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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="ami_1.html#NM2">Instruction-Location</a> of <a href="scmfsa_2.html#K1" target="_self">SCM+FSA</a> ;<br/>

<b>assume </b><a NAME="E1:98"/><i><font color="Green">E66</font></i>: 
 <a href="scmfsa_2.html#K13" target="_self">goto</a> <font color="Maroon">c<sub>1</sub></font> is <a href="ami_1.html#V3">halting</a>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>2</sub></font> = <font color="Maroon">c<sub>1</sub></font> as    <a href="subset_1.html#M2">Element</a> of  <a href="scmfsa_1.html#K3">SCM+FSA-Instr-Loc</a>  ;<br/>
<b>consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="scmfsa_1.html#NM1">SCM+FSA-State</a>;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></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="scmfsa_1.html#K17">Next</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> = the <a href="ami_1.html#U5">Object-Kind</a> of <a href="scmfsa_2.html#K1" target="_self">SCM+FSA</a> ;<br/>
<b>consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="subset_1.html#M2">Element</a> of  <a href="scmfsa_1.html#K3">SCM+FSA-Instr-Loc</a> <b> such that </b><br/><a NAME="E5:98"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="scmfsa_1.html#K17">Next</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="scmfsa_1.html#K17">Next</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/>
<b>consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="subset_1.html#M2">Element</a> of  <a href="ami_2.html#K3">SCM-Instr-Loc</a> <b> such that </b><br/><a NAME="E7:98"/><i><font color="Green">E68</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp;  <a href="scmfsa_1.html#K17">Next</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="ami_2.html#K15">Next</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="ref" href="scmfsa_1.html#D15">SCMFSA_1:def 15</a></i>;<br/>
<a NAME="E8:98"/><i><font color="Green">E69</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="scmfsa_1.html#K17">Next</a> <font color="Maroon">c<sub>2</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="E9:98"/><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="scmfsa_1.html#K17">Next</a> <font color="Maroon">c<sub>2</sub></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">E70</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></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="scmfsa_1.html#K17">Next</a> <font color="Maroon">c<sub>2</sub></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="scmfsa_1.html#K17">Next</a> <font color="Maroon">c<sub>2</sub></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="scmfsa_1.html#K17">Next</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="cqc_lang.html#T6">CQC_LANG:6</a></i>
;<br/>
<a NAME="E11:98"/><b>then </b><i><font color="Green">E71</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></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="scmfsa_1.html#K17">Next</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 2
 
<b>by </b><i><a class="txt" href="scmfsa_2.html#E5:98"><i><font color="Green">E67</font></i></a>, <a class="txt" href="scmfsa_2.html#E7:98"><i><font color="Green">E68</font></i></a>, <a class="ref" href="ami_2.html#D15">AMI_2:def 15</a></i>;<br/>
<a NAME="E12:98"/>
0 <a href="hidden.html#R2">in</a>  <a href="numbers.html#K4">INT</a> 
 
<b>by </b><i><a class="ref" href="int_1.html#D2">INT_1:def 2</a></i>;<br/>
<a NAME="E13:98"/><b>then </b><i><font color="Green">E72</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#K4">INT</a> 
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E14:98"/><i><font color="Green">E73</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <a href="scmfsa_1.html#K6">SCM+FSA-OK</a> 
 
