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

<span class="kw">let </span><font color="Maroon" title="c1">N</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="memstr_0.html#V2" title="MEMSTR_0:attr.2">IC-Ins-separated</a>   <a href="extpro_1.html#V3" title="EXTPRO_1:attr.3">halting</a>   <a href="amistd_1.html#V3" title="AMISTD_1:attr.3">standard</a>   <a href="extpro_1.html#L1" title="EXTPRO_1:struct.1">AMI-Struct</a> of <font color="Maroon" title="c1">N</font> holds   <a href="amistd_3.html#K2" title="AMISTD_3:func.2">ExecTree</a> <span class="p1">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1: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="trees_1.html#K8" title="TREES_1:func.8">TrivialInfiniteTree</a> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><br/>

<span class="kw">set </span><font color="Maroon" title="c2">D</font> =  <a href="trees_1.html#K8" title="TREES_1:func.8">TrivialInfiniteTree</a> ;<br/>
<span class="kw">let </span><font color="Maroon" title="c3">S</font> be   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="memstr_0.html#V2" title="MEMSTR_0:attr.2">IC-Ins-separated</a>   <a href="extpro_1.html#V3" title="EXTPRO_1:attr.3">halting</a>   <a href="amistd_1.html#V3" title="AMISTD_1:attr.3">standard</a>   <a href="extpro_1.html#L1" title="EXTPRO_1:struct.1">AMI-Struct</a> of <font color="Maroon" title="c1">N</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="amistd_3.html#K2" title="AMISTD_3:func.2">ExecTree</a> <span class="p1">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="trees_1.html#K8" title="TREES_1:func.8">TrivialInfiniteTree</a> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><br/>

<span class="kw">set </span><font color="Maroon" title="c4">M</font> =  <a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">S</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">E</font> =  <a href="amistd_3.html#K2" title="AMISTD_3:func.2">ExecTree</a> <span class="p1">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">S</font></span>)</span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span><span class="p1">(<span class="default"><a href="amistd_3.html#K2" title="AMISTD_3:func.2">ExecTree</a> <span class="p2">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">S</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">S</font></span>)</span>;<br/>
<span class="kw">defpred </span><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> ] <span class="kw">means </span><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#K2" title="AMISTD_3:func.2">ExecTree</a> <span class="p3">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">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="trees_1.html#K8" title="TREES_1:func.8">TrivialInfiniteTree</a> <a href="trees_2.html#K2" title="TREES_2:func.2">-level</a> $1;<br/>
<a NAME="E1:5"/><span class="lab"><font color="Green" title="E3">A1</font></span>: 
 <a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="compos_1.html#K2" title="COMPOS_1:func.2">halt</a> <font color="Maroon" title="c3">S</font></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="compos_1.html#D10" title="COMPOS_1:def.10">COMPOS_1:def 10</a></span>;<br/>
<a NAME="E2:5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E4">A2</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">S</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"><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 class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>;<br/>
<a NAME="E3:5"/><span class="lab"><font color="Green" title="E5">A3</font></span>: 
<span class="p1">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">S</font></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="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="compos_1.html#K2" title="COMPOS_1:func.2">halt</a> <font color="Maroon" title="c3">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:5"><span class="lab"><font color="Green" title="E3">A1</font></span></a>, <a class="ref" href="funcop_1.html#T72" title="FUNCOP_1:th.72">FUNCOP_1:72</a></span>;<br/>
<a NAME="E4:5"/><span class="lab"><font color="Green" title="E6">A4</font></span>: 
 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#K2" title="AMISTD_3:func.2">ExecTree</a> <span class="p2">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">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="compos_1.html#K2" title="COMPOS_1:func.2">halt</a> <font color="Maroon" title="c3">S</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="amistd_3.html#K2" title="AMISTD_3:func.2">ExecTree</a> <span class="p4">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">S</font></span>)</span></span>)</span> <a href="trees_2.html#K3" title="TREES_2:func.3">.</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="5_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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#K2" title="AMISTD_3:func.2">ExecTree</a> <span class="p2">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">S</font></span>)</span></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="compos_1.html#K2" title="COMPOS_1:func.2">halt</a> <font color="Maroon" title="c3">S</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="amistd_3.html#K2" title="AMISTD_3:func.2">ExecTree</a> <span class="p4">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">S</font></span>)</span></span>)</span> <a href="trees_2.html#K3" title="TREES_2:func.3">.</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/>

