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

<b>let </b><font color="Maroon" title="c1">N</font> be  <a href="setfam_1.html#V1" title="SETFAM_1:attr.1">with_non-empty_elements</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">S</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="ami_1.html#V2" title="AMI_1:attr.2">stored-program</a> <a href="ami_1.html#V4" title="AMI_1:attr.4">halting</a> <a href="ami_1.html#V5" title="AMI_1:attr.5">IC-Ins-separated</a> <a href="ami_1.html#V7" title="AMI_1:attr.7">definite</a> <a href="ami_1.html#V9" title="AMI_1:attr.9">realistic</a> <a href="amistd_1.html#V4" title="AMISTD_1:attr.4">standard</a>  <a href="ami_1.html#L1" title="AMI_1:struct.1">AMI-Struct</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<font color="Maroon" title="c1">N</font> holds   <a href="amistd_3.html#K5" title="AMISTD_3:func.5">ExecTree</a> <span class="p1">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Olive" title="b1">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="amistd_3.html#K1" title="AMISTD_3:func.1">TrivialInfiniteTree</a>  <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <span class="p1">(<span class="default"><a href="amistd_1.html#K5" title="AMISTD_1:func.5">il.</a> <font color="Olive" title="b1">S</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span><br/><b>let </b><font color="Maroon" title="c2">S</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="ami_1.html#V2" title="AMI_1:attr.2">stored-program</a> <a href="ami_1.html#V4" title="AMI_1:attr.4">halting</a> <a href="ami_1.html#V5" title="AMI_1:attr.5">IC-Ins-separated</a> <a href="ami_1.html#V7" title="AMI_1:attr.7">definite</a> <a href="ami_1.html#V9" title="AMI_1:attr.9">realistic</a> <a href="amistd_1.html#V4" title="AMISTD_1:attr.4">standard</a>  <a href="ami_1.html#L1" title="AMI_1:struct.1">AMI-Struct</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<font color="Maroon" title="c1">N</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="amistd_3.html#K5" title="AMISTD_3:func.5">ExecTree</a> <span class="p1">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="amistd_3.html#K1" title="AMISTD_3:func.1">TrivialInfiniteTree</a>  <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <span class="p1">(<span class="default"><a href="amistd_1.html#K5" title="AMISTD_1:func.5">il.</a> <font color="Maroon" title="c2">S</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span><br/>



<b>set </b><font color="Maroon" title="c3">D</font> =  <a href="amistd_3.html#K1" title="AMISTD_3:func.1">TrivialInfiniteTree</a> ;<br/>
<b>set </b><font color="Maroon" title="c4">M</font> =  <a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font>;<br/>
<b>set </b><font color="Maroon" title="c5">E</font> =  <a href="amistd_3.html#K5" title="AMISTD_3:func.5">ExecTree</a> <span class="p1">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span>;<br/>
<a NAME="E1:52"/><i><font color="Green" title="E26">A1</font></i>: 
 <a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="amistd_1.html#K5" title="AMISTD_1:func.5">il.</a> <font color="Maroon" title="c2">S</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="ami_1.html#K30" title="AMI_1:func.30">.--&gt;</a> <span class="p1">(<span class="default"><a href="ami_1.html#K5" title="AMI_1:func.5">halt</a> <font color="Maroon" title="c2">S</font></span>)</span>
 
<b>by </b><i><a class="ref" href="amistd_2.html#D13" title="AMISTD_2:def.13">AMISTD_2:def 13</a></i>;<br/>
<a NAME="E2:52"/><b>then </b><i><font color="Green" title="E27">A2</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="amistd_1.html#K5" title="AMISTD_1:func.5">il.</a> <font color="Maroon" title="c2">S</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></i>;<br/>
<a NAME="E3:52"/><i><font color="Green" title="E28">A3</font></i>: 
<span class="p1">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="amistd_1.html#K5" title="AMISTD_1:func.5">il.</a> <font color="Maroon" title="c2">S</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K5" title="AMI_1:func.5">halt</a> <font color="Maroon" title="c2">S</font>
 
<b>by </b><i><a class="txt" href="amistd_3.html#E1:52"><i><font color="Green" title="E26">A1</font></i></a>, <a class="ref" href="funcop_1.html#T87" title="FUNCOP_1:th.87">FUNCOP_1:87</a></i>;<br/>
<a NAME="E4:52"/><i><font color="Green" title="E29">A4</font></i>: 
<span class="p1">(<span class="default"><a href="amistd_3.html#K5" title="AMISTD_3:func.5">ExecTree</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="ami_1.html#K27" title="AMI_1:func.27">.</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> <span class="p1">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span>
 
<b>by </b><i><a class="ref" href="amistd_3.html#D5" target="_self" title="AMISTD_3:def.5">Def5</a></i>;<br/>
<a NAME="E5:52"/><i><font color="Green" title="E30">A5</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> <span class="p1">(<span class="default"><a href="amistd_3.html#K5" title="AMISTD_3:func.5">ExecTree</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> holds   <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#K5" title="AMI_1:func.5">halt</a> <font color="Maroon" title="c2">S</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="amistd_3.html#K5" title="AMISTD_3:func.5">ExecTree</a> <span class="p4">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="ami_1.html#K27" title="AMI_1:func.27">.</a> <font color="Olive" title="b1">t</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">t</font> be    <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> <span class="p1">(<span class="default"><a href="amistd_3.html#K5" title="AMISTD_3:func.5">ExecTree</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <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#K5" title="AMI_1:func.5">halt</a> <font color="Maroon" title="c2">S</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="amistd_3.html#K5" title="AMISTD_3:func.5">ExecTree</a> <span class="p4">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="ami_1.html#K27" title="AMI_1:func.27">.</a> <font color="Maroon" title="c6">t</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span></span><br/>

<a NAME="E1:52_1"/>
 <a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K5" title="AMI_1:func.5">halt</a> <font color="Maroon" title="c2">S</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="amistd_3.html#K5" title="AMISTD_3:func.5">ExecTree</a> <span class="p3">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="ami_1.html#K27" title="AMI_1:func.27">.</a> <font color="Maroon" title="c6">t</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="amistd_3.html#K5" title="AMISTD_3:func.5">ExecTree</a> <span class="p4">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="ami_1.html#K27" title="AMI_1:func.27">.</a> <font color="Maroon" title="c6">t</font></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="amistd_1.html#T15" title="AMISTD_1:th.15">AMISTD_1:15</a></i>;<br/>
<b>hence </b><a NAME="E2:52_1"/>
 <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#K5" title="AMI_1:func.5">halt</a> <font color="Maroon" title="c2">S</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="amistd_3.html#K5" title="AMISTD_3:func.5">ExecTree</a> <span class="p4">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="ami_1.html#K27" title="AMI_1:func.27">.</a> <font color="Maroon" title="c6">t</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="card_1.html#T50" title="CARD_1:th.50">CARD_1:50</a>, <a class="ref" href="card_5.html#T1" title="CARD_5:th.1">CARD_5:1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b><span class="p1">(<span class="default"><a href="amistd_3.html#K5" title="AMISTD_3:func.5">ExecTree</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="ami_1.html#K27" title="AMI_1:func.27">.</a> $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span>;<br/>
<a NAME="E6:52"/><i><font color="Green" title="E31">A6</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ]
 
