_4_1_1"/><b>then </b><i><font color="Green">E131</font></i>: 
<font color="Maroon">t0</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><a href="amistd_3.html#K5" target="_self">ExecTree</a> <span class="p3">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span></span>)</span></span>)</span> <a href="trees_2.html#K2">-level</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="amistd_3.html#T2" target="_self">Th2</a></i>;<br/><a NAME="E21:54_4_1_1"/>
<font color="Maroon">t1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t0</font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">t1</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="amistd_3.html#E5:54"><i><font color="Green">E42</font></i></a>, <a class="ref" href="finseq_3.html#T61">FINSEQ_3:61</a></i>;<br/><a NAME="E22:54_4_1_1"/><b>then </b><i><font color="Green">E132</font></i>: 
<font color="Maroon">t0</font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">t1</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="hidden.html#R2">in</a>  <a href="trees_2.html#K1">succ</a> <font color="Maroon">t0</font>
 <b>by </b><i>, <a class="ref" href="trees_2.html#T14">TREES_2:14</a></i>;<br/><a NAME="E23:54_4_1_1"/>
 <a href="trees_2.html#K1">succ</a> <font color="Maroon">t0</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">t0</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_4_1_1_1"><b>proof </b></a><div class="add">

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a>,<a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a>
<div class="add"><b>let </b><font color="Maroon">s</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:54_4_1_1_1_1"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="trees_2.html#K1">succ</a> <font color="Maroon">t0</font>
 ;<br/><b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:54_4_1_1_1_1"/><i><font color="Green">E133</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t0</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">k</font></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>and </b><br/><a NAME="E4:54_4_1_1_1_1"/><i><font color="Green">E134</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="amistd_1.html#K1">NIC</a> <span class="p2">(<span class="default"><a href="ami_5.html#K5">pi</a> <span class="p3">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="amistd_3.html#K5" target="_self">ExecTree</a> <span class="p5">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">t0</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="amistd_3.html#K5" target="_self">ExecTree</a> <span class="p4">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">t0</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="amistd_3.html#T3" target="_self">Th3</a></i>;<br/><a NAME="E5:54_4_1_1_1_1"/>
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, , <a class="ref" href="amistd_3.html#T5" target="_self">Th5</a>, <a class="ref" href="amistd_3.html#T6" target="_self">Th6</a>, <a class="ref" href="amistd_3.html#T13" target="_self">Th13</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E6:54_4_1_1_1_1"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">t0</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="amistd_3.html#E13"><i><font color="Green">Lemma54</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div>
<b>end;</b></div>

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

<b>assume </b><a NAME="E2:54_4_1_1_1"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">t0</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<a NAME="E3:54_4_1_1_1"/><b>then </b>
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t0</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:54_4_1_1_1"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="trees_2.html#K1">succ</a> <font color="Maroon">t0</font>
 <b>by </b><i><a class="ref" href="amistd_3.html#T3" target="_self">Th3</a>, <a class="ref" href="amistd_3.html#T7" target="_self">Th7</a></i>;<br/>


</div><b>end;</b></div><a NAME="E24:54_4_1_1"/><b>then </b><i><font color="Green">E135</font></i>: 
<font color="Maroon">t0</font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">t1</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">t0</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="amistd_3.html#T10" target="_self">Th10</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E25:54_4_1_1"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">t1</font> holds <br/><font color="Maroon">t1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="finseq_2.html#K2">|-&gt;</a> 0</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_4_1_1_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:54_4_1_1_2"/><i><font color="Green">E136</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">t1</font> )
 ;<br/>

<a NAME="E2:54_4_1_1_2"/><i><font color="Green">E137</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="amistd_3.html#E5:54"><i><font color="Green">E42</font></i></a>, <a class="ref" href="amistd_3.html#T13" target="_self">Th13</a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_4_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_4_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:54_4_1_1_2_1_1"/>
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> or <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 )
 </span><b>by </b><i><a class="ref" href="amistd_3.html#T14" target="_self">Th14</a>, <a class="ref" href="finseq_2.html#T8">FINSEQ_2:8</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_4_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:54_4_1_1_2_1_1_1"/><i><font color="Green">E138</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><b>hence </b><font color="Maroon">t1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> = 
<font color="Maroon">t0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
<b>by </b><i><a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="finseq_2.html#K2">|-&gt;</a> 0</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
<b>by </b><i>, <a class="ref" href="amistd_3.html#T9" target="_self">Th9</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>
<br/>.= 
0
<b>by </b><i><a class="ref" href="amistd_3.html#T15" target="_self">Th15</a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_4_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:54_4_1_1_2_1_1_2"/>
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:54_4_1_1_2_1_1_2"/>
<font color="Maroon">t1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="amistd_3.html#E13"><i><font color="Green">Lemma54</font></i></a>, <a class="ref" href="finseq_2.html#T20">FINSEQ_2:20</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:54_4_1_1_2"/>
<font color="Maroon">t1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="finseq_2.html#K2">|-&gt;</a> 0</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="amistd_3.html#T14" target="_self">Th14</a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>