<span class="kw">reconsider </span><font color="Maroon" title="c7">Et</font> = <span class="p1">(<span class="default"><a href="amistd_3.html#K2" title="AMISTD_3:func.2">ExecTree</a> <span class="p2">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">S</font></span>)</span></span>)</span> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c6">t</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/>
<a NAME="E2:5_1"/>
 <a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> (<span class="p1">(<span class="default"><a href="compos_1.html#K2" title="COMPOS_1:func.2">halt</a> <font color="Maroon" title="c3">S</font></span>)</span>,<font color="Maroon" title="c7">Et</font>) <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"><font color="Maroon" title="c7">Et</font></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="amistd_1.html#T2" title="AMISTD_1:th.2">AMISTD_1:2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:5_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="compos_1.html#K2" title="COMPOS_1:func.2">halt</a> <font color="Maroon" title="c3">S</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="amistd_3.html#K2" title="AMISTD_3:func.2">ExecTree</a> <span class="p4">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">S</font></span>)</span></span>)</span> <a href="trees_2.html#K3" title="TREES_2:func.3">.</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 class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T30" title="CARD_1:th.30">CARD_1:30</a>, <a class="ref" href="card_1.html#T49" title="CARD_1:th.49">CARD_1:49</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E5:5"/><span class="lab"><font color="Green" title="E7">A5</font></span>: 
 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#K2" title="AMISTD_3:func.2">ExecTree</a> <span class="p2">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">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#K2" title="AMISTD_3:func.2">ExecTree</a> <span class="p2">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">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="5_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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#K2" title="AMISTD_3:func.2">ExecTree</a> <span class="p2">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">S</font></span>)</span></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K2" title="AMISTD_3:func.2">ExecTree</a> <span class="p2">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">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/>

<span class="kw">assume </span><a NAME="E1:5_2"/><span class="lab"><font color="Green" title="E8">A6</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">f</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K2" title="AMISTD_3:func.2">ExecTree</a> <span class="p2">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">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/>

<a NAME="E2:5_2"/><span class="lab"><font color="Green" title="E9">A7</font></span>: 
<span class="p1">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">S</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="amistd_3.html#K2" title="AMISTD_3:func.2">ExecTree</a> <span class="p3">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">S</font></span>)</span></span>)</span> <a href="trees_2.html#K3" title="TREES_2:func.3">.</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="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">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#K2" title="AMISTD_3:func.2">ExecTree</a> <span class="p3">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">S</font></span>)</span></span>)</span> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c6">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:5_2"><span class="lab"><font color="Green" title="E8">A6</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">Ef</font> = <span class="p1">(<span class="default"><a href="amistd_3.html#K2" title="AMISTD_3:func.2">ExecTree</a> <span class="p2">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">S</font></span>)</span></span>)</span> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c6">f</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/>
<a NAME="E4:5_2"/><span class="lab"><font color="Green" title="E10">A8</font></span>: 
<span class="p1">(<span class="default"><a href="amistd_3.html#K2" title="AMISTD_3:func.2">ExecTree</a> <span class="p2">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">S</font></span>)</span></span>)</span> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:5"><span class="lab"><font color="Green" title="E4">A2</font></span></a>, <a class="txt" href="#E1:5_2"><span class="lab"><font color="Green" title="E8">A6</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E5:5_2"/><span class="kw">then </span>
 <a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> (<span class="p1">(<span class="default"><a href="compos_1.html#K2" title="COMPOS_1:func.2">halt</a> <font color="Maroon" title="c3">S</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="amistd_3.html#K2" title="AMISTD_3:func.2">ExecTree</a> <span class="p3">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">S</font></span>)</span></span>)</span> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c6">f</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"><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 class="kw">by</span> <span class="lab"><a class="ref" href="amistd_1.html#T2" title="AMISTD_1:th.2">AMISTD_1:2</a></span>;<br/>
<a NAME="E6:5_2"/><span class="kw">then </span>
 <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_1.html#K1" title="AMISTD_1:func.1">NIC</a> (<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">S</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="amistd_3.html#K2" title="AMISTD_3:func.2">ExecTree</a> <span class="p2">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">S</font></span>)</span></span>)</span> <a href="trees_2.html#K3" title="TREES_2:func.3">.</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#K2" title="AMISTD_3:func.2">ExecTree</a> <span class="p1">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">S</font></span>)</span></span>)</span> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c6">f</font></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_1.html#K1" title="AMISTD_1:func.1">NIC</a> (<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">S</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="amistd_3.html#K2" title="AMISTD_3:func.2">ExecTree</a> <span class="p0">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">S</font></span>)</span></span>)</span> <a href="trees_2.html#K3" title="TREES_2:func.3">.</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#K2" title="AMISTD_3:func.2">ExecTree</a> <span class="p5">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">S</font></span>)</span></span>)</span> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c6">f</font></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> <font color="Maroon" title="c7">Ef</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:5"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="txt" href="#E4:5_2"><span class="lab"><font color="Green" title="E10">A8</font></span></a>, <a class="txt" href="#E2:5_2"><span class="lab"><font color="Green" title="E9">A7</font></span></a>, <a class="ref" href="card_5.html#T38" title="CARD_5:th.38">CARD_5:38</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E11">A9</font></span>: <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_1.html#K1" title="AMISTD_1:func.1">NIC</a> (<span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">S</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="amistd_3.html#K2" title="AMISTD_3:func.2">ExecTree</a> <span class="p3">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">S</font></span>)</span></span>)</span> <a href="trees_2.html#K3" title="TREES_2:func.3">.</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#K2" title="AMISTD_3:func.2">ExecTree</a> <span class="p2">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">S</font></span>)</span></span>)</span> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c6">f</font></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_1.html#K1" title="AMISTD_1:func.1">NIC</a> (<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">S</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="amistd_3.html#K2" title="AMISTD_3:func.2">ExecTree</a> <span class="p1">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">S</font></span>)</span></span>)</span> <a href="trees_2.html#K3" title="TREES_2:func.3">.</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#K2" title="AMISTD_3:func.2">ExecTree</a> <span class="p0">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">S</font></span>)</span></span>)</span> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c6">f</font></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> = 
<font color="Maroon" title="c7">Ef</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T72" title="FUNCOP_1:th.72">FUNCOP_1:72</a></span>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:5"><span class="lab"><font color="Green" title="E4">A2</font></span></a>, <a class="txt" href="#E1:5_2"><span class="lab"><font color="Green" title="E8">A6</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>
;<br/>
<a NAME="E8:5_2"/><span class="lab"><font color="Green" title="E12">A10</font></span>: 
 <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="compos_1.html#K2" title="COMPOS_1:func.2">halt</a> <font color="Maroon" title="c3">S</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="amistd_3.html#K2" title="AMISTD_3:func.2">ExecTree</a> <span class="p4">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">S</font></span>)</span></span>)</span> <a href="trees_2.html#K3" title="TREES_2:func.3">.</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:5"><span class="lab"><font color="Green" title="E6">A4</font></span></a></span>;<br/>
<a NAME="E9:5_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E13">A11</font></span>: 
 <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"><span class="p3">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">S</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="amistd_3.html#K2" title="AMISTD_3:func.2">ExecTree</a> <span class="p5">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">S</font></span>)</span></span>)</span> <a href="trees_2.html#K3" title="TREES_2:func.3">.</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#K2" title="AMISTD_3:func.2">ExecTree</a> <span class="p4">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">S</font></span>)</span></span>)</span> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c6">f</font></span>)</span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:5"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="txt" href="#E4:5_2"><span class="lab"><font color="Green" title="E10">A8</font></span></a>, <a class="txt" href="#E2:5_2"><span class="lab"><font color="Green" title="E9">A7</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E10:5_2"/><span class="lab"><font color="Green" title="E14">A12</font></span>: 
 <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"><span class="p3">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">S</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="amistd_3.html#K2" title="AMISTD_3:func.2">ExecTree</a> <span class="p5">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">S</font></span>)</span></span>)</span> <a href="trees_2.html#K3" title="TREES_2:func.3">.</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#K2" title="AMISTD_3:func.2">ExecTree</a> <span class="p4">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">S</font></span>)</span></span>)</span> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c6">f</font></span>)</span>)</span>)</span> </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="amistd_3.html#D2" target="_self" title="AMISTD_3:def.2">Def4</a></span>;<br/>
<a NAME="E11:5_2"/><span class="lab"><font color="Green" title="E15">A13</font></span>: 
 <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="5_2_2"><span class="kw">proof </span></a><div class="add">

