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

<b>let </b><font color="Maroon">R</font> be  <a href="scmring1.html#V1">good</a> <a href="funcsdom.html#NM1">Ring</a>;<br/><b>let </b><font color="Maroon">i1</font> be   <a href="ami_1.html#NM2">Instruction-Location</a> of <span class="p1">(<span class="default"><a href="scmring2.html#K1" target="_self">SCM</a> <font color="Maroon">R</font></span>)</span>;<br/>



<b>assume </b><a NAME="E1:36"/><i><font color="Green">E33</font></i>: 
 <a href="scmring2.html#K9" target="_self">goto</a> <font color="Maroon">i1</font> is <a href="ami_1.html#V3">halting</a>
 ;<br/>

<b>reconsider </b><font color="Maroon">V</font> =  <a href="scmring2.html#K9" target="_self">goto</a> <font color="Maroon">i1</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="scmring1.html#K1">SCM-Instr</a> <font color="Maroon">R</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">i5</font> = <font color="Maroon">i1</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="ami_2.html#K3">SCM-Instr-Loc</a>  <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">s</font> being   <a href="scmring1.html#NM1">SCM-State</a> of <font color="Maroon">R</font>;<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="scmring2.html#K11" target="_self">Next</a> <font color="Maroon">i1</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 <span class="p1">(<span class="default"><a href="scmring2.html#K1" target="_self">SCM</a> <font color="Maroon">R</font></span>)</span>;<br/>
<a NAME="E5:36"/><i><font color="Green">E35</font></i>: 
the <a href="ami_1.html#U5">Object-Kind</a> of <span class="p1">(<span class="default"><a href="scmring2.html#K1" target="_self">SCM</a> <font color="Maroon">R</font></span>)</span> <a href="funct_2.html#R1">=</a>  <a href="scmring1.html#K2">SCM-OK</a> <font color="Maroon">R</font>
 
<b>by </b><i/>;<br/>
<a NAME="E6:36"/><i><font color="Green">E37</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="scmring2.html#K11" target="_self">Next</a> <font color="Maroon">i1</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="E7:36"/><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="scmring2.html#K11" target="_self">Next</a> <font color="Maroon">i1</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">E38</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="scmring2.html#K11" target="_self">Next</a> <font color="Maroon">i1</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="scmring2.html#K11" target="_self">Next</a> <font color="Maroon">i1</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="scmring2.html#K11" target="_self">Next</a> <font color="Maroon">i1</font>
<b>by </b><i><a class="ref" href="cqc_lang.html#T6">CQC_LANG:6</a></i>
<br/>.= 
 <a href="ami_2.html#K15">Next</a> <font color="Maroon">i5</font>
<b>by </b><i/>
;<br/>
<a NAME="E9:36"/><b>then </b><i><font color="Green">E40</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="scmring2.html#K11" target="_self">Next</a> <font color="Maroon">i1</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">i5</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="E10:36"/><i><font color="Green">E62</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="E11:36"/><i><font color="Green">E63</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="scmring1.html#K2">SCM-OK</a> <font color="Maroon">R</font></span>)</span>
 