<b>by </b><i><a class="txt" href="amistd_3.html#E4:52"><i><font color="Green" title="E29">A4</font></i></a>, <a class="ref" href="amistd_3.html#T21" target="_self" title="AMISTD_3:th.21">Th21</a></i>;<br/>
<a NAME="E7:52"/><i><font color="Green" title="E32">A7</font></i>: 
for <font color="Olive" title="b1">f</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> <span class="p1">(<span class="default"><a href="amistd_3.html#K5" title="AMISTD_3:func.5">ExecTree</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">f</font>] holds <br/>for <font color="Olive" title="b2">a</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="b1">f</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="Olive" title="b2">a</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></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> <span class="p1">(<span class="default"><a href="amistd_3.html#K5" title="AMISTD_3:func.5">ExecTree</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">f</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="Olive" title="b2">a</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">f</font> be    <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> <span class="p1">(<span class="default"><a href="amistd_3.html#K5" title="AMISTD_3:func.5">ExecTree</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">f</font>] implies for <font color="Olive" title="b1">a</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" title="c6">f</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="Olive" title="b1">a</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></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> <span class="p1">(<span class="default"><a href="amistd_3.html#K5" title="AMISTD_3:func.5">ExecTree</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">f</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="Olive" title="b1">a</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>] )</span><br/>

<b>assume </b><a NAME="E1:52_2"/><i><font color="Green" title="E33">A8</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">f</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">a</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" title="c6">f</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="Olive" title="b1">a</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></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> <span class="p1">(<span class="default"><a href="amistd_3.html#K5" title="AMISTD_3:func.5">ExecTree</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">f</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="Olive" title="b1">a</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>]</span><br/>

<b>let </b><font color="Maroon" title="c7">a</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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">f</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="c7">a</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></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> <span class="p1">(<span class="default"><a href="amistd_3.html#K5" title="AMISTD_3:func.5">ExecTree</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">f</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="c7">a</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>] )</span><br/>

<b>assume </b><a NAME="E2:52_2"/><i><font color="Green" title="E34">A9</font></i>: 
<font color="Maroon" title="c6">f</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="c7">a</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></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> <span class="p1">(<span class="default"><a href="amistd_3.html#K5" title="AMISTD_3:func.5">ExecTree</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">f</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="c7">a</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>]</span><br/>

<a NAME="E3:52_2"/><i><font color="Green" title="E35">A10</font></i>: 
 <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#K5" title="AMI_1:func.5">halt</a> <font color="Maroon" title="c2">S</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="amistd_3.html#K5" title="AMISTD_3:func.5">ExecTree</a> <span class="p4">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="ami_1.html#K27" title="AMI_1:func.27">.</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="amistd_3.html#E5:52"><i><font color="Green" title="E30">A5</font></i></a></i>;<br/>
<a NAME="E4:52_2"/><i><font color="Green" title="E36">A11</font></i>: 
 <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c6">f</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="c6">f</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#K32" title="AMI_1:func.32">pi</a> <span class="p3">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="amistd_3.html#K5" title="AMISTD_3:func.5">ExecTree</a> <span class="p5">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="ami_1.html#K27" title="AMI_1:func.27">.</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="amistd_3.html#K5" title="AMISTD_3:func.5">ExecTree</a> <span class="p4">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="ami_1.html#K27" title="AMI_1:func.27">.</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span> </span>}</span> 
 
<b>by </b><i><a class="ref" href="amistd_3.html#D5" target="_self" title="AMISTD_3:def.5">Def5</a></i>;<br/>
<a NAME="E5:52_2"/><i><font color="Green" title="E37">A12</font></i>: 
<span class="p1">(<span class="default"><a href="amistd_3.html#K5" title="AMISTD_3:func.5">ExecTree</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="ami_1.html#K27" title="AMI_1:func.27">.</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="amistd_1.html#K5" title="AMISTD_1:func.5">il.</a> <font color="Maroon" title="c2">S</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="amistd_3.html#E2:52"><i><font color="Green" title="E27">A2</font></i></a>, <a class="txt" href="amistd_3.html#E1:52_2"><i><font color="Green" title="E33">A8</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E6:52_2"/><b>then </b><i><font color="Green" title="E38">A13</font></i>: 
 <a href="amistd_1.html#K7" title="AMISTD_1:func.7">locnum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="amistd_3.html#K5" title="AMISTD_3:func.5">ExecTree</a> <span class="p3">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="ami_1.html#K27" title="AMI_1:func.27">.</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="amistd_1.html#D13" title="AMISTD_1:def.13">AMISTD_1:def 13</a></i>;<br/>
<a NAME="E7:52_2"/><i><font color="Green" title="E39">A14</font></i>: 
 <a href="ami_1.html#K32" title="AMI_1:func.32">pi</a> <span class="p1">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="amistd_3.html#K5" title="AMISTD_3:func.5">ExecTree</a> <span class="p3">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="ami_1.html#K27" title="AMI_1:func.27">.</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></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"><a href="amistd_3.html#K5" title="AMISTD_3:func.5">ExecTree</a> <span class="p3">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="ami_1.html#K27" title="AMI_1:func.27">.</a> <font color="Maroon" title="c6">f</font></span>)</span>
 