<div><span class="kw">hereby </span> <span class="comment"><font color="firebrick">:: according to </font></span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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"><span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/><span class="kw">assume </span><a NAME="E1:5_2_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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/><span class="kw">then </span><span class="kw">consider </span><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> <span class="kw"> such that </span><br/><a NAME="E3:5_2_2_1"/><span class="lab"><font color="Green" title="E16">A14</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E4:5_2_2_1"/><span class="lab"><font color="Green" title="E17">A15</font></span>: 
<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"><span class="p3">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">S</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="amistd_3.html#K2" title="AMISTD_3:func.2">ExecTree</a> <span class="p5">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">S</font></span>)</span></span>)</span> <a href="trees_2.html#K3" title="TREES_2:func.3">.</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#K2" title="AMISTD_3:func.2">ExecTree</a> <span class="p4">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">S</font></span>)</span></span>)</span> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c6">f</font></span>)</span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:5_2"><span class="lab"><font color="Green" title="E14">A12</font></span></a></span>;<br/><a NAME="E5:5_2_2_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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:5"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="txt" href="#E8:5_2"><span class="lab"><font color="Green" title="E12">A10</font></span></a>, <a class="txt" href="#E4:5_2"><span class="lab"><font color="Green" title="E10">A8</font></span></a>, <a class="txt" href="#E2:5_2"><span class="lab"><font color="Green" title="E9">A7</font></span></a>, <a class="txt" href="#E4:5_2_2_1"><span class="lab"><font color="Green" title="E17">A15</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E6:5_2_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:5_2_2_1"><span class="lab"><font color="Green" title="E16">A14</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div>

<span class="kw">let </span><font color="Maroon" title="c8">s</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">assume </span><a NAME="E2:5_2_2"/>
<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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="E3:5_2_2"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:5_2_2"/>
<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 class="kw">by</span> <span class="lab"><a class="txt" href="#E10:5_2"><span class="lab"><font color="Green" title="E14">A12</font></span></a>, <a class="txt" href="#E9:5_2"><span class="lab"><font color="Green" title="E13">A11</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">let </span><font color="Maroon" title="c8">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="c8">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#K2" title="AMISTD_3:func.2">ExecTree</a> <span class="p2">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">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="c8">a</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>] )</span><br/>