</div><b>end;</b></div><a NAME="E26:54_4_1_1"/><b>then </b>
<font color="Maroon">t1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="finseq_2.html#K2">|-&gt;</a> 0
 <b>by </b><i><a class="txt" href="amistd_3.html#E4:54"><i><font color="Green">E40</font></i></a>, <a class="txt" href="amistd_3.html#E5:54"><i><font color="Green">E42</font></i></a>, <a class="ref" href="finseq_1.html#T18">FINSEQ_1:18</a></i>;<br/><b>hence </b><a NAME="E27:54_4_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="finseq_2.html#K2">|-&gt;</a> 0</span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div>
<b>end;</b></div>

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

<b>assume </b><a NAME="E2:54_4_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="finseq_2.html#K2">|-&gt;</a> 0</span>)</span></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<a NAME="E3:54_4_1"/><b>then </b><i><font color="Green">E140</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="finseq_2.html#K2">|-&gt;</a> 0
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b><font color="Maroon">a<sub>1</sub></font> <a href="finseq_2.html#K2">|-&gt;</a> 0 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="amistd_3.html#K5" target="_self">ExecTree</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span></span>)</span>;<br/>
<a NAME="E4:54_4_1"/>
0 <a href="finseq_2.html#K2">|-&gt;</a> 0 <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="finseq_2.html#T72">FINSEQ_2:72</a></i>;<br/>
<a NAME="E5:54_4_1"/><b>then </b><i><font color="Green">E141</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[0]
 
<b>by </b><i><a class="ref" href="trees_1.html#T47">TREES_1:47</a></i>;<br/>
<a NAME="E6:54_4_1"/><i><font color="Green">E142</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_4_1_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:54_4_1_2"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">n</font>]
 ;<br/>

<b>then reconsider </b><font color="Maroon">t</font> = <font color="Maroon">n</font> <a href="finseq_2.html#K2">|-&gt;</a> 0 as    <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="amistd_3.html#K5" target="_self">ExecTree</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span></span>)</span> ;<br/>
<a NAME="E3:54_4_1_2"/><i><font color="Green">E143</font></i>: 
 <a href="trees_2.html#K1">succ</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Olive">k</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="amistd_1.html#K1">NIC</a> <span class="p2">(<span class="default"><a href="ami_5.html#K5">pi</a> <span class="p3">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="amistd_3.html#K5" target="_self">ExecTree</a> <span class="p5">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">t</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="amistd_3.html#K5" target="_self">ExecTree</a> <span class="p4">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">t</font></span>)</span></span>)</span> </span>}</span> 
 
<b>by </b><i/>;<br/>
<a NAME="E4:54_4_1_2"/><i><font color="Green">E144</font></i>: 
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="amistd_1.html#K1">NIC</a> <span class="p2">(<span class="default"><a href="ami_1.html#K5">halt</a> <font color="Maroon">S</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="amistd_3.html#K5" target="_self">ExecTree</a> <span class="p4">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">t</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E5:54_4_1_2"/><i><font color="Green">E145</font></i>: 
<span class="p1">(<span class="default"><a href="amistd_3.html#K5" target="_self">ExecTree</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span>
 
