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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  <a href="setfam_1.html#V1">with_non-empty_elements</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be  non <a href="struct_0.html#V3">empty</a> non <a href="ami_1.html#V2">void</a> <a href="ami_1.html#V5">IC-Ins-separated</a> <a href="ami_1.html#V8">definite</a> <a href="amistd_1.html#V4">standard</a>  <a href="ami_1.html#L1">AMI-Struct</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="subset_1.html#NM2">Subset</a> of the <a href="ami_1.html#U2">Instruction-Locations</a> of <font color="Maroon">c<sub>3</sub></font>;<br/>







<b>assume </b><a NAME="E1:51"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="amistd_1.html#K5">il.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<a NAME="E2:51"/><b>then </b><i><font color="Green">E28</font></i>: 
 <a href="amistd_3.html#K3" target="_self">LocNums</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="amistd_3.html#T26" target="_self">Th26</a></i>;<br/>
<a NAME="E3:51"/>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="ordinal1.html#K5">omega</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:51_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<a NAME="E2:51_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:51_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>


</div><b>end;</b></div>
<b>then </b><i><font color="Green">E29</font></i>: <span class="p1">(<span class="default"><a href="wellord1.html#K3">canonical_isomorphism_of</a> <span class="p2">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <span class="p3">(<span class="default"><a href="wellord2.html#K2">order_type_of</a> <span class="p4">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <span class="p5"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> 0 = 
<span class="p1">(<span class="default">0 <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> 0
<b>by </b><i><a class="ref" href="amistd_3.html#T15" target="_self">Th15</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="cqc_lang.html#T6">CQC_LANG:6</a></i>
;<br/>
<a NAME="E5:51"/><i><font color="Green">E30</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="amistd_3.html#K4" target="_self">LocSeq</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="amistd_3.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E6:51"/><i><font color="Green">E31</font></i>: 
 <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="txt" href="amistd_3.html#E1:51"><i><font color="Green">E27</font></i></a>, <a class="ref" href="card_1.html#T50">CARD_1:50</a>, <a class="ref" href="card_5.html#T1">CARD_5:1</a></i>;<br/>
<a NAME="E7:51"/><i><font color="Green">E32</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">0 <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><a href="amistd_1.html#K5">il.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>;<br/>
<a NAME="E8:51"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="amistd_3.html#K4" target="_self">LocSeq</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="amistd_3.html#K4" target="_self">LocSeq</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">0 <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><a href="amistd_1.html#K5">il.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:51_3"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="amistd_3.html#K4" target="_self">LocSeq</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 ;<br/>

<a NAME="E2:51_3"/><b>then </b><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="txt" href="amistd_3.html#E5:51"><i><font color="Green">E30</font></i></a>, <a class="txt" href="amistd_3.html#E6:51"><i><font color="Green">E31</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><a href="amistd_3.html#K4" target="_self">LocSeq</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> = 
 <a href="amistd_1.html#K5">il.</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wellord1.html#K3">canonical_isomorphism_of</a> <span class="p3">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <span class="p4">(<span class="default"><a href="wellord2.html#K2">order_type_of</a> <span class="p5">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <span class="p0">(<span class="default"><a href="amistd_3.html#K3" target="_self">LocNums</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <span class="p4">(<span class="default"><a href="amistd_3.html#K3" target="_self">LocNums</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="amistd_3.html#E5:51"><i><font color="Green">E30</font></i></a>, <a class="txt" href="amistd_3.html#E1:51_3"><i><font color="Green">E33</font></i></a>, <a class="ref" href="amistd_3.html#D4" target="_self">Def4</a></i>
<br/>.= 
<span class="p1">(<span class="default">0 <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><a href="amistd_1.html#K5">il.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="amistd_3.html#E2:51"><i><font color="Green">E28</font></i></a>, <a class="txt" href="amistd_3.html#E4:51"><i><font color="Green">E29</font></i></a>, <a class="txt" href="amistd_3.html#E2:51_3"><i><font color="Green">E34</font></i></a>, <a class="ref" href="cqc_lang.html#T6">CQC_LANG:6</a></i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:51"/>
 <a href="amistd_3.html#K4" target="_self">LocSeq</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 0 <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p1">(<span class="default"><a href="amistd_1.html#K5">il.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="amistd_3.html#E5:51"><i><font color="Green">E30</font></i></a>, <a class="txt" href="amistd_3.html#E6:51"><i><font color="Green">E31</font></i></a>, <a class="txt" href="amistd_3.html#E7:51"><i><font color="Green">E32</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


</div>