<span class="kw">assume </span><a NAME="E12:5_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="c8">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#K2" title="AMISTD_3:func.2">ExecTree</a> <span class="p2">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">S</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="c8">a</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>]</span><br/>

<a NAME="E13:5_2"/><span class="kw">then </span>
<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="c8">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="trees_2.html#T12" title="TREES_2:th.12">TREES_2:12</a></span>;<br/>
<a NAME="E14:5_2"/><span class="kw">then </span>
<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="c8">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:5_2"><span class="lab"><font color="Green" title="E15">A13</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<span class="kw">then </span><span class="p1">(<span class="default"><a href="amistd_3.html#K2" title="AMISTD_3:func.2">ExecTree</a> <span class="p2">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">S</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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="c8">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#K1" title="AMISTD_3:func.1">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"><span class="p4">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">S</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="amistd_3.html#K2" title="AMISTD_3:func.2">ExecTree</a> <span class="p0">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">S</font></span>)</span></span>)</span> <a href="trees_2.html#K3" title="TREES_2:func.3">.</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#K2" title="AMISTD_3:func.2">ExecTree</a> <span class="p5">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">S</font></span>)</span></span>)</span> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c6">f</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c3">S</font>)</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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:5_2"><span class="lab"><font color="Green" title="E13">A11</font></span></a>, <a class="ref" href="amistd_3.html#D2" target="_self" title="AMISTD_3:def.2">Def4</a></span>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:5_2"><span class="lab"><font color="Green" title="E13">A11</font></span></a>, <a class="txt" href="#E7:5_2"><span class="lab"><font color="Green" title="E11">A9</font></span></a>, <a class="ref" href="amistd_3.html#D1" target="_self" title="AMISTD_3:def.1">Def3</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E16:5_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="c8">a</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:5"><span class="lab"><font color="Green" title="E4">A2</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E6:5"/>
<span class="p1">(<span class="default"><a href="amistd_3.html#K2" title="AMISTD_3:func.2">ExecTree</a> <span class="p2">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">S</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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="valued_1.html#K64" title="VALUED_1:func.64">FirstLoc</a> <span class="p1">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">S</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="amistd_3.html#D2" target="_self" title="AMISTD_3:def.2">Def4</a></span>;<br/>
<a NAME="E7:5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E8">A16</font></span>: 
<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>]
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T33" title="VALUED_1:th.33">VALUED_1:33</a></span>;<br/>
<a NAME="E8:5"/><span class="lab"><font color="Green" title="E9">A17</font></span>: 
 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#K2" title="AMISTD_3:func.2">ExecTree</a> <span class="p2">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">S</font></span>)</span></span>)</span> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">f</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="hilbert2.html#S1" title="HILBERT2:sch.1">HILBERT2:sch 1</a>(<a class="txt" href="#E7:5"><span class="lab"><font color="Green" title="E8">A16</font></span></a>, <a class="txt" href="#E5:5"><span class="lab"><font color="Green" title="E7">A5</font></span></a>);<br/></span>
<a NAME="E9:5"/><span class="lab"><font color="Green" title="E10">A18</font></span>: 
 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#K2" title="AMISTD_3:func.2">ExecTree</a> <span class="p2">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">S</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="amistd_3.html#K2" title="AMISTD_3:func.2">ExecTree</a> <span class="p2">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">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="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K2" title="AMISTD_3:func.2">ExecTree</a> <span class="p2">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">S</font></span>)</span></span>)</span> implies <span class="p1">(<span class="default"><a href="amistd_3.html#K2" title="AMISTD_3:func.2">ExecTree</a> <span class="p2">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">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="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:5_3"/>
<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#K2" title="AMISTD_3:func.2">ExecTree</a> <span class="p2">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">S</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="amistd_3.html#K2" title="AMISTD_3:func.2">ExecTree</a> <span class="p2">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">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="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><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#K2" title="AMISTD_3:func.2">ExecTree</a> <span class="p2">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">S</font></span>)</span></span>)</span> ;<br/>
<a NAME="E3:5_3"/>
<span class="p1">(<span class="default"><a href="amistd_3.html#K2" title="AMISTD_3:func.2">ExecTree</a> <span class="p2">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">S</font></span>)</span></span>)</span> <a href="trees_2.html#K3" title="TREES_2:func.3">.</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="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">S</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:5"><span class="lab"><font color="Green" title="E9">A17</font></span></a></span>;<br/>
<a NAME="E4:5_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="amistd_3.html#K2" title="AMISTD_3:func.2">ExecTree</a> <span class="p2">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">S</font></span>)</span></span>)</span> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E2:5"><span class="lab"><font color="Green" title="E4">A2</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:5_3"/>
<span class="p1">(<span class="default"><a href="amistd_3.html#K2" title="AMISTD_3:func.2">ExecTree</a> <span class="p2">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">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="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E10:5"/><span class="lab"><font color="Green" title="E11">A19</font></span>: 
 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="5_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">set </span><font color="Maroon" title="c7">f0</font> = <font color="Maroon" title="c6">n</font> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>;<br/>
<span class="kw">set </span><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#K5" title="FINSEQ_2:func.5">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>;<br/>
<a NAME="E1:5_4"/><span class="lab"><font color="Green" title="E12">A20</font></span>: 
<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#K2" title="AMISTD_3:func.2">ExecTree</a> <span class="p3">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">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#K2" title="AMISTD_3:func.2">ExecTree</a> <span class="p2">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="trees_2.html#D6" title="TREES_2:def.6">TREES_2:def 6</a></span>;<br/>
<a NAME="E2:5_4"/><span class="lab"><font color="Green" title="E13">A21</font></span>: 
 <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#K5" title="FINSEQ_2:func.5">|-&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
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#D7" title="CARD_1:def.7">CARD_1:def 7</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:5_4"/><span class="lab"><font color="Green" title="E14">A22</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c6">n</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<a NAME="E4:5_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#K2" title="AMISTD_3:func.2">ExecTree</a> <span class="p3">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">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#K5" title="FINSEQ_2:func.5">|-&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="5_4_1"><span class="kw">proof </span></a><div class="add">

