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

<b>let </b><font color="Maroon">loc</font> be   <a href="ami_1.html#NM2">Instruction-Location</a> of <a href="ami_3.html#K1" target="_self">SCM</a> ;<br/>

<b>assume </b><a NAME="E1:26"/><i><font color="Green">E24</font></i>: 
 <a href="ami_3.html#K8" target="_self">goto</a> <font color="Maroon">loc</font> is <a href="ami_1.html#V3">halting</a>
 ;<br/>

<b>reconsider </b><font color="Maroon">V</font> =  <a href="ami_3.html#K8" target="_self">goto</a> <font color="Maroon">loc</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="ami_2.html#K4">SCM-Instr</a>  ;<br/>
<b>reconsider </b><font color="Maroon">a3</font> = <font color="Maroon">loc</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="ami_2.html#K3">SCM-Instr-Loc</a>  ;<br/>
<b>consider </b><font color="Maroon">s</font> being   <a href="ami_2.html#NM1">SCM-State</a>;<br/>
<b>set </b><font color="Maroon">t</font> = <font color="Maroon">s</font> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default">0 <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><a href="ami_2.html#K15">Next</a> <font color="Maroon">a3</font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">f</font> = the <a href="ami_1.html#U5">Object-Kind</a> of <a href="ami_3.html#K1" target="_self">SCM</a> ;<br/>
<a NAME="E5:26"/><i><font color="Green">E25</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="ami_2.html#K15">Next</a> <font color="Maroon">a3</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="E6:26"/><b>then </b>
0 <a href="hidden.html#R2">in</a>  <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="ami_2.html#K15">Next</a> <font color="Maroon">a3</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>then </b><i><font color="Green">E48</font></i>: <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default">0 <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p3">(<span class="default"><a href="ami_2.html#K15">Next</a> <font color="Maroon">a3</font></span>)</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> <span class="p2">(<span class="default"><a href="ami_2.html#K15">Next</a> <font color="Maroon">a3</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> 0
<b>by </b><i><a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
 <a href="ami_2.html#K15">Next</a> <font color="Maroon">a3</font>
<b>by </b><i><a class="ref" href="cqc_lang.html#T6">CQC_LANG:6</a></i>
;<br/>
<a NAME="E8:26"/><b>then </b><i><font color="Green">E49</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default">0 <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p3">(<span class="default"><a href="ami_2.html#K15">Next</a> <font color="Maroon">a3</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">a3</font> <a href="nat_1.html#K1">+</a> 2
 
<b>by </b><i><a class="ref" href="ami_2.html#D15">AMI_2:def 15</a></i>;<br/>
<a NAME="E9:26"/><i><font color="Green">E51</font></i>: 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E10:26"/><i><font color="Green">E57</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <a href="ami_2.html#K5">SCM-OK</a> 
 