<b>by </b><i><a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>
<i><font color="Green">E74</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></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="scmfsa_1.html#K17">Next</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></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="scmfsa_1.html#K17">Next</a> <font color="Maroon">c<sub>2</sub></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#K4">INT</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="scmfsa_1.html#K17">Next</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="scmfsa_2.html#E14:98"><i><font color="Green">E73</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
<a href="numbers.html#K4">INT</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#K4">INT</a> 
<b>by </b><i><a class="txt" href="scmfsa_2.html#E13:98"><i><font color="Green">E72</font></i></a>, <a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>
;<br/>
<a NAME="E16:98"/><i><font color="Green">E75</font></i>: 
 <a href="relset_1.html#K4">dom</a> the <a href="ami_1.html#U5">Object-Kind</a> of <a href="scmfsa_2.html#K1" target="_self">SCM+FSA</a>  <a href="hidden.html#R1">=</a>  <a href="numbers.html#K4">INT</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E17:98"/>
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="relset_1.html#K4">dom</a> the <a href="ami_1.html#U5">Object-Kind</a> of <a href="scmfsa_2.html#K1" target="_self">SCM+FSA</a>  holds <br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></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="scmfsa_1.html#K17">Next</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="ami_1.html#U5">Object-Kind</a> of <a href="scmfsa_2.html#K1" target="_self">SCM+FSA</a>  <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="98_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:98_3"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">c<sub>8</sub></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 <a href="scmfsa_2.html#K1" target="_self">SCM+FSA</a> 
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:98_3_1"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>8</sub></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="98_3_1_1"><b>suppose </b></a><a NAME="E1:98_3_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:98_3_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></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="scmfsa_1.html#K17">Next</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> the <a href="ami_1.html#U5">Object-Kind</a> of <a href="scmfsa_2.html#K1" target="_self">SCM+FSA</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="scmfsa_2.html#E10:98"><i><font color="Green">E70</font></i></a>, <a class="ref" href="scmfsa_1.html#T9">SCMFSA_1:9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_3_1_2"><b>suppose </b></a><a NAME="E1:98_3_1_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:98_3_1_2"/><b>then </b>
not <font color="Maroon">c<sub>8</sub></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="scmfsa_1.html#K17">Next</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="scmfsa_2.html#E8:98"><i><font color="Green">E69</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:98_3_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></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="scmfsa_1.html#K17">Next</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>;<br/><b>hence </b><a NAME="E4:98_3_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></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="scmfsa_1.html#K17">Next</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> the <a href="ami_1.html#U5">Object-Kind</a> of <a href="scmfsa_2.html#K1" target="_self">SCM+FSA</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="scmfsa_2.html#E1:98_3"><i><font color="Green">E76</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">c<sub>8</sub></font> = <font color="Maroon">c<sub>3</sub></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="scmfsa_1.html#K17">Next</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> as   <a href="ami_1.html#NM4">State</a> of <a href="scmfsa_2.html#K1" target="_self">SCM+FSA</a>  <b>by </b><i><a class="txt" href="scmfsa_2.html#E15:98"><i><font color="Green">E74</font></i></a>, <a class="txt" href="scmfsa_2.html#E16:98"><i><font color="Green">E75</font></i></a>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="scmfsa_1.html#NM1">SCM+FSA-State</a> ;<br/>
<a NAME="E20:98"/>
 <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">c<sub>1</sub></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="E21:98"/><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">c<sub>1</sub></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">E76</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></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">c<sub>1</sub></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">c<sub>1</sub></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">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="cqc_lang.html#T6">CQC_LANG:6</a></i>
;<br/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></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">c<sub>1</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> 0 = 
<span class="p1">(<span class="default"><a href="scmfsa_1.html#K7">SCM+FSA-Chg</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> 0
<b>by </b><i><a class="ref" href="scmfsa_1.html#D7">SCMFSA_1:def 7</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="scmfsa_1.html#T20">SCMFSA_1:20</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="ami_1.html#K4">Exec</a> <span class="p2">(<span class="default"><a href="scmfsa_2.html#K13" target="_self">goto</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> 0
<b>by </b><i><a class="ref" href="scmfsa_2.html#T7" target="_self">Th7</a>, <a class="ref" href="scmfsa_2.html#T95" target="_self">Th95</a></i>
<br/>.= 
<font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> 0
<b>by </b><i><a class="txt" href="scmfsa_2.html#E1:98"><i><font color="Green">E66</font></i></a>, <a class="ref" href="ami_1.html#D8">AMI_1:def 8</a></i>
;<br/>
<b>hence </b><a NAME="E24:98"/>
contradiction
 <b>by </b><i><a class="txt" href="scmfsa_2.html#E5:98"><i><font color="Green">E67</font></i></a>, <a class="txt" href="scmfsa_2.html#E7:98"><i><font color="Green">E68</font></i></a>, <a class="txt" href="scmfsa_2.html#E11:98"><i><font color="Green">E71</font></i></a>, <a class="txt" href="scmfsa_2.html#E22:98"><i><font color="Green">E76</font></i></a></i>;<br/>


</div>