<div><span class="kw">hereby </span> <span class="comment"><font color="firebrick">:: according to </font></span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K5" title="FINSEQ_2:func.5">|-&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#K2" title="AMISTD_3:func.2">ExecTree</a> <span class="p3">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">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"><span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K2" title="AMISTD_3:func.2">ExecTree</a> <span class="p3">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">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#K5" title="FINSEQ_2:func.5">|-&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/><span class="kw">assume </span><a NAME="E1:5_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#K2" title="AMISTD_3:func.2">ExecTree</a> <span class="p3">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K5" title="FINSEQ_2:func.5">|-&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/><span class="kw">then </span><span class="kw">consider </span><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#K2" title="AMISTD_3:func.2">ExecTree</a> <span class="p2">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">S</font></span>)</span></span>)</span><span class="kw"> such that </span><br/><a NAME="E3:5_4_1_1"/><span class="lab"><font color="Green" title="E15">A23</font></span>: 
<font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">t1</font>
 <span class="kw">and </span><br/><a NAME="E4:5_4_1_1"/><span class="lab"><font color="Green" title="E16">A24</font></span>: 
 <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
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:5_4"><span class="lab"><font color="Green" title="E12">A20</font></span></a></span>;<br/><span class="kw">reconsider </span><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#K2" title="AMISTD_3:func.2">ExecTree</a> <span class="p2">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">S</font></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T59" title="RELAT_1:th.59">RELAT_1:59</a>, <a class="ref" href="trees_1.html#T20" title="TREES_1:th.20">TREES_1:20</a></span>;<br/><a NAME="E6:5_4_1_1"/><span class="lab"><font color="Green" title="E17">A25</font></span>: 
 <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"><span class="p3">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">S</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="amistd_3.html#K2" title="AMISTD_3:func.2">ExecTree</a> <span class="p5">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">S</font></span>)</span></span>)</span> <a href="trees_2.html#K3" title="TREES_2:func.3">.</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#K2" title="AMISTD_3:func.2">ExecTree</a> <span class="p4">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">S</font></span>)</span></span>)</span> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c11">t0</font></span>)</span>)</span>)</span> </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="amistd_3.html#D2" target="_self" title="AMISTD_3:def.2">Def4</a></span>;<br/><a NAME="E7:5_4_1_1"/>
<span class="p1">(<span class="default"><a href="amistd_3.html#K2" title="AMISTD_3:func.2">ExecTree</a> <span class="p2">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">S</font></span>)</span></span>)</span> <a href="trees_2.html#K3" title="TREES_2:func.3">.</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="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">S</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:5"><span class="lab"><font color="Green" title="E9">A17</font></span></a></span>;<br/><a NAME="E8:5_4_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E18">A26</font></span>: 
<span class="p1">(<span class="default"><a href="amistd_3.html#K2" title="AMISTD_3:func.2">ExecTree</a> <span class="p2">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">S</font></span>)</span></span>)</span> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c11">t0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:5"><span class="lab"><font color="Green" title="E4">A2</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E9:5_4_1_1"/><span class="lab"><font color="Green" title="E19">A27</font></span>: 
(  <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="compos_1.html#K2" title="COMPOS_1:func.2">halt</a> <font color="Maroon" title="c3">S</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="amistd_3.html#K2" title="AMISTD_3:func.2">ExecTree</a> <span class="p4">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">S</font></span>)</span></span>)</span> <a href="trees_2.html#K3" title="TREES_2:func.3">.</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> &amp; <span class="p1">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">S</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="amistd_3.html#K2" title="AMISTD_3:func.2">ExecTree</a> <span class="p3">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">S</font></span>)</span></span>)</span> <a href="trees_2.html#K3" title="TREES_2:func.3">.</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="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">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#K2" title="AMISTD_3:func.2">ExecTree</a> <span class="p3">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">S</font></span>)</span></span>)</span> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c11">t0</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:5"><span class="lab"><font color="Green" title="E6">A4</font></span></a>, <a class="txt" href="#E8:5"><span class="lab"><font color="Green" title="E9">A17</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>;<br/><a NAME="E10:5_4_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A28</font></span>: 
 <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"><span class="p3">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">S</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="amistd_3.html#K2" title="AMISTD_3:func.2">ExecTree</a> <span class="p5">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">S</font></span>)</span></span>)</span> <a href="trees_2.html#K3" title="TREES_2:func.3">.</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#K2" title="AMISTD_3:func.2">ExecTree</a> <span class="p4">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">S</font></span>)</span></span>)</span> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c11">t0</font></span>)</span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:5"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="txt" href="#E8:5_4_1_1"><span class="lab"><font color="Green" title="E18">A26</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E11:5_4_1_1"/><span class="lab"><font color="Green" title="E21">A29</font></span>: 
 <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="5_4_1_1_1"><span class="kw">proof </span></a><div class="add">