<b>by </b><i><a class="txt" href="amistd_3.html#E1:52_2"><i><font color="Green" title="E33">A8</font></i></a>, <a class="ref" href="ami_1.html#D47" title="AMI_1:def.47">AMI_1:def 47</a></i>;<br/>
<a NAME="E8:52_2"/><b>then </b><i><font color="Green" title="E40">A15</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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#K32" title="AMI_1:func.32">pi</a> <span class="p3">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="amistd_3.html#K5" title="AMISTD_3:func.5">ExecTree</a> <span class="p5">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="ami_1.html#K27" title="AMI_1:func.27">.</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="amistd_3.html#K5" title="AMISTD_3:func.5">ExecTree</a> <span class="p4">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="ami_1.html#K27" title="AMI_1:func.27">.</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="amistd_3.html#E3:52"><i><font color="Green" title="E28">A3</font></i></a>, <a class="txt" href="amistd_3.html#E3:52_2"><i><font color="Green" title="E35">A10</font></i></a>, <a class="txt" href="amistd_3.html#E5:52_2"><i><font color="Green" title="E37">A12</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E9:52_2"/><i><font color="Green" title="E41">A16</font></i>: 
 <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">f</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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_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> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">f</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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c6">f</font></span>
<div class="add"><b>let </b><font color="Maroon" title="c8">s</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c8">s</font> <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="c6">f</font> implies <font color="Maroon" title="c8">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">f</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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )</span><br/><b>assume </b><a NAME="E1:52_2_1_1"/>
<font color="Maroon" title="c8">s</font> <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="c6">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">f</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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/><b>then consider </b><font color="Maroon" title="c9">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="E3:52_2_1_1"/><i><font color="Green" title="E42">A17</font></i>: 
<font color="Maroon" title="c8">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</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="c9">k</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <b>and </b><br/><a NAME="E4:52_2_1_1"/><i><font color="Green" title="E43">A18</font></i>: 
<font color="Maroon" title="c9">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#K32" title="AMI_1:func.32">pi</a> <span class="p3">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="amistd_3.html#K5" title="AMISTD_3:func.5">ExecTree</a> <span class="p5">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="ami_1.html#K27" title="AMI_1:func.27">.</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="amistd_3.html#K5" title="AMISTD_3:func.5">ExecTree</a> <span class="p4">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="ami_1.html#K27" title="AMI_1:func.27">.</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="amistd_3.html#E4:52_2"><i><font color="Green" title="E36">A11</font></i></a></i>;<br/><a NAME="E5:52_2_1_1"/>
<font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="amistd_3.html#E3:52"><i><font color="Green" title="E28">A3</font></i></a>, <a class="txt" href="amistd_3.html#E3:52_2"><i><font color="Green" title="E35">A10</font></i></a>, <a class="txt" href="amistd_3.html#E5:52_2"><i><font color="Green" title="E37">A12</font></i></a>, <a class="txt" href="amistd_3.html#E7:52_2"><i><font color="Green" title="E39">A14</font></i></a>, <a class="txt" href="amistd_3.html#E4:52_2_1_1"><i><font color="Green" title="E43">A18</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E6:52_2_1_1"/>
<font color="Maroon" title="c8">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">f</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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="amistd_3.html#E3:52_2_1_1"><i><font color="Green" title="E42">A17</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon" title="c8">s</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> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c8">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">f</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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> or <font color="Maroon" title="c8">s</font> <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="c6">f</font> )</span><br/>

<b>assume </b><a NAME="E2:52_2_1"/>
<font color="Maroon" title="c8">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">f</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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">s</font> <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="c6">f</font></span><br/>

<a NAME="E3:52_2_1"/><b>then </b>
<font color="Maroon" title="c8">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:52_2_1"/>
<font color="Maroon" title="c8">s</font> <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="c6">f</font>
 <b>by </b><i><a class="txt" href="amistd_3.html#E4:52_2"><i><font color="Green" title="E36">A11</font></i></a>, <a class="txt" href="amistd_3.html#E8:52_2"><i><font color="Green" title="E40">A15</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E10:52_2"/>
<font color="Maroon" title="c6">f</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="c7">a</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&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="c6">f</font>
 
<b>by </b><i><a class="txt" href="amistd_3.html#E2:52_2"><i><font color="Green" title="E34">A9</font></i></a>, <a class="ref" href="trees_2.html#T14" title="TREES_2:th.14">TREES_2:14</a></i>;<br/>
<a NAME="E11:52_2"/><b>then </b><i><font color="Green" title="E42">A19</font></i>: 
<font color="Maroon" title="c6">f</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="c7">a</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="amistd_3.html#E9:52_2"><i><font color="Green" title="E41">A16</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E12:52_2"/>
 <a href="amistd_3.html#K3" title="AMISTD_3:func.3">LocNums</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#K5" title="AMI_1:func.5">halt</a> <font color="Maroon" title="c2">S</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="amistd_3.html#K5" title="AMISTD_3:func.5">ExecTree</a> <span class="p4">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="ami_1.html#K27" title="AMI_1:func.27">.</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="amistd_3.html#E6:52_2"><i><font color="Green" title="E38">A13</font></i></a>, <a class="ref" href="amistd_3.html#T29" target="_self" title="AMISTD_3:th.29">Th29</a></i>;<br/>
<a NAME="E13:52_2"/><b>then </b>
 <a href="wellord1.html#K3" title="WELLORD1:func.3">canonical_isomorphism_of</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p2">(<span class="default"><a href="wellord2.html#K2" title="WELLORD2:func.2">order_type_of</a> <span class="p3">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p4">(<span class="default"><a href="amistd_3.html#K3" title="AMISTD_3:func.3">LocNums</a> <span class="p5">(<span class="default"><a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> <span class="p0">(<span class="default"><a href="ami_1.html#K32" title="AMI_1:func.32">pi</a> <span class="p1">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="amistd_3.html#K5" title="AMISTD_3:func.5">ExecTree</a> <span class="p3">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="ami_1.html#K27" title="AMI_1:func.27">.</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="amistd_3.html#K5" title="AMISTD_3:func.5">ExecTree</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="ami_1.html#K27" title="AMI_1:func.27">.</a> <font color="Maroon" title="c6">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" title="WELLORD2:func.1">RelIncl</a> <span class="p2">(<span class="default"><a href="amistd_3.html#K3" title="AMISTD_3:func.3">LocNums</a> <span class="p3">(<span class="default"><a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> <span class="p4">(<span class="default"><a href="ami_1.html#K32" title="AMI_1:func.32">pi</a> <span class="p5">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="amistd_3.html#K5" title="AMISTD_3:func.5">ExecTree</a> <span class="p1">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="ami_1.html#K27" title="AMI_1:func.27">.</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="amistd_3.html#K5" title="AMISTD_3:func.5">ExecTree</a> <span class="p0">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="ami_1.html#K27" title="AMI_1:func.27">.</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p1">(<span class="default"><a href="amistd_1.html#K7" title="AMISTD_1:func.7">locnum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="amistd_3.html#K5" title="AMISTD_3:func.5">ExecTree</a> <span class="p4">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="ami_1.html#K27" title="AMI_1:func.27">.</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="amistd_3.html#E3:52"><i><font color="Green" title="E28">A3</font></i></a>, <a class="txt" href="amistd_3.html#E5:52_2"><i><font color="Green" title="E37">A12</font></i></a>, <a class="txt" href="amistd_3.html#E6:52_2"><i><font color="Green" title="E38">A13</font></i></a>, <a class="txt" href="amistd_3.html#E7:52_2"><i><font color="Green" title="E39">A14</font></i></a>, <a class="ref" href="amistd_3.html#T15" target="_self" title="AMISTD_3:th.15">Th15</a></i>;<br/>
<b>then </b><i><font color="Green" title="E43">A20</font></i>: <span class="p1">(<span class="default"><a href="wellord1.html#K3" title="WELLORD1:func.3">canonical_isomorphism_of</a> <span class="p2">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p3">(<span class="default"><a href="wellord2.html#K2" title="WELLORD2:func.2">order_type_of</a> <span class="p4">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p5">(<span class="default"><a href="amistd_3.html#K3" title="AMISTD_3:func.3">LocNums</a> <span class="p0">(<span class="default"><a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K32" title="AMI_1:func.32">pi</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="amistd_3.html#K5" title="AMISTD_3:func.5">ExecTree</a> <span class="p4">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="ami_1.html#K27" title="AMI_1:func.27">.</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="amistd_3.html#K5" title="AMISTD_3:func.5">ExecTree</a> <span class="p3">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="ami_1.html#K27" title="AMI_1:func.27">.</a> <font color="Maroon" title="c6">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" title="WELLORD2:func.1">RelIncl</a> <span class="p3">(<span class="default"><a href="amistd_3.html#K3" title="AMISTD_3:func.3">LocNums</a> <span class="p4">(<span class="default"><a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> <span class="p5">(<span class="default"><a href="ami_1.html#K32" title="AMI_1:func.32">pi</a> <span class="p0">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="amistd_3.html#K5" title="AMISTD_3:func.5">ExecTree</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="ami_1.html#K27" title="AMI_1:func.27">.</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="amistd_3.html#K5" title="AMISTD_3:func.5">ExecTree</a> <span class="p1">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="ami_1.html#K27" title="AMI_1:func.27">.</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  = 
 <a href="amistd_1.html#K7" title="AMISTD_1:func.7">locnum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="amistd_3.html#K5" title="AMISTD_3:func.5">ExecTree</a> <span class="p3">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="ami_1.html#K27" title="AMI_1:func.27">.</a> <font color="Maroon" title="c6">f</font></span>)</span>
<b>by </b><i><a class="ref" href="funcop_1.html#T87" title="FUNCOP_1:th.87">FUNCOP_1:87</a></i>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="txt" href="amistd_3.html#E5:52_2"><i><font color="Green" title="E37">A12</font></i></a>, <a class="ref" href="amistd_1.html#D13" title="AMISTD_1:def.13">AMISTD_1:def 13</a></i>
;<br/>
<span class="p1">(<span class="default"><a href="amistd_3.html#K5" title="AMISTD_3:func.5">ExecTree</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="ami_1.html#K27" title="AMI_1:func.27">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</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="Maroon" title="c7">a</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></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#K32" title="AMI_1:func.32">pi</a> <span class="p4">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="amistd_3.html#K5" title="AMISTD_3:func.5">ExecTree</a> <span class="p0">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="ami_1.html#K27" title="AMI_1:func.27">.</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="amistd_3.html#K5" title="AMISTD_3:func.5">ExecTree</a> <span class="p5">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="ami_1.html#K27" title="AMI_1:func.27">.</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="txt" href="amistd_3.html#E8:52_2"><i><font color="Green" title="E40">A15</font></i></a>, <a class="txt" href="amistd_3.html#E11:52_2"><i><font color="Green" title="E42">A19</font></i></a>, <a class="ref" href="amistd_3.html#D5" target="_self" title="AMISTD_3:def.5">Def5</a></i>
<br/>.= 
 <a href="amistd_1.html#K5" title="AMISTD_1:func.5">il.</a> <font color="Maroon" title="c2">S</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="txt" href="amistd_3.html#E8:52_2"><i><font color="Green" title="E40">A15</font></i></a>, <a class="txt" href="amistd_3.html#E14:52_2"><i><font color="Green" title="E43">A20</font></i></a>, <a class="ref" href="amistd_3.html#D4" target="_self" title="AMISTD_3:def.4">Def4</a></i>
;<br/>
<b>hence </b><a NAME="E16:52_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">f</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="c7">a</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>]
 <b>by </b><i><a class="txt" href="amistd_3.html#E2:52"><i><font color="Green" title="E27">A2</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E8:52"/><i><font color="Green" title="E33">A21</font></i>: 
for <font color="Olive" title="b1">f</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> <span class="p1">(<span class="default"><a href="amistd_3.html#K5" title="AMISTD_3:func.5">ExecTree</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">f</font>]
 
