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<b>let </b><font color="Maroon" title="c1">I</font> be   <a href="scmfsa6a.html#NM1" title="SCMFSA6A:NM.1">Macro-Instruction</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:31_1_1"/><i><font color="Green" title="E22">A1</font></i>: 
<font color="Maroon" title="c1">I</font> is <a href="scmfsa6b.html#V2" title="SCMFSA6B:attr.2">parahalting</a>
 <b>and </b><br/><a NAME="E2:31_1_1"/><i><font color="Green" title="E23">A2</font></i>: 
<font color="Maroon" title="c1">I</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c2">I</font> = <font color="Maroon" title="c1">I</font> as  <a href="scmfsa6b.html#V2" title="SCMFSA6B:attr.2">parahalting</a> <a href="scmfsa6a.html#NM1" title="SCMFSA6A:NM.1">Macro-Instruction</a> <b>by </b><i><a class="txt" href="scmfsa6b.html#E1:31_1_1"><i><font color="Green" title="E22">A1</font></i></a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of the <a href="ami_1.html#U4" title="AMI_1:sel.4">Instructions</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a>  =  <a href="scmfsa_2.html#K13" title="SCMFSA_2:func.13">goto</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  = 1;<br/>
<b>deffunc </b><font color="Maroon">H<sub>3</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  =  <a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> ;<br/>
<b>consider </b><font color="Maroon" title="c3">S</font> being   <a href="ami_1.html#NM4" title="AMI_1:NM.4">State</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a> <b> such that </b><br/><a NAME="E5:31_1_1"/><i><font color="Green" title="E24">A3</font></i>: 
 <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c3">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0
 <b>and </b><br/><a NAME="E6:31_1_1"/><i><font color="Green" title="E25">A4</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/> ( <font color="Maroon" title="c3">S</font> <a href="ami_1.html#K13" title="AMI_1:func.13">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">i</font>) &amp; <font color="Maroon" title="c3">S</font> <a href="scmfsa_2.html#K20" title="SCMFSA_2:func.20">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">i</font>) &amp; <font color="Maroon" title="c3">S</font> <a href="scmfsa_2.html#K21" title="SCMFSA_2:func.21">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K6" title="SCMFSA_2:func.6">fsloc</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive" title="b1">i</font>) )
 <b>from </b><i><a class="ref" href="scmfsa6a.html#S1" title="SCMFSA6A:sch.1">SCMFSA6A:sch 1</a>();<br/></i>
<a NAME="E7:31_1_1"/><i><font color="Green" title="E26">A5</font></i>: 
<font color="Maroon" title="c2">I</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">S</font>
 
<b>by </b><i><a class="txt" href="scmfsa6b.html#E2:31_1_1"><i><font color="Green" title="E23">A2</font></i></a>, <a class="ref" href="xboole_1.html#T2" title="XBOOLE_1:th.2">XBOOLE_1:2</a></i>;<br/>
<a NAME="E8:31_1_1"/><i><font color="Green" title="E27">A6</font></i>: 
 <a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> 0 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">S</font>
 
<b>by </b><i><a class="ref" href="scmfsa_2.html#T66" title="SCMFSA_2:th.66">SCMFSA_2:66</a></i>;<br/>
<a NAME="E9:31_1_1"/>
<font color="Maroon" title="c3">S</font> <a href="scmfsa_2.html#K20" title="SCMFSA_2:func.20">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> 0</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="scmfsa6b.html#E6:31_1_1"><i><font color="Green" title="E25">A4</font></i></a></i>;<br/>
<a NAME="E10:31_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> 0</span>)</span> <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">.--&gt;</a> 1 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">S</font>
 
<b>by </b><i><a class="txt" href="scmfsa6b.html#E8:31_1_1"><i><font color="Green" title="E27">A6</font></i></a>, <a class="ref" href="scmfsa6a.html#T7" title="SCMFSA6A:th.7">SCMFSA6A:7</a></i>;<br/>
<a NAME="E11:31_1_1"/><b>then </b><i><font color="Green" title="E28">A7</font></i>: 
<font color="Maroon" title="c2">I</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> 0</span>)</span> <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">.--&gt;</a> 1</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">S</font>
 