<div><span class="kw">hereby </span> <span class="comment"><font color="firebrick">:: according to </font></span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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"><span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/><span class="kw">assume </span><a NAME="E1:5_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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/><span class="kw">then </span><span class="kw">consider </span><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> <span class="kw"> such that </span><br/><a NAME="E3:5_4_1_1_1_1"/><span class="lab"><font color="Green" title="E22">A30</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E4:5_4_1_1_1_1"/><span class="lab"><font color="Green" title="E23">A31</font></span>: 
<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"><span class="p3">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">S</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="amistd_3.html#K2" title="AMISTD_3:func.2">ExecTree</a> <span class="p5">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">S</font></span>)</span></span>)</span> <a href="trees_2.html#K3" title="TREES_2:func.3">.</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#K2" title="AMISTD_3:func.2">ExecTree</a> <span class="p4">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">S</font></span>)</span></span>)</span> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c11">t0</font></span>)</span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:5_4_1_1"><span class="lab"><font color="Green" title="E17">A25</font></span></a></span>;<br/><a NAME="E5:5_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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:5"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="txt" href="#E8:5_4_1_1"><span class="lab"><font color="Green" title="E18">A26</font></span></a>, <a class="txt" href="#E9:5_4_1_1"><span class="lab"><font color="Green" title="E19">A27</font></span></a>, <a class="txt" href="#E4:5_4_1_1_1_1"><span class="lab"><font color="Green" title="E23">A31</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E6:5_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:5_4_1_1_1_1"><span class="lab"><font color="Green" title="E22">A30</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div>

<span class="kw">let </span><font color="Maroon" title="c12">s</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">assume </span><a NAME="E2:5_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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="E3:5_4_1_1_1"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:5_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:5_4_1_1"><span class="lab"><font color="Green" title="E17">A25</font></span></a>, <a class="txt" href="#E10:5_4_1_1"><span class="lab"><font color="Green" title="E20">A28</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E12:5_4_1_1"/>
( <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>  &amp; <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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:5_4_1_1"><span class="lab"><font color="Green" title="E16">A24</font></span></a>, <a class="ref" href="finseq_3.html#T55" title="FINSEQ_3:th.55">FINSEQ_3:55</a>, <a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/><a NAME="E13:5_4_1_1"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="trees_2.html#T12" title="TREES_2:th.12">TREES_2:12</a></span>;<br/><a NAME="E14:5_4_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A32</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:5_4_1_1"><span class="lab"><font color="Green" title="E21">A29</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E15:5_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
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/><a NAME="E16:5_4_1_1"/><span class="kw">then </span>
 <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T5" title="FINSEQ_1:th.5">FINSEQ_1:5</a></span>;<br/><a NAME="E17:5_4_1_1"/><span class="kw">then </span>
 <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:5_4_1_1"><span class="lab"><font color="Green" title="E16">A24</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E18:5_4_1_1"/><span class="kw">then </span>
 <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T62" title="RELAT_1:th.62">RELAT_1:62</a></span>;<br/><a NAME="E19:5_4_1_1"/><span class="kw">then </span>
( <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#K2" title="AMISTD_3:func.2">ExecTree</a> <span class="p3">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">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#K2" title="AMISTD_3:func.2">ExecTree</a> <span class="p2">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">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>  &amp;  <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> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a>, <a class="ref" href="trees_2.html#D6" title="TREES_2:def.6">TREES_2:def 6</a></span>;<br/><a NAME="E20:5_4_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A33</font></span>: 
<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#K2" title="AMISTD_3:func.2">ExecTree</a> <span class="p3">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">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>
 ;<br/><a NAME="E21:5_4_1_1"/><span class="lab"><font color="Green" title="E24">A34</font></span>: 
<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#K2" title="AMISTD_3:func.2">ExecTree</a> <span class="p3">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">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#K5" title="FINSEQ_2:func.5">|-&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 class="kw">by</span> <span class="lab"><a class="txt" href="#E3:5_4"><span class="lab"><font color="Green" title="E14">A22</font></span></a>, <a class="ref" href="trees_2.html#T39" title="TREES_2:th.39">TREES_2:39</a></span>;<br/><a NAME="E22:5_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#K5" title="FINSEQ_2:func.5">|-&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="5_4_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K5" title="FINSEQ_2:func.5">|-&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/>