<b>from </b><i><a class="ref" href="hilbert2.html#S1" title="HILBERT2:sch.1">HILBERT2:sch 1</a>(<a class="txt" href="amistd_3.html#E6:52"><i><font color="Green" title="E31">A6</font></i></a>, <a class="txt" href="amistd_3.html#E7:52"><i><font color="Green" title="E32">A7</font></i></a>);<br/></i>
<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><span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><a href="amistd_3.html#K5" title="AMISTD_3:func.5">ExecTree</a> <span class="p3">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span></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> <a href="amistd_3.html#K1" title="AMISTD_3:func.1">TrivialInfiniteTree</a>  <a href="trees_2.html#K2" title="TREES_2:func.2">-level</a> $1;<br/>
<a NAME="E9:52"/><i><font color="Green" title="E34">A22</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="52_3"><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> <span class="p2">(<span class="default"><a href="amistd_3.html#K5" title="AMISTD_3:func.5">ExecTree</a> <span class="p3">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span></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/>.= 
<a href="amistd_3.html#K1" title="AMISTD_3:func.1">TrivialInfiniteTree</a>  <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>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E10:52"/><i><font color="Green" title="E35">A23</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="52_4"><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c6">n</font>] implies <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c6">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/>

<b>assume </b><a NAME="E1:52_4"/><i><font color="Green" title="E36">A24</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c6">n</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c6">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/>