<b>by </b><i><a class="txt" href="amistd_3.html#E1:54"><i><font color="Green">E35</font></i></a></i>;<br/>
<a NAME="E6:54_4_1_2"/><b>then </b><i><font color="Green">E146</font></i>: 
<span class="p1">(<span class="default"><a href="amistd_3.html#K5" target="_self">ExecTree</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a>  <a href="amistd_1.html#K5">il.</a> <font color="Maroon">S</font>,0
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E7:54_4_1_2"/>
 <a href="ami_5.html#K5">pi</a> <span class="p1">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="amistd_3.html#K5" target="_self">ExecTree</a> <span class="p3">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">t</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="amistd_3.html#K5" target="_self">ExecTree</a> <span class="p3">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">t</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="ami_5.html#D5">AMI_5:def 5</a></i>;<br/>
<a NAME="E8:54_4_1_2"/><b>then </b>
0 <a href="hidden.html#R2">in</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="amistd_1.html#K1">NIC</a> <span class="p2">(<span class="default"><a href="ami_5.html#K5">pi</a> <span class="p3">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="amistd_3.html#K5" target="_self">ExecTree</a> <span class="p5">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">t</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="amistd_3.html#K5" target="_self">ExecTree</a> <span class="p4">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">t</font></span>)</span></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="amistd_3.html#T5" target="_self">Th5</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E9:54_4_1_2"/><b>then </b>
<font color="Maroon">t</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R2">in</a>  <a href="trees_2.html#K1">succ</a> <font color="Maroon">t</font>
 
<b>by </b><i><a class="ref" href="amistd_3.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E10:54_4_1_2"/><b>then </b>
<font color="Maroon">t</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="amistd_3.html#K5" target="_self">ExecTree</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span></span>)</span>
 
;<br/>
<b>hence </b><a NAME="E11:54_4_1_2"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i><a class="ref" href="finseq_2.html#T74">FINSEQ_2:74</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:54_4_1"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="amistd_3.html#E5:54"><i><font color="Green">E42</font></i></a>, <a class="ref" href="amistd_3.html#T2" target="_self">Th2</a>);<br/></i>
<a NAME="E8:54_4_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="finseq_2.html#K2">|-&gt;</a> 0 is    <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="amistd_3.html#K5" target="_self">ExecTree</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span></span>)</span>
 
;<br/>
<b>hence </b><a NAME="E9:54_4_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><a href="amistd_3.html#K5" target="_self">ExecTree</a> <span class="p3">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span></span>)</span></span>)</span> <a href="trees_2.html#K2">-level</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, <a class="txt" href="amistd_3.html#E4:54"><i><font color="Green">E40</font></i></a>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:54_4"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E11:54"/>
for <font color="Olive">x</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="amistd_3.html#E2:54"><i><font color="Green">E36</font></i></a>, );<br/></i>
<a NAME="E12:54"/><b>then </b><i><font color="Green">E147</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="amistd_3.html#K5" target="_self">ExecTree</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="amistd_3.html#K1" target="_self">TrivialInfiniteTree</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E13:54"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="amistd_3.html#K5" target="_self">ExecTree</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="amistd_3.html#K5" target="_self">ExecTree</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="amistd_1.html#K5">il.</a> <font color="Maroon">S</font>,0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:54_5"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="amistd_3.html#K5" target="_self">ExecTree</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as    <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="amistd_3.html#K5" target="_self">ExecTree</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span></span>)</span> ;<br/>
<a NAME="E3:54_5"/>
<span class="p1">(<span class="default"><a href="amistd_3.html#K5" target="_self">ExecTree</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span>
 
<b>by </b><i><a class="txt" href="amistd_3.html#E1:54"><i><font color="Green">E35</font></i></a></i>;<br/>
<a NAME="E4:54_5"/><b>then </b>
<span class="p1">(<span class="default"><a href="amistd_3.html#K5" target="_self">ExecTree</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="amistd_1.html#K5">il.</a> <font color="Maroon">S</font>,0</span>)</span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i>, <a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>;<br/>
<b>hence </b><a NAME="E5:54_5"/>
<span class="p1">(<span class="default"><a href="amistd_3.html#K5" target="_self">ExecTree</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="amistd_1.html#K5">il.</a> <font color="Maroon">S</font>,0
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E14:54"/>
 <a href="amistd_3.html#K5" target="_self">ExecTree</a> <span class="p1">(<span class="default"><a href="amistd_2.html#K7">Stop</a> <font color="Maroon">S</font></span>)</span> <a href="hidden.html#R1">=</a> <a href="amistd_3.html#K1" target="_self">TrivialInfiniteTree</a>  <a href="trees_2.html#K9">--&gt;</a> <span class="p1">(<span class="default"><a href="amistd_1.html#K5">il.</a> <font color="Maroon">S</font>,0</span>)</span>
 <b>by </b><i><a class="txt" href="amistd_3.html#E3:54"><i><font color="Green">E38</font></i></a>, <a class="ref" href="funcop_1.html#T17">FUNCOP_1:17</a></i>;<br/>


</div>