<span class="kw">assume </span><a NAME="E1:5_4_1_1_2"/>
( 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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K5" title="FINSEQ_2:func.5">|-&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="E2:5_4_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A35</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:5_4_1_1"><span class="lab"><font color="Green" title="E16">A24</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_4_1_1_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_4_1_1_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:5_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><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:5_4_1_1_2"><span class="lab"><font color="Green" title="E25">A35</font></span></a>, <a class="ref" href="finseq_2.html#T7" title="FINSEQ_2:th.7">FINSEQ_2:7</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_4_1_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:5_4_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E26">A36</font></span>: 
<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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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"><span class="kw">hence </span><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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T49" title="FUNCT_1:th.49">FUNCT_1:49</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&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 class="kw">by</span> <span class="lab"><a class="txt" href="#E21:5_4_1_1"><span class="lab"><font color="Green" title="E24">A34</font></span></a>, <a class="txt" href="#E20:5_4_1_1"><span class="lab"><font color="Green" title="E23">A33</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:5_4_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E26">A36</font></span></a>, <a class="ref" href="funcop_1.html#T7" title="FUNCOP_1:th.7">FUNCOP_1:7</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_4_1_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:5_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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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"><span class="kw">hence </span><a NAME="E2:5_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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:5_4_1_1"><span class="lab"><font color="Green" title="E22">A32</font></span></a>, <a class="ref" href="finseq_2.html#T17" title="FINSEQ_2:th.17">FINSEQ_2:17</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:5_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#K5" title="FINSEQ_2:func.5">|-&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 class="kw">by</span> <span class="lab"><a class="txt" href="#E2:5_4_1_1_2"><span class="lab"><font color="Green" title="E25">A35</font></span></a>, <a class="ref" href="funcop_1.html#T7" title="FUNCOP_1:th.7">FUNCOP_1:7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E23:5_4_1_1"/><span class="kw">then </span>
<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#K5" title="FINSEQ_2:func.5">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:5_4"><span class="lab"><font color="Green" title="E13">A21</font></span></a>, <a class="txt" href="#E4:5_4_1_1"><span class="lab"><font color="Green" title="E16">A24</font></span></a>, <a class="ref" href="finseq_1.html#T14" title="FINSEQ_1:th.14">FINSEQ_1:14</a></span>;<br/><span class="kw">hence </span><a NAME="E24:5_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#K5" title="FINSEQ_2:func.5">|-&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 class="kw">by</span> <span class="lab"><a class="txt" href="#E3:5_4_1_1"><span class="lab"><font color="Green" title="E15">A23</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div>

<span class="kw">defpred </span><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> ] <span class="kw">means </span>$1 <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&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#K2" title="AMISTD_3:func.2">ExecTree</a> <span class="p2">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">S</font></span>)</span></span>)</span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c9">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K5" title="FINSEQ_2:func.5">|-&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#K2" title="AMISTD_3:func.2">ExecTree</a> <span class="p3">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">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="E2:5_4_1"/><span class="lab"><font color="Green" title="E15">A37</font></span>: 
 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="5_4_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">assume </span><a NAME="E1:5_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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">t</font> = <font color="Maroon" title="c10">n</font> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&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#K2" title="AMISTD_3:func.2">ExecTree</a> <span class="p2">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">S</font></span>)</span></span>)</span> ;<br/>
<a NAME="E3:5_4_1_2"/><span class="lab"><font color="Green" title="E16">A38</font></span>: 
 <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"><span class="p3">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">S</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="amistd_3.html#K2" title="AMISTD_3:func.2">ExecTree</a> <span class="p5">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">S</font></span>)</span></span>)</span> <a href="trees_2.html#K3" title="TREES_2:func.3">.</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#K2" title="AMISTD_3:func.2">ExecTree</a> <span class="p4">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">S</font></span>)</span></span>)</span> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c11">t</font></span>)</span>)</span>)</span> </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="amistd_3.html#D2" target="_self" title="AMISTD_3:def.2">Def4</a></span>;<br/>
<a NAME="E4:5_4_1_2"/>
<span class="p1">(<span class="default"><a href="amistd_3.html#K2" title="AMISTD_3:func.2">ExecTree</a> <span class="p2">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">S</font></span>)</span></span>)</span> <a href="trees_2.html#K3" title="TREES_2:func.3">.</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="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">S</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:5"><span class="lab"><font color="Green" title="E9">A17</font></span></a></span>;<br/>
<a NAME="E5:5_4_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A39</font></span>: 
<span class="p1">(<span class="default"><a href="amistd_3.html#K2" title="AMISTD_3:func.2">ExecTree</a> <span class="p2">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">S</font></span>)</span></span>)</span> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c11">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:5"><span class="lab"><font color="Green" title="E4">A2</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E6:5_4_1_2"/>
(  <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="compos_1.html#K2" title="COMPOS_1:func.2">halt</a> <font color="Maroon" title="c3">S</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="amistd_3.html#K2" title="AMISTD_3:func.2">ExecTree</a> <span class="p4">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">S</font></span>)</span></span>)</span> <a href="trees_2.html#K3" title="TREES_2:func.3">.</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> &amp; <span class="p1">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">S</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="amistd_3.html#K2" title="AMISTD_3:func.2">ExecTree</a> <span class="p3">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">S</font></span>)</span></span>)</span> <a href="trees_2.html#K3" title="TREES_2:func.3">.</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="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">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#K2" title="AMISTD_3:func.2">ExecTree</a> <span class="p3">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">S</font></span>)</span></span>)</span> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c11">t</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:5"><span class="lab"><font color="Green" title="E6">A4</font></span></a>, <a class="txt" href="#E8:5"><span class="lab"><font color="Green" title="E9">A17</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>;<br/>
<a NAME="E7:5_4_1_2"/><span class="kw">then </span>
 <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"><span class="p3">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">S</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="amistd_3.html#K2" title="AMISTD_3:func.2">ExecTree</a> <span class="p5">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">S</font></span>)</span></span>)</span> <a href="trees_2.html#K3" title="TREES_2:func.3">.</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#K2" title="AMISTD_3:func.2">ExecTree</a> <span class="p4">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">S</font></span>)</span></span>)</span> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c11">t</font></span>)</span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:5"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="txt" href="#E5:5_4_1_2"><span class="lab"><font color="Green" title="E17">A39</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E8:5_4_1_2"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:5_4_1_2"><span class="lab"><font color="Green" title="E16">A38</font></span></a></span>;<br/>
<a NAME="E9:5_4_1_2"/><span class="kw">then </span>
<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#K2" title="AMISTD_3:func.2">ExecTree</a> <span class="p2">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">S</font></span>)</span></span>)</span>
 