<b>by </b><i><a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>
<i><font color="Green">E92</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="scmring2.html#K11" target="_self">Next</a> <font color="Maroon">i1</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="scmring2.html#K11" target="_self">Next</a> <font color="Maroon">i1</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="scmring2.html#K11" target="_self">Next</a> <font color="Maroon">i1</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="scmring2.html#E5:36"><i><font color="Green">E35</font></i></a>, <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="E13:36"/><i><font color="Green">E93</font></i>: 
 <a href="relset_1.html#K4">dom</a> the <a href="ami_1.html#U5">Object-Kind</a> of <span class="p1">(<span class="default"><a href="scmring2.html#K1" target="_self">SCM</a> <font color="Maroon">R</font></span>)</span> <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="E14:36"/>
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="relset_1.html#K4">dom</a> the <a href="ami_1.html#U5">Object-Kind</a> of <span class="p1">(<span class="default"><a href="scmring2.html#K1" target="_self">SCM</a> <font color="Maroon">R</font></span>)</span> 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="scmring2.html#K11" target="_self">Next</a> <font color="Maroon">i1</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 <span class="p1">(<span class="default"><a href="scmring2.html#K1" target="_self">SCM</a> <font color="Maroon">R</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_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="E1:36_3"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> the <a href="ami_1.html#U5">Object-Kind</a> of <span class="p1">(<span class="default"><a href="scmring2.html#K1" target="_self">SCM</a> <font color="Maroon">R</font></span>)</span>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:36_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="36_3_1_1"><b>suppose </b></a><a NAME="E1:36_3_1_1"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:36_3_1_1"/><b>then </b>
the <a href="ami_1.html#U5">Object-Kind</a> of <span class="p1">(<span class="default"><a href="scmring2.html#K1" target="_self">SCM</a> <font color="Maroon">R</font></span>)</span> <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="scmring1.html#T2">SCMRING1:2</a></i>;<br/><b>hence </b><a NAME="E3:36_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="scmring2.html#K11" target="_self">Next</a> <font color="Maroon">i1</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 <span class="p1">(<span class="default"><a href="scmring2.html#K1" target="_self">SCM</a> <font color="Maroon">R</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="scmring2.html#D2" target="_self">Def2</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_3_1_2"><b>suppose </b></a><a NAME="E1:36_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="E2:36_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="scmring2.html#K11" target="_self">Next</a> <font color="Maroon">i1</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:36_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="scmring2.html#K11" target="_self">Next</a> <font color="Maroon">i1</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="E4:36_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="scmring2.html#K11" target="_self">Next</a> <font color="Maroon">i1</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 <span class="p1">(<span class="default"><a href="scmring2.html#K1" target="_self">SCM</a> <font color="Maroon">R</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="txt" href="scmring2.html#E8:36"><i><font color="Green">E38</font></i></a>, <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="scmring2.html#K11" target="_self">Next</a> <font color="Maroon">i1</font></span>)</span></span>)</span> as   <a href="ami_1.html#NM4">State</a> of <span class="p1">(<span class="default"><a href="scmring2.html#K1" target="_self">SCM</a> <font color="Maroon">R</font></span>)</span> <b>by </b><i>, <a class="txt" href="scmring2.html#E6:36"><i><font color="Green">E37</font></i></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="scmring1.html#NM1">SCM-State</a> of <font color="Maroon">R</font> <b>by </b><i/>;<br/>
<a NAME="E17:36"/>
 <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">i1</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="E18:36"/><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">i1</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">E96</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">i1</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">i1</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">i1</font>
<b>by </b><i><a class="ref" href="cqc_lang.html#T6">CQC_LANG:6</a></i>
;<br/>
<a NAME="E20:36"/><i><font color="Green">E97</font></i>: 
6 is    <a href="subset_1.html#M2">Element</a> of  <a href="gr_cy_1.html#K1">Segm</a> 8
 
<b>by </b><i><a class="ref" href="gr_cy_1.html#T10">GR_CY_1:10</a></i>;<br/>
<a NAME="E21:36"/><i><font color="Green">E98</font></i>: 
the <a href="ami_1.html#U2">Instruction-Locations</a> of <span class="p1">(<span class="default"><a href="scmring2.html#K1" target="_self">SCM</a> <font color="Maroon">R</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="ami_2.html#K3">SCM-Instr-Loc</a> 
 
<b>by </b><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">i1</font></span>)</span> = 
 <a href="scmring1.html#K4">SCM-Chg</a> <font color="Maroon">w</font>,<font color="Maroon">i5</font>
<b>by </b><i><a class="ref" href="scmring1.html#D5">SCMRING1:def 5</a></i>
<br/>.= 
 <a href="scmring1.html#K4">SCM-Chg</a> <font color="Maroon">w</font>,<span class="p1">(<span class="default"><font color="Maroon">V</font> <a href="scmring1.html#K9">jump_address</a> </span>)</span>
<b>by </b><i>, <a class="ref" href="scmring1.html#T18">SCMRING1:18</a></i>
<br/>.= 
 <a href="scmring1.html#K15">SCM-Exec-Res</a> <font color="Maroon">V</font>,<font color="Maroon">w</font>
<b>by </b><i><a class="txt" href="scmring2.html#E9:36"><i><font color="Green">E40</font></i></a>, <a class="ref" href="scmring1.html#D14">SCMRING1:def 14</a></i>
<br/>.= 
 <a href="ami_1.html#K4">Exec</a> <span class="p1">(<span class="default"><a href="scmring2.html#K9" target="_self">goto</a> <font color="Maroon">i1</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="scmring2.html#D1" target="_self">Def1</a>, <a class="ref" href="ami_1.html#D8">AMI_1:def 8</a></i>
;<br/>
<b>hence </b><a NAME="E23:36"/>
contradiction
 <b>by </b><i><a class="txt" href="scmring2.html#E1:36"><i><font color="Green">E33</font></i></a>, <a class="txt" href="scmring2.html#E8:36"><i><font color="Green">E38</font></i></a></i>;<br/>


</div>