<b>set </b><font color="Maroon" title="c7">f0</font> = <font color="Maroon" title="c6">n</font> <a href="finseq_2.html#K2" title="FINSEQ_2:func.2">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ;<br/>
<b>set </b><font color="Maroon" title="c8">f1</font> = <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="finseq_2.html#K2" title="FINSEQ_2:func.2">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ;<br/>
<a NAME="E2:52_4"/><i><font color="Green" title="E37">A25</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><a href="amistd_3.html#K5" title="AMISTD_3:func.5">ExecTree</a> <span class="p3">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span></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> <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> 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> <span class="p1">(<span class="default"><a href="amistd_3.html#K5" title="AMISTD_3:func.5">ExecTree</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> :  <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> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 </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="E3:52_4"/><i><font color="Green" title="E38">A26</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<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="finseq_2.html#K2" title="FINSEQ_2:func.2">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="ref" href="finseq_2.html#T69" title="FINSEQ_2:th.69">FINSEQ_2:69</a></i>;<br/>
<a NAME="E4:52_4"/>
<span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><a href="amistd_3.html#K5" title="AMISTD_3:func.5">ExecTree</a> <span class="p3">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span></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> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="finseq_2.html#K2" title="FINSEQ_2:func.2">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_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" 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> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="finseq_2.html#K2" title="FINSEQ_2:func.2">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><a href="amistd_3.html#K5" title="AMISTD_3:func.5">ExecTree</a> <span class="p3">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span></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></span>
<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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c9">a</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> <span class="p2">(<span class="default"><a href="amistd_3.html#K5" title="AMISTD_3:func.5">ExecTree</a> <span class="p3">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span></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> implies <font color="Maroon" title="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="finseq_2.html#K2" title="FINSEQ_2:func.2">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )</span><br/><b>assume </b><a NAME="E1:52_4_1_1"/>
<font color="Maroon" title="c9">a</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> <span class="p2">(<span class="default"><a href="amistd_3.html#K5" title="AMISTD_3:func.5">ExecTree</a> <span class="p3">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span></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 class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="finseq_2.html#K2" title="FINSEQ_2:func.2">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/><b>then consider </b><font color="Maroon" title="c10">t1</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> <span class="p1">(<span class="default"><a href="amistd_3.html#K5" title="AMISTD_3:func.5">ExecTree</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E3:52_4_1_1"/><i><font color="Green" title="E39">A27</font></i>: 
<font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">t1</font>
 <b>and </b><br/><a NAME="E4:52_4_1_1"/><i><font color="Green" title="E40">A28</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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#E2:52_4"><i><font color="Green" title="E37">A25</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c11">t0</font> = <font color="Maroon" title="c10">t1</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="c6">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> <span class="p1">(<span class="default"><a href="amistd_3.html#K5" title="AMISTD_3:func.5">ExecTree</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="relat_1.html#T88" title="RELAT_1:th.88">RELAT_1:88</a>, <a class="ref" href="trees_1.html#T45" title="TREES_1:th.45">TREES_1:45</a></i>;<br/><a NAME="E6:52_4_1_1"/><i><font color="Green" title="E41">A29</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><a href="amistd_3.html#K5" title="AMISTD_3:func.5">ExecTree</a> <span class="p3">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span></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">w</font> 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> <span class="p1">(<span class="default"><a href="amistd_3.html#K5" title="AMISTD_3:func.5">ExecTree</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> :  <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="trees_2.html#D6" title="TREES_2:def.6">TREES_2:def 6</a></i>;<br/><a NAME="E7:52_4_1_1"/><i><font color="Green" title="E42">A30</font></i>: 
 <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c11">t0</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">t0</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#K32" title="AMI_1:func.32">pi</a> <span class="p3">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="amistd_3.html#K5" title="AMISTD_3:func.5">ExecTree</a> <span class="p5">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="ami_1.html#K27" title="AMI_1:func.27">.</a> <font color="Maroon" title="c11">t0</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="amistd_3.html#K5" title="AMISTD_3:func.5">ExecTree</a> <span class="p4">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="ami_1.html#K27" title="AMI_1:func.27">.</a> <font color="Maroon" title="c11">t0</font></span>)</span></span>)</span> </span>}</span> 
 <b>by </b><i><a class="ref" href="amistd_3.html#D5" target="_self" title="AMISTD_3:def.5">Def5</a></i>;<br/><a NAME="E8:52_4_1_1"/><i><font color="Green" title="E43">A31</font></i>: 
 <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#K5" title="AMI_1:func.5">halt</a> <font color="Maroon" title="c2">S</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="amistd_3.html#K5" title="AMISTD_3:func.5">ExecTree</a> <span class="p4">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="ami_1.html#K27" title="AMI_1:func.27">.</a> <font color="Maroon" title="c11">t0</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="amistd_3.html#E5:52"><i><font color="Green" title="E30">A5</font></i></a></i>;<br/><a NAME="E9:52_4_1_1"/>
<span class="p1">(<span class="default"><a href="amistd_3.html#K5" title="AMISTD_3:func.5">ExecTree</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="ami_1.html#K27" title="AMI_1:func.27">.</a> <font color="Maroon" title="c11">t0</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> <span class="p1">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span>
 <b>by </b><i><a class="txt" href="amistd_3.html#E8:52"><i><font color="Green" title="E33">A21</font></i></a></i>;<br/><a NAME="E10:52_4_1_1"/><b>then </b><i><font color="Green" title="E44">A32</font></i>: 
<span class="p1">(<span class="default"><a href="amistd_3.html#K5" title="AMISTD_3:func.5">ExecTree</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="ami_1.html#K27" title="AMI_1:func.27">.</a> <font color="Maroon" title="c11">t0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="amistd_1.html#K5" title="AMISTD_1:func.5">il.</a> <font color="Maroon" title="c2">S</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="amistd_3.html#E2:52"><i><font color="Green" title="E27">A2</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E11:52_4_1_1"/><i><font color="Green" title="E45">A33</font></i>: 
 <a href="ami_1.html#K32" title="AMI_1:func.32">pi</a> <span class="p1">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="amistd_3.html#K5" title="AMISTD_3:func.5">ExecTree</a> <span class="p3">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="ami_1.html#K27" title="AMI_1:func.27">.</a> <font color="Maroon" title="c11">t0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></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"><a href="amistd_3.html#K5" title="AMISTD_3:func.5">ExecTree</a> <span class="p3">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="ami_1.html#K27" title="AMI_1:func.27">.</a> <font color="Maroon" title="c11">t0</font></span>)</span>
 <b>by </b><i><a class="txt" href="amistd_3.html#E8:52"><i><font color="Green" title="E33">A21</font></i></a>, <a class="ref" href="ami_1.html#D47" title="AMI_1:def.47">AMI_1:def 47</a></i>;<br/><a NAME="E12:52_4_1_1"/><b>then </b><i><font color="Green" title="E46">A34</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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#K32" title="AMI_1:func.32">pi</a> <span class="p3">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="amistd_3.html#K5" title="AMISTD_3:func.5">ExecTree</a> <span class="p5">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="ami_1.html#K27" title="AMI_1:func.27">.</a> <font color="Maroon" title="c11">t0</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="amistd_3.html#K5" title="AMISTD_3:func.5">ExecTree</a> <span class="p4">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="ami_1.html#K27" title="AMI_1:func.27">.</a> <font color="Maroon" title="c11">t0</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="amistd_3.html#E3:52"><i><font color="Green" title="E28">A3</font></i></a>, <a class="txt" href="amistd_3.html#E8:52_4_1_1"><i><font color="Green" title="E43">A31</font></i></a>, <a class="txt" href="amistd_3.html#E10:52_4_1_1"><i><font color="Green" title="E44">A32</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E13:52_4_1_1"/><i><font color="Green" title="E47">A35</font></i>: 
<font color="Maroon" title="c10">t1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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> 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="E14:52_4_1_1"/><i><font color="Green" title="E48">A36</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><a href="amistd_3.html#K5" title="AMISTD_3:func.5">ExecTree</a> <span class="p3">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span></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"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="finseq_2.html#K2" title="FINSEQ_2:func.2">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="amistd_3.html#E1:52_4"><i><font color="Green" title="E36">A24</font></i></a>, <a class="ref" href="amistd_3.html#T20" target="_self" title="AMISTD_3:th.20">Th20</a></i>;<br/><a NAME="E15:52_4_1_1"/>
<font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <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="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E16:52_4_1_1"/><b>then </b>
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c6">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="c6">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="E17:52_4_1_1"/><b>then </b>
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c6">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c10">t1</font>
 <b>by </b><i><a class="txt" href="amistd_3.html#E4:52_4_1_1"><i><font color="Green" title="E40">A28</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E18:52_4_1_1"/><b>then </b>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">t0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c6">n</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></i>;<br/><a NAME="E19:52_4_1_1"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">t0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">n</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E20:52_4_1_1"/><b>then </b><i><font color="Green" title="E49">A37</font></i>: 
<font color="Maroon" title="c11">t0</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> <span class="p2">(<span class="default"><a href="amistd_3.html#K5" title="AMISTD_3:func.5">ExecTree</a> <span class="p3">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span></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#E6:52_4_1_1"><i><font color="Green" title="E41">A29</font></i></a></i>;<br/><a NAME="E21:52_4_1_1"/>
<font color="Maroon" title="c10">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">t0</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"><font color="Maroon" title="c10">t1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<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></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#E4:52_4_1_1"><i><font color="Green" title="E40">A28</font></i></a>, <a class="ref" href="finseq_3.html#T61" title="FINSEQ_3:th.61">FINSEQ_3:61</a></i>;<br/><a NAME="E22:52_4_1_1"/><b>then </b><i><font color="Green" title="E50">A38</font></i>: 
<font color="Maroon" title="c11">t0</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"><font color="Maroon" title="c10">t1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<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></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="c11">t0</font>
 <b>by </b><i><a class="txt" href="amistd_3.html#E13:52_4_1_1"><i><font color="Green" title="E47">A35</font></i></a>, <a class="ref" href="trees_2.html#T14" title="TREES_2:th.14">TREES_2:14</a></i>;<br/><a NAME="E23:52_4_1_1"/>
 <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c11">t0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">t0</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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_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" 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> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">t0</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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c11">t0</font></span>
<div class="add"><b>let </b><font color="Maroon" title="c12">s</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c12">s</font> <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="c11">t0</font> implies <font color="Maroon" title="c12">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">t0</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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )</span><br/><b>assume </b><a NAME="E1:52_4_1_1_1_1"/>
<font color="Maroon" title="c12">s</font> <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="c11">t0</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c12">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">t0</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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/><b>then consider </b><font color="Maroon" title="c13">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="E3:52_4_1_1_1_1"/><i><font color="Green" title="E51">A39</font></i>: 
<font color="Maroon" title="c12">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">t0</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="c13">k</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <b>and </b><br/><a NAME="E4:52_4_1_1_1_1"/><i><font color="Green" title="E52">A40</font></i>: 
<font color="Maroon" title="c13">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#K32" title="AMI_1:func.32">pi</a> <span class="p3">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="amistd_3.html#K5" title="AMISTD_3:func.5">ExecTree</a> <span class="p5">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="ami_1.html#K27" title="AMI_1:func.27">.</a> <font color="Maroon" title="c11">t0</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="amistd_3.html#K5" title="AMISTD_3:func.5">ExecTree</a> <span class="p4">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="ami_1.html#K27" title="AMI_1:func.27">.</a> <font color="Maroon" title="c11">t0</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="amistd_3.html#E7:52_4_1_1"><i><font color="Green" title="E42">A30</font></i></a></i>;<br/><a NAME="E5:52_4_1_1_1_1"/>
<font color="Maroon" title="c13">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="amistd_3.html#E3:52"><i><font color="Green" title="E28">A3</font></i></a>, <a class="txt" href="amistd_3.html#E8:52_4_1_1"><i><font color="Green" title="E43">A31</font></i></a>, <a class="txt" href="amistd_3.html#E10:52_4_1_1"><i><font color="Green" title="E44">A32</font></i></a>, <a class="txt" href="amistd_3.html#E11:52_4_1_1"><i><font color="Green" title="E45">A33</font></i></a>, <a class="txt" href="amistd_3.html#E4:52_4_1_1_1_1"><i><font color="Green" title="E52">A40</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E6:52_4_1_1_1_1"/>
<font color="Maroon" title="c12">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">t0</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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="amistd_3.html#E3:52_4_1_1_1_1"><i><font color="Green" title="E51">A39</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon" title="c12">s</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> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c12">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">t0</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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> or <font color="Maroon" title="c12">s</font> <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="c11">t0</font> )</span><br/>