<b>by </b><i><a class="txt" href="scmfsa6b.html#E7:31_1_1"><i><font color="Green" title="E26">A5</font></i></a>, <a class="ref" href="scmfsa6a.html#T6" title="SCMFSA6A:th.6">SCMFSA6A:6</a></i>;<br/>
<a NAME="E12:31_1_1"/><i><font color="Green" title="E29">A8</font></i>: 
 <a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">S</font>
 
<b>by </b><i><a class="ref" href="ami_5.html#T25" title="AMI_5:th.25">AMI_5:25</a></i>;<br/>
<a NAME="E13:31_1_1"/>
<font color="Maroon" title="c3">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0
 
<b>by </b><i><a class="txt" href="scmfsa6b.html#E5:31_1_1"><i><font color="Green" title="E24">A3</font></i></a></i>;<br/>
<a NAME="E14:31_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a> </span>)</span> <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">.--&gt;</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">S</font>
 
<b>by </b><i><a class="txt" href="scmfsa6b.html#E12:31_1_1"><i><font color="Green" title="E29">A8</font></i></a>, <a class="ref" href="scmfsa6a.html#T7" title="SCMFSA6A:th.7">SCMFSA6A:7</a></i>;<br/>
<a NAME="E15:31_1_1"/><b>then </b><i><font color="Green" title="E30">A9</font></i>: 
 <a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">S</font>
 
;<br/>
<a NAME="E16:31_1_1"/>
 <a href="scmfsa6a.html#K3" title="SCMFSA6A:func.3">Initialized</a> <font color="Maroon" title="c2">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> 0</span>)</span> <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">.--&gt;</a> 1</span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p2">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="scmfsa6a.html#D3" title="SCMFSA6A:def.3">SCMFSA6A:def 3</a></i>;<br/>
<a NAME="E17:31_1_1"/><b>then </b>
 <a href="scmfsa6a.html#K3" title="SCMFSA6A:func.3">Initialized</a> <font color="Maroon" title="c2">I</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">S</font>
 
<b>by </b><i><a class="txt" href="scmfsa6b.html#E11:31_1_1"><i><font color="Green" title="E28">A7</font></i></a>, <a class="txt" href="scmfsa6b.html#E15:31_1_1"><i><font color="Green" title="E30">A9</font></i></a>, <a class="ref" href="scmfsa6a.html#T6" title="SCMFSA6A:th.6">SCMFSA6A:6</a></i>;<br/>
<a NAME="E18:31_1_1"/><b>then </b><i><font color="Green" title="E31">A10</font></i>: 
<font color="Maroon" title="c3">S</font> is <a href="ami_1.html#V9" title="AMI_1:attr.9">halting</a>
 
<b>by </b><i><a class="ref" href="ami_1.html#D26" title="AMI_1:def.26">AMI_1:def 26</a></i>;<br/>
<a NAME="E19:31_1_1"/>
<font color="Maroon" title="c3">S</font> <a href="ami_1.html#K13" title="AMI_1:func.13">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmfsa_2.html#K13" title="SCMFSA_2:func.13">goto</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span>
 
<b>by </b><i><a class="txt" href="scmfsa6b.html#E6:31_1_1"><i><font color="Green" title="E25">A4</font></i></a></i>;<br/>
<b>hence </b><a NAME="E20:31_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="scmfsa6b.html#E5:31_1_1"><i><font color="Green" title="E24">A3</font></i></a>, <a class="txt" href="scmfsa6b.html#E18:31_1_1"><i><font color="Green" title="E31">A10</font></i></a>, <a class="ref" href="scmfsa6b.html#T24" target="_self" title="SCMFSA6B:th.24">Th24</a></i>;<br/>


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