<b>by </b><i><a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>
<i><font color="Green">E84</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default">0 <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p3">(<span class="default"><a href="ami_2.html#K15">Next</a> <font color="Maroon">a3</font></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">s</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default">0 <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p3">(<span class="default"><a href="ami_2.html#K15">Next</a> <font color="Maroon">a3</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>
<br/>.= 
<a href="numbers.html#K5">NAT</a>  <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default">0 <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p3">(<span class="default"><a href="ami_2.html#K15">Next</a> <font color="Maroon">a3</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
<a href="numbers.html#K5">NAT</a>  <a href="xboole_0.html#K2">\/</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 href="numbers.html#K5">NAT</a> 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>
;<br/>
<a NAME="E12:26"/><i><font color="Green">E85</font></i>: 
 <a href="relat_1.html#K1">dom</a> the <a href="ami_1.html#U5">Object-Kind</a> of <a href="ami_3.html#K1" target="_self">SCM</a>  <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E13:26"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="ami_1.html#U5">Object-Kind</a> of <a href="ami_3.html#K1" target="_self">SCM</a>  holds <br/><span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default">0 <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p3">(<span class="default"><a href="ami_2.html#K15">Next</a> <font color="Maroon">a3</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="ami_1.html#U5">Object-Kind</a> of <a href="ami_3.html#K1" target="_self">SCM</a>  <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_3"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:26_3"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="ami_1.html#U5">Object-Kind</a> of <a href="ami_3.html#K1" target="_self">SCM</a> 
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:26_3_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_3_1_1"><b>suppose </b></a><a NAME="E1:26_3_1_1"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:26_3_1_1"/><b>then </b>
the <a href="ami_1.html#U5">Object-Kind</a> of <a href="ami_3.html#K1" target="_self">SCM</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="ami_2.html#K3">SCM-Instr-Loc</a> 
 <b>by </b><i><a class="ref" href="ami_2.html#T7">AMI_2:7</a></i>;<br/><b>hence </b><a NAME="E3:26_3_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default">0 <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p3">(<span class="default"><a href="ami_2.html#K15">Next</a> <font color="Maroon">a3</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="ami_1.html#U5">Object-Kind</a> of <a href="ami_3.html#K1" target="_self">SCM</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="txt" href="ami_3.html#E5:26"><i><font color="Green">E25</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_3_1_2"><b>suppose </b></a><a NAME="E1:26_3_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:26_3_1_2"/><b>then </b>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <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="ami_2.html#K15">Next</a> <font color="Maroon">a3</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="ami_3.html#E1:26"><i><font color="Green">E24</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:26_3_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default">0 <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p3">(<span class="default"><a href="ami_2.html#K15">Next</a> <font color="Maroon">a3</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>;<br/><b>hence </b><a NAME="E4:26_3_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default">0 <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p3">(<span class="default"><a href="ami_2.html#K15">Next</a> <font color="Maroon">a3</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="ami_1.html#U5">Object-Kind</a> of <a href="ami_3.html#K1" target="_self">SCM</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">t</font> = <font color="Maroon">s</font> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default">0 <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><a href="ami_2.html#K15">Next</a> <font color="Maroon">a3</font></span>)</span></span>)</span> as   <a href="ami_1.html#NM4">State</a> of <a href="ami_3.html#K1" target="_self">SCM</a>  <b>by </b><i><a class="ref" href="ami_3.html#D2" target="_self">Def2</a>, , <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>
<b>reconsider </b><font color="Maroon">w</font> = <font color="Maroon">t</font> as   <a href="ami_2.html#NM1">SCM-State</a> ;<br/>
<a NAME="E16:26"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">0 <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">loc</font></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="E17:26"/><b>then </b>
0 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">0 <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">loc</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>then </b><i><font color="Green">E88</font></i>: <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default">0 <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">loc</font></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">loc</font></span>)</span> <a href="funct_1.html#K1">.</a> 0
<b>by </b><i><a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
<font color="Maroon">loc</font>
<b>by </b><i><a class="ref" href="cqc_lang.html#T6">CQC_LANG:6</a></i>
;<br/>
<a NAME="E19:26"/><i><font color="Green">E89</font></i>: 
6 is    <a href="subset_1.html#M2">Element</a> of  <a href="gr_cy_1.html#K1">Segm</a> 9
 
<b>by </b><i><a class="ref" href="gr_cy_1.html#T10">GR_CY_1:10</a></i>;<br/>
<font color="Maroon">w</font> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default">0 <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">loc</font></span>)</span> = 
 <a href="ami_2.html#K7">SCM-Chg</a> <font color="Maroon">w</font>,<font color="Maroon">a3</font>
<b>by </b><i><a class="ref" href="ami_2.html#D7">AMI_2:def 7</a></i>
<br/>.= 
 <a href="ami_2.html#K7">SCM-Chg</a> <font color="Maroon">w</font>,<span class="p1">(<span class="default"><font color="Maroon">V</font> <a href="ami_2.html#K11">jump_address</a> </span>)</span>
<b>by </b><i>, <a class="ref" href="ami_2.html#T24">AMI_2:24</a></i>
<br/>.= 
 <a href="ami_2.html#K16">SCM-Exec-Res</a> <font color="Maroon">V</font>,<font color="Maroon">w</font>
<b>by </b><i><a class="ref" href="ami_2.html#D16">AMI_2:def 16</a></i>
<br/>.= 
 <a href="ami_1.html#K4">Exec</a> <span class="p1">(<span class="default"><a href="ami_3.html#K8" target="_self">goto</a> <font color="Maroon">loc</font></span>)</span>,<font color="Maroon">t</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">t</font>
<b>by </b><i>, <a class="ref" href="ami_1.html#D8">AMI_1:def 8</a></i>
;<br/>
<b>hence </b><a NAME="E21:26"/>
contradiction
 <b>by </b><i><a class="ref" href="ami_3.html#T2" target="_self">Th2</a>, </i>;<br/>


</div>