<b>assume </b><a NAME="E2:52_4_1_1_1"/>
<font color="Maroon" title="c12">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">t0</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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c12">s</font> <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="c11">t0</font></span><br/>

<a NAME="E3:52_4_1_1_1"/><b>then </b>
<font color="Maroon" title="c12">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">t0</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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:52_4_1_1_1"/>
<font color="Maroon" title="c12">s</font> <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="c11">t0</font>
 <b>by </b><i><a class="txt" href="amistd_3.html#E7:52_4_1_1"><i><font color="Green" title="E42">A30</font></i></a>, <a class="txt" href="amistd_3.html#E12:52_4_1_1"><i><font color="Green" title="E46">A34</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E24:52_4_1_1"/><b>then </b><i><font color="Green" title="E51">A41</font></i>: 
<font color="Maroon" title="c11">t0</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"><font color="Maroon" title="c10">t1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<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></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="c11">t0</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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="amistd_3.html#E22:52_4_1_1"><i><font color="Green" title="E50">A38</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E25:52_4_1_1"/>
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <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="c10">t1</font> holds <br/><font color="Maroon" title="c10">t1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</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="c6">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="finseq_2.html#K2" title="FINSEQ_2:func.2">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_4_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">k</font> &amp; <font color="Maroon" title="c12">k</font> <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="c10">t1</font> implies <font color="Maroon" title="c10">t1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k</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="c6">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="finseq_2.html#K2" title="FINSEQ_2:func.2">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k</font> )</span><br/>

<b>assume </b><a NAME="E1:52_4_1_1_2"/><i><font color="Green" title="E52">A42</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">k</font> &amp; <font color="Maroon" title="c12">k</font> <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="c10">t1</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">t1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k</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="c6">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="finseq_2.html#K2" title="FINSEQ_2:func.2">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k</font></span><br/>

<a NAME="E2:52_4_1_1_2"/><i><font color="Green" title="E53">A43</font></i>: 
<font color="Maroon" title="c12">k</font> <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="c6">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#E4:52_4_1_1"><i><font color="Green" title="E40">A28</font></i></a>, <a class="txt" href="amistd_3.html#E1:52_4_1_1_2"><i><font color="Green" title="E52">A42</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_4_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_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="E1:52_4_1_1_2_1_1"/>
( <font color="Maroon" title="c12">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c6">n</font> or <font color="Maroon" title="c12">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 </span><b>by </b><i><a class="txt" href="amistd_3.html#E2:52_4_1_1_2"><i><font color="Green" title="E53">A43</font></i></a>, <a class="ref" href="finseq_2.html#T8" title="FINSEQ_2:th.8">FINSEQ_2:8</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_4_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:52_4_1_1_2_1_1_1"/><i><font color="Green" title="E54">A44</font></i>: 
<font color="Maroon" title="c12">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c6">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">t1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/><div class="add"><b>hence </b><font color="Maroon" title="c10">t1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k</font> = 
<font color="Maroon" title="c11">t0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k</font>
<b>by </b><i><a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="finseq_2.html#K2" title="FINSEQ_2:func.2">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k</font>
<b>by </b><i><a class="txt" href="amistd_3.html#E14:52_4_1_1"><i><font color="Green" title="E48">A36</font></i></a>, <a class="txt" href="amistd_3.html#E20:52_4_1_1"><i><font color="Green" title="E49">A37</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="txt" href="amistd_3.html#E1:52_4_1_1_2_1_1_1"><i><font color="Green" title="E54">A44</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_4_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:52_4_1_1_2_1_1_2"/>
<font color="Maroon" title="c12">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">t1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/><div class="add"><b>hence </b><a NAME="E2:52_4_1_1_2_1_1_2"/>
<font color="Maroon" title="c10">t1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="amistd_3.html#E24:52_4_1_1"><i><font color="Green" title="E51">A41</font></i></a>, <a class="ref" href="finseq_2.html#T20" title="FINSEQ_2:th.20">FINSEQ_2:20</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:52_4_1_1_2"/>
<font color="Maroon" title="c10">t1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k</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="c6">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="finseq_2.html#K2" title="FINSEQ_2:func.2">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k</font>
 <b>by </b><i><a class="txt" href="amistd_3.html#E2:52_4_1_1_2"><i><font color="Green" title="E53">A43</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E26:52_4_1_1"/><b>then </b>
<font color="Maroon" title="c10">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="finseq_2.html#K2" title="FINSEQ_2:func.2">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="amistd_3.html#E3:52_4"><i><font color="Green" title="E38">A26</font></i></a>, <a class="txt" href="amistd_3.html#E4:52_4_1_1"><i><font color="Green" title="E40">A28</font></i></a>, <a class="ref" href="finseq_1.html#T18" title="FINSEQ_1:th.18">FINSEQ_1:18</a></i>;<br/><b>hence </b><a NAME="E27:52_4_1_1"/>
<font color="Maroon" title="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="finseq_2.html#K2" title="FINSEQ_2:func.2">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="amistd_3.html#E3:52_4_1_1"><i><font color="Green" title="E39">A27</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><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> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="finseq_2.html#K2" title="FINSEQ_2:func.2">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> or <font color="Maroon" title="c9">a</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> <span class="p2">(<span class="default"><a href="amistd_3.html#K5" title="AMISTD_3:func.5">ExecTree</a> <span class="p3">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span></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> )</span><br/>

