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

<b>let </b><font color="Maroon" title="c1">k</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#V3" title="STRUCT_0:attr.3">empty</a> non <a href="ami_1.html#V2" title="AMI_1:attr.2">void</a> <a href="ami_1.html#V4" title="AMI_1:attr.4">halting</a> <a href="ami_1.html#V5" title="AMI_1:attr.5">IC-Ins-separated</a> <a href="ami_1.html#V8" title="AMI_1:attr.8">definite</a> <a href="ami_1.html#V10" title="AMI_1:attr.10">realistic</a> <a href="amistd_1.html#V4" title="AMISTD_1:attr.4">standard</a> <a href="amistd_2.html#V6" title="AMISTD_2:attr.6">without_implicit_jumps</a> <a href="amistd_2.html#V8" title="AMISTD_2:attr.8">regular</a>  <a href="ami_1.html#L1" title="AMI_1:struct.1">AMI-Struct</a> of <font color="Maroon" title="c2">N</font>;<br/>





<i><font color="Green" title="E73">A1</font></i>:  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="amistd_2.html#K9" title="AMISTD_2:func.9">IncAddr</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K7" title="AMISTD_2:func.7">Stop</a> <font color="Maroon" title="c3">S</font></span>)</span>,<font color="Maroon" title="c1">k</font></span>)</span> = 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="amistd_2.html#K7" title="AMISTD_2:func.7">Stop</a> <font color="Maroon" title="c3">S</font></span>)</span>
<b>by </b><i><a class="ref" href="amistd_2.html#D15" target="_self" title="AMISTD_2:def.15">Def15</a></i>
<br/>.= 
<span class="p1"><a href="realset1.html#K5" title="REALSET1:func.5">{</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>,0</span>)</span></span><a href="realset1.html#K5" title="REALSET1:func.5">}</a></span>
<b>by </b><i><a class="txt" href="amistd_2.html#E44"><i><font color="Green" title="E34">Lm7</font></i></a></i>
;<br/>
<a NAME="E2:133"/><i><font color="Green" title="E74">A2</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="amistd_2.html#K7" title="AMISTD_2:func.7">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="realset1.html#K5" title="REALSET1:func.5">{</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>,0</span>)</span></span><a href="realset1.html#K5" title="REALSET1:func.5">}</a></span>
 
<b>by </b><i><a class="txt" href="amistd_2.html#E44"><i><font color="Green" title="E34">Lm7</font></i></a></i>;<br/>
<a NAME="E3:133"/>
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> <span class="p1"><a href="realset1.html#K5" title="REALSET1:func.5">{</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>,0</span>)</span></span><a href="realset1.html#K5" title="REALSET1:func.5">}</a></span> holds <br/><span class="p1">(<span class="default"><a href="amistd_2.html#K9" title="AMISTD_2:func.9">IncAddr</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K7" title="AMISTD_2:func.7">Stop</a> <font color="Maroon" title="c3">S</font></span>)</span>,<font color="Maroon" title="c1">k</font></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> <span class="p1">(<span class="default"><a href="amistd_2.html#K7" title="AMISTD_2:func.7">Stop</a> <font color="Maroon" title="c3">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="133_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:133_2"/><i><font color="Green" title="E75">A3</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="realset1.html#K5" title="REALSET1:func.5">{</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>,0</span>)</span></span><a href="realset1.html#K5" title="REALSET1:func.5">}</a></span>
 ;<br/>

<a NAME="E2:133_2"/><b>then </b><i><font color="Green" title="E76">A4</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="amistd_1.html#K5" title="AMISTD_1:func.5">il.</a> <font color="Maroon" title="c3">S</font>,0
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>then </b><i><font color="Green" title="E77">A5</font></i>:  <a href="ami_5.html#K5" title="AMI_5:func.5">pi</a> <span class="p1">(<span class="default"><a href="amistd_2.html#K7" title="AMISTD_2:func.7">Stop</a> <font color="Maroon" title="c3">S</font></span>)</span>,<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>,0</span>)</span> = 
<span class="p1">(<span class="default"><a href="amistd_2.html#K7" title="AMISTD_2:func.7">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"><a href="amistd_1.html#K5" title="AMISTD_1:func.5">il.</a> <font color="Maroon" title="c3">S</font>,0</span>)</span>
<b>by </b><i><a class="txt" href="amistd_2.html#E2:133"><i><font color="Green" title="E74">A2</font></i></a>, <a class="txt" href="amistd_2.html#E1:133_2"><i><font color="Green" title="E75">A3</font></i></a>, <a class="ref" href="ami_5.html#D5" title="AMI_5:def.5">AMI_5:def 5</a></i>
<br/>.= 
 <a href="ami_1.html#K5" title="AMI_1:func.5">halt</a> <font color="Maroon" title="c3">S</font>
<b>by </b><i><a class="ref" href="cqc_lang.html#T6" title="CQC_LANG:th.6">CQC_LANG:6</a></i>
;<br/>
<b>thus </b><span class="p1">(<span class="default"><a href="amistd_2.html#K9" title="AMISTD_2:func.9">IncAddr</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K7" title="AMISTD_2:func.7">Stop</a> <font color="Maroon" title="c3">S</font></span>)</span>,<font color="Maroon" title="c1">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> = 
 <a href="amistd_2.html#K8" title="AMISTD_2:func.8">IncAddr</a> <span class="p1">(<span class="default"><a href="ami_5.html#K5" title="AMI_5:func.5">pi</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K7" title="AMISTD_2:func.7">Stop</a> <font color="Maroon" title="c3">S</font></span>)</span>,<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>,0</span>)</span></span>)</span>,<font color="Maroon" title="c1">k</font>
<b>by </b><i><a class="txt" href="amistd_2.html#E2:133"><i><font color="Green" title="E74">A2</font></i></a>, <a class="txt" href="amistd_2.html#E1:133_2"><i><font color="Green" title="E75">A3</font></i></a>, <a class="txt" href="amistd_2.html#E2:133_2"><i><font color="Green" title="E76">A4</font></i></a>, <a class="ref" href="amistd_2.html#D15" target="_self" title="AMISTD_2:def.15">Def15</a></i>
<br/>.= 
 <a href="ami_1.html#K5" title="AMI_1:func.5">halt</a> <font color="Maroon" title="c3">S</font>
<b>by </b><i><a class="txt" href="amistd_2.html#E3:133_2"><i><font color="Green" title="E77">A5</font></i></a>, <a class="ref" href="amistd_2.html#T29" target="_self" title="AMISTD_2:th.29">Th29</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="amistd_2.html#K7" title="AMISTD_2:func.7">Stop</a> <font color="Maroon" title="c3">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font>
<b>by </b><i><a class="txt" href="amistd_2.html#E2:133_2"><i><font color="Green" title="E76">A4</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="E4:133"/>
 <a href="amistd_2.html#K9" title="AMISTD_2:func.9">IncAddr</a> <span class="p1">(<span class="default"><a href="amistd_2.html#K7" title="AMISTD_2:func.7">Stop</a> <font color="Maroon" title="c3">S</font></span>)</span>,<font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="amistd_2.html#K7" title="AMISTD_2:func.7">Stop</a> <font color="Maroon" title="c3">S</font>
 <b>by </b><i><a class="txt" href="amistd_2.html#E1:133"><i><font color="Green" title="E73">A1</font></i></a>, <a class="txt" href="amistd_2.html#E2:133"><i><font color="Green" title="E74">A2</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/>


</div>