;<br/>
<span class="kw">hence </span><a NAME="E10:5_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]
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T60" title="FINSEQ_2:th.60">FINSEQ_2:60</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E3:5_4_1"/><span class="lab"><font color="Green" title="E16">A40</font></span>: 
<font color="Maroon">S<sub>3</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="trees_1.html#T22" title="TREES_1:th.22">TREES_1:22</a></span>;<br/>
<a NAME="E4:5_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>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E3:5_4_1"><span class="lab"><font color="Green" title="E16">A40</font></span></a>, <a class="txt" href="#E2:5_4_1"><span class="lab"><font color="Green" title="E15">A37</font></span></a>);<br/></span>
<a NAME="E5:5_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A41</font></span>: 
<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#K5" title="FINSEQ_2:func.5">|-&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#K2" title="AMISTD_3:func.2">ExecTree</a> <span class="p2">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">S</font></span>)</span></span>)</span>
 
;<br/>
<span class="kw">assume </span><a NAME="E6:5_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#K5" title="FINSEQ_2:func.5">|-&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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K2" title="AMISTD_3:func.2">ExecTree</a> <span class="p3">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">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="E7:5_4_1"/><span class="kw">then </span>
<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#K5" title="FINSEQ_2:func.5">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:5_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#K2" title="AMISTD_3:func.2">ExecTree</a> <span class="p3">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E1:5_4"><span class="lab"><font color="Green" title="E12">A20</font></span></a>, <a class="txt" href="#E2:5_4"><span class="lab"><font color="Green" title="E13">A21</font></span></a>, <a class="txt" href="#E5:5_4_1"><span class="lab"><font color="Green" title="E17">A41</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:5_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]
 <span class="kw">by</span> <span class="lab"><a class="ref" href="trees_2.html#T39" title="TREES_2:th.39">TREES_2:39</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<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#K2" title="AMISTD_3:func.2">ExecTree</a> <span class="p3">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="trees_9.html#T44" title="TREES_9:th.44">TREES_9:44</a></span>
<br/>.= 
<a href="trees_1.html#K8" title="TREES_1:func.8">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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="trees_9.html#T44" title="TREES_9:th.44">TREES_9:44</a></span>
;<br/>
<a NAME="E12:5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E12">A42</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
;<br/>
<a NAME="E13:5"/>
 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>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E12:5"><span class="lab"><font color="Green" title="E12">A42</font></span></a>, <a class="txt" href="#E10:5"><span class="lab"><font color="Green" title="E11">A19</font></span></a>);<br/></span>
<a NAME="E14:5"/><span class="kw">then </span>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="amistd_3.html#K2" title="AMISTD_3:func.2">ExecTree</a> <span class="p2">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">S</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_1.html#K8" title="TREES_1:func.8">TrivialInfiniteTree</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="trees_2.html#T38" title="TREES_2:th.38">TREES_2:38</a></span>;<br/>
<span class="kw">hence </span><a NAME="E15:5"/>
 <a href="amistd_3.html#K2" title="AMISTD_3:func.2">ExecTree</a> <span class="p1">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c3">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="trees_1.html#K8" title="TREES_1:func.8">TrivialInfiniteTree</a> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:5"><span class="lab"><font color="Green" title="E10">A18</font></span></a>, <a class="ref" href="funcop_1.html#T11" title="FUNCOP_1:th.11">FUNCOP_1:11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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