<b>assume </b><a NAME="E2:52_4_1"/>
<font color="Maroon" title="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="finseq_2.html#K2" title="FINSEQ_2:func.2">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">a</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> <span class="p2">(<span class="default"><a href="amistd_3.html#K5" title="AMISTD_3:func.5">ExecTree</a> <span class="p3">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span></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></span><br/>

<a NAME="E3:52_4_1"/><b>then </b><i><font color="Green" title="E39">A45</font></i>: 
<font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="finseq_2.html#K2" title="FINSEQ_2:func.2">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>3</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="finseq_2.html#K2" title="FINSEQ_2:func.2">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="amistd_3.html#K5" title="AMISTD_3:func.5">ExecTree</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span>;<br/>
<a NAME="E4:52_4_1"/>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="finseq_2.html#K2" title="FINSEQ_2:func.2">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</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="finseq_2.html#T72" title="FINSEQ_2:th.72">FINSEQ_2:72</a></i>;<br/>
<a NAME="E5:52_4_1"/><b>then </b><i><font color="Green" title="E40">A46</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<b>by </b><i><a class="ref" href="trees_1.html#T47" title="TREES_1:th.47">TREES_1:47</a></i>;<br/>
<a NAME="E6:52_4_1"/><i><font color="Green" title="E41">A47</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>3</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>3</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="52_4_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c10">n</font>] implies <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c10">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/>

<b>assume </b><a NAME="E1:52_4_1_2"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c10">n</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c10">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/>

<b>then reconsider </b><font color="Maroon" title="c11">t</font> = <font color="Maroon" title="c10">n</font> <a href="finseq_2.html#K2" title="FINSEQ_2:func.2">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  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> <span class="p1">(<span class="default"><a href="amistd_3.html#K5" title="AMISTD_3:func.5">ExecTree</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> ;<br/>
<a NAME="E3:52_4_1_2"/><i><font color="Green" title="E42">A48</font></i>: 
 <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c11">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="Maroon" title="c11">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="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#K32" title="AMI_1:func.32">pi</a> <span class="p3">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="amistd_3.html#K5" title="AMISTD_3:func.5">ExecTree</a> <span class="p5">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="ami_1.html#K27" title="AMI_1:func.27">.</a> <font color="Maroon" title="c11">t</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="amistd_3.html#K5" title="AMISTD_3:func.5">ExecTree</a> <span class="p4">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="ami_1.html#K27" title="AMI_1:func.27">.</a> <font color="Maroon" title="c11">t</font></span>)</span></span>)</span> </span>}</span> 
 
<b>by </b><i><a class="ref" href="amistd_3.html#D5" target="_self" title="AMISTD_3:def.5">Def5</a></i>;<br/>
<a NAME="E4:52_4_1_2"/><i><font color="Green" title="E43">A49</font></i>: 
 <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#K5" title="AMI_1:func.5">halt</a> <font color="Maroon" title="c2">S</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="amistd_3.html#K5" title="AMISTD_3:func.5">ExecTree</a> <span class="p4">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="ami_1.html#K27" title="AMI_1:func.27">.</a> <font color="Maroon" title="c11">t</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="amistd_3.html#E5:52"><i><font color="Green" title="E30">A5</font></i></a></i>;<br/>
<a NAME="E5:52_4_1_2"/>
<span class="p1">(<span class="default"><a href="amistd_3.html#K5" title="AMISTD_3:func.5">ExecTree</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="ami_1.html#K27" title="AMI_1:func.27">.</a> <font color="Maroon" title="c11">t</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> <span class="p1">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span>
 
<b>by </b><i><a class="txt" href="amistd_3.html#E8:52"><i><font color="Green" title="E33">A21</font></i></a></i>;<br/>
<a NAME="E6:52_4_1_2"/><b>then </b><i><font color="Green" title="E44">A50</font></i>: 
<span class="p1">(<span class="default"><a href="amistd_3.html#K5" title="AMISTD_3:func.5">ExecTree</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="ami_1.html#K27" title="AMI_1:func.27">.</a> <font color="Maroon" title="c11">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="amistd_1.html#K5" title="AMISTD_1:func.5">il.</a> <font color="Maroon" title="c2">S</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="amistd_3.html#E2:52"><i><font color="Green" title="E27">A2</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E7:52_4_1_2"/>
 <a href="ami_1.html#K32" title="AMI_1:func.32">pi</a> <span class="p1">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="amistd_3.html#K5" title="AMISTD_3:func.5">ExecTree</a> <span class="p3">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="ami_1.html#K27" title="AMI_1:func.27">.</a> <font color="Maroon" title="c11">t</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></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"><a href="amistd_3.html#K5" title="AMISTD_3:func.5">ExecTree</a> <span class="p3">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="ami_1.html#K27" title="AMI_1:func.27">.</a> <font color="Maroon" title="c11">t</font></span>)</span>
 
