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

<b>let </b><font color="Maroon" title="c1">n</font> be  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/><b>let </b><font color="Maroon" title="c2">N</font> be  <a href="setfam_1.html#V1" title="SETFAM_1:attr.1">with_non-empty_elements</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c3">S</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> non <a href="ami_1.html#V2" title="AMI_1:attr.2">void</a> <a href="ami_1.html#V5" title="AMI_1:attr.5">IC-Ins-separated</a> <a href="ami_1.html#V7" title="AMI_1:attr.7">definite</a> <a href="amistd_1.html#V4" title="AMISTD_1:attr.4">standard</a>  <a href="ami_1.html#L1" title="AMI_1:struct.1">AMI-Struct</a> of <font color="Maroon" title="c2">N</font>;<br/><b>let </b><font color="Maroon" title="c4">F</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="ami_1.html#U2" title="AMI_1:sel.2">Instruction-Locations</a> of <font color="Maroon" title="c3">S</font>;<br/>







<b>assume </b><a NAME="E1:50"/><i><font color="Green" title="E26">A1</font></i>: 
<font color="Maroon" title="c4">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"><a href="amistd_1.html#K5" title="AMISTD_1:func.5">il.</a> <font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c1">n</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/>

<a NAME="E2:50"/><b>then </b><i><font color="Green" title="E27">A2</font></i>: 
 <a href="amistd_3.html#K3" title="AMISTD_3:func.3">LocNums</a> <font color="Maroon" title="c4">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"><font color="Maroon" title="c1">n</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="amistd_3.html#T26" target="_self" title="AMISTD_3:th.26">Th26</a></i>;<br/>
<a NAME="E3:50"/>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">n</font></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="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:50_1"/>
<font color="Maroon" title="c5">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"><font color="Maroon" title="c1">n</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/>

<a NAME="E2:50_1"/><b>then </b>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:50_1"/>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>


</div><b>end;</b></div>
<b>then </b><i><font color="Green" title="E28">A3</font></i>: <span class="p1">(<span class="default"><a href="wellord1.html#K3" title="WELLORD1:func.3">canonical_isomorphism_of</a> <span class="p2">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p3">(<span class="default"><a href="wellord2.html#K2" title="WELLORD2:func.2">order_type_of</a> <span class="p4">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p5"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">n</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></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"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">n</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></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>  = 
<span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c1">n</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> 
<b>by </b><i><a class="ref" href="amistd_3.html#T15" target="_self" title="AMISTD_3:th.15">Th15</a></i>
<br/>.= 
<font color="Maroon" title="c1">n</font>
<b>by </b><i><a class="ref" href="cqc_lang.html#T6" title="CQC_LANG:th.6">CQC_LANG:6</a></i>
;<br/>
<a NAME="E5:50"/><i><font color="Green" title="E29">A4</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="amistd_3.html#K4" title="AMISTD_3:func.4">LocSeq</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c4">F</font>
 
<b>by </b><i><a class="ref" href="amistd_3.html#D4" target="_self" title="AMISTD_3:def.4">Def4</a></i>;<br/>
<a NAME="E6:50"/><i><font color="Green" title="E30">A5</font></i>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c4">F</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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="amistd_3.html#E1:50"><i><font color="Green" title="E26">A1</font></i></a>, <a class="ref" href="card_1.html#T50" title="CARD_1:th.50">CARD_1:50</a>, <a class="ref" href="card_5.html#T1" title="CARD_5:th.1">CARD_5:1</a></i>;<br/>
<a NAME="E7:50"/><i><font color="Green" title="E31">A6</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <span class="p2">(<span class="default"><a href="amistd_1.html#K5" title="AMISTD_1:func.5">il.</a> <font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="cqc_lang.html#T5" title="CQC_LANG:th.5">CQC_LANG:5</a></i>;<br/>
<a NAME="E8:50"/>
for <font color="Olive" title="b1">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="amistd_3.html#K4" title="AMISTD_3:func.4">LocSeq</a> <font color="Maroon" title="c4">F</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="amistd_3.html#K4" title="AMISTD_3:func.4">LocSeq</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <span class="p2">(<span class="default"><a href="amistd_1.html#K5" title="AMISTD_1:func.5">il.</a> <font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:50_3"/><i><font color="Green" title="E32">A7</font></i>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="amistd_3.html#K4" title="AMISTD_3:func.4">LocSeq</a> <font color="Maroon" title="c4">F</font></span>)</span>
 ;<br/>

<a NAME="E2:50_3"/><b>then </b><i><font color="Green" title="E33">A8</font></i>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="amistd_3.html#E5:50"><i><font color="Green" title="E29">A4</font></i></a>, <a class="txt" href="amistd_3.html#E6:50"><i><font color="Green" title="E30">A5</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><a href="amistd_3.html#K4" title="AMISTD_3:func.4">LocSeq</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font> = 
 <a href="amistd_1.html#K5" title="AMISTD_1:func.5">il.</a> <font color="Maroon" title="c3">S</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wellord1.html#K3" title="WELLORD1:func.3">canonical_isomorphism_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="wellord2.html#K2" title="WELLORD2:func.2">order_type_of</a> <span class="p5">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p0">(<span class="default"><a href="amistd_3.html#K3" title="AMISTD_3:func.3">LocNums</a> <font color="Maroon" title="c4">F</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p4">(<span class="default"><a href="amistd_3.html#K3" title="AMISTD_3:func.3">LocNums</a> <font color="Maroon" title="c4">F</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font></span>)</span>
<b>by </b><i><a class="txt" href="amistd_3.html#E5:50"><i><font color="Green" title="E29">A4</font></i></a>, <a class="txt" href="amistd_3.html#E1:50_3"><i><font color="Green" title="E32">A7</font></i></a>, <a class="ref" href="amistd_3.html#D4" target="_self" title="AMISTD_3:def.4">Def4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <span class="p2">(<span class="default"><a href="amistd_1.html#K5" title="AMISTD_1:func.5">il.</a> <font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font>
<b>by </b><i><a class="txt" href="amistd_3.html#E2:50"><i><font color="Green" title="E27">A2</font></i></a>, <a class="txt" href="amistd_3.html#E4:50"><i><font color="Green" title="E28">A3</font></i></a>, <a class="txt" href="amistd_3.html#E2:50_3"><i><font color="Green" title="E33">A8</font></i></a>, <a class="ref" href="cqc_lang.html#T6" title="CQC_LANG:th.6">CQC_LANG:6</a></i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:50"/>
 <a href="amistd_3.html#K4" title="AMISTD_3:func.4">LocSeq</a> <font color="Maroon" title="c4">F</font> <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#K14" title="FUNCOP_1:func.14">.--&gt;</a> <span class="p1">(<span class="default"><a href="amistd_1.html#K5" title="AMISTD_1:func.5">il.</a> <font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c1">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="amistd_3.html#E5:50"><i><font color="Green" title="E29">A4</font></i></a>, <a class="txt" href="amistd_3.html#E6:50"><i><font color="Green" title="E30">A5</font></i></a>, <a class="txt" href="amistd_3.html#E7:50"><i><font color="Green" title="E31">A6</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/>


</div>