<b>by </b><i><a class="txt" href="amistd_3.html#E8:52"><i><font color="Green" title="E33">A21</font></i></a>, <a class="ref" href="ami_1.html#D47" title="AMI_1:def.47">AMI_1:def 47</a></i>;<br/>
<a NAME="E8:52_4_1_2"/><b>then </b>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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#K32" title="AMI_1:func.32">pi</a> <span class="p3">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="amistd_3.html#K5" title="AMISTD_3:func.5">ExecTree</a> <span class="p5">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="ami_1.html#K27" title="AMI_1:func.27">.</a> <font color="Maroon" title="c11">t</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="amistd_3.html#K5" title="AMISTD_3:func.5">ExecTree</a> <span class="p4">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="ami_1.html#K27" title="AMI_1:func.27">.</a> <font color="Maroon" title="c11">t</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="amistd_3.html#E3:52"><i><font color="Green" title="E28">A3</font></i></a>, <a class="txt" href="amistd_3.html#E4:52_4_1_2"><i><font color="Green" title="E43">A49</font></i></a>, <a class="txt" href="amistd_3.html#E6:52_4_1_2"><i><font color="Green" title="E44">A50</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E9:52_4_1_2"/><b>then </b>
<font color="Maroon" title="c11">t</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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&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="c11">t</font>
 
<b>by </b><i><a class="txt" href="amistd_3.html#E3:52_4_1_2"><i><font color="Green" title="E42">A48</font></i></a></i>;<br/>
<a NAME="E10:52_4_1_2"/><b>then </b>
<font color="Maroon" title="c11">t</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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></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> <span class="p1">(<span class="default"><a href="amistd_3.html#K5" title="AMISTD_3:func.5">ExecTree</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span>
 
;<br/>
<b>hence </b><a NAME="E11:52_4_1_2"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c10">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <b>by </b><i><a class="ref" href="finseq_2.html#T74" title="FINSEQ_2:th.74">FINSEQ_2:74</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E7:52_4_1"/>
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>3</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#E5:52_4_1"><i><font color="Green" title="E40">A46</font></i></a>, <a class="txt" href="amistd_3.html#E6:52_4_1"><i><font color="Green" title="E41">A47</font></i></a>);<br/></i>
<a NAME="E8:52_4_1"/><b>then </b>
<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="finseq_2.html#K2" title="FINSEQ_2:func.2">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  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> <span class="p1">(<span class="default"><a href="amistd_3.html#K5" title="AMISTD_3:func.5">ExecTree</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span>
 
;<br/>
<b>hence </b><a NAME="E9:52_4_1"/>
<font color="Maroon" title="c9">a</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> <span class="p2">(<span class="default"><a href="amistd_3.html#K5" title="AMISTD_3:func.5">ExecTree</a> <span class="p3">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span></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>
 <b>by </b><i><a class="txt" href="amistd_3.html#E2:52_4"><i><font color="Green" title="E37">A25</font></i></a>, <a class="txt" href="amistd_3.html#E3:52_4"><i><font color="Green" title="E38">A26</font></i></a>, <a class="txt" href="amistd_3.html#E3:52_4_1"><i><font color="Green" title="E39">A45</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:52_4"/>
<font color="Maroon">S<sub>2</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="ref" href="amistd_3.html#T20" target="_self" title="AMISTD_3:th.20">Th20</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E11:52"/>
for <font color="Olive" title="b1">x</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">x</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#E9:52"><i><font color="Green" title="E34">A22</font></i></a>, <a class="txt" href="amistd_3.html#E10:52"><i><font color="Green" title="E35">A23</font></i></a>);<br/></i>
<a NAME="E12:52"/><b>then </b><i><font color="Green" title="E36">A51</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="amistd_3.html#K5" title="AMISTD_3:func.5">ExecTree</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="amistd_3.html#K1" title="AMISTD_3:func.1">TrivialInfiniteTree</a> 
 
<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="E13:52"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</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> <span class="p1">(<span class="default"><a href="amistd_3.html#K5" title="AMISTD_3:func.5">ExecTree</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="amistd_3.html#K5" title="AMISTD_3:func.5">ExecTree</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="amistd_1.html#K5" title="AMISTD_1:func.5">il.</a> <font color="Maroon" title="c2">S</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">x</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> <span class="p1">(<span class="default"><a href="amistd_3.html#K5" title="AMISTD_3:func.5">ExecTree</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> implies <span class="p1">(<span class="default"><a href="amistd_3.html#K5" title="AMISTD_3:func.5">ExecTree</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="amistd_1.html#K5" title="AMISTD_1:func.5">il.</a> <font color="Maroon" title="c2">S</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/>

<b>assume </b><a NAME="E1:52_5"/>
<font color="Maroon" title="c6">x</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> <span class="p1">(<span class="default"><a href="amistd_3.html#K5" title="AMISTD_3:func.5">ExecTree</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="amistd_3.html#K5" title="AMISTD_3:func.5">ExecTree</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="amistd_1.html#K5" title="AMISTD_1:func.5">il.</a> <font color="Maroon" title="c2">S</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/>

<b>then reconsider </b><font color="Maroon" title="c7">x</font> = <font color="Maroon" title="c6">x</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> <span class="p1">(<span class="default"><a href="amistd_3.html#K5" title="AMISTD_3:func.5">ExecTree</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> ;<br/>
<a NAME="E3:52_5"/>
<span class="p1">(<span class="default"><a href="amistd_3.html#K5" title="AMISTD_3:func.5">ExecTree</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="ami_1.html#K27" title="AMI_1:func.27">.</a> <font color="Maroon" title="c7">x</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> <span class="p1">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span>
 
<b>by </b><i><a class="txt" href="amistd_3.html#E8:52"><i><font color="Green" title="E33">A21</font></i></a></i>;<br/>
<a NAME="E4:52_5"/><b>then </b>
<span class="p1">(<span class="default"><a href="amistd_3.html#K5" title="AMISTD_3:func.5">ExecTree</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="ami_1.html#K27" title="AMI_1:func.27">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="amistd_1.html#K5" title="AMISTD_1:func.5">il.</a> <font color="Maroon" title="c2">S</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="amistd_3.html#E1:52"><i><font color="Green" title="E26">A1</font></i></a>, <a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></i>;<br/>
<a NAME="E5:52_5"/><b>then </b>
<span class="p1">(<span class="default"><a href="amistd_3.html#K5" title="AMISTD_3:func.5">ExecTree</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="ami_1.html#K27" title="AMI_1:func.27">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="amistd_1.html#K5" title="AMISTD_1:func.5">il.</a> <font color="Maroon" title="c2">S</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E6:52_5"/>
<span class="p1">(<span class="default"><a href="amistd_3.html#K5" title="AMISTD_3:func.5">ExecTree</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="amistd_1.html#K5" title="AMISTD_1:func.5">il.</a> <font color="Maroon" title="c2">S</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="ami_1.html#D38" title="AMI_1:def.38">AMI_1:def 38</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E14:52"/>
 <a href="amistd_3.html#K5" title="AMISTD_3:func.5">ExecTree</a> <span class="p1">(<span class="default"><a href="amistd_2.html#K6" title="AMISTD_2:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="amistd_3.html#K1" title="AMISTD_3:func.1">TrivialInfiniteTree</a>  <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <span class="p1">(<span class="default"><a href="amistd_1.html#K5" title="AMISTD_1:func.5">il.</a> <font color="Maroon" title="c2">S</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
 <b>by </b><i><a class="txt" href="amistd_3.html#E12:52"><i><font color="Green" title="E36">A51</font></i></a>, <a class="ref" href="funcop_1.html#T17" title="FUNCOP_1:th.17">FUNCOP_1:17</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
