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

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="ami_1.html#D29">AMI_1:def 29</a><br/>

<b>assume </b><a NAME="E1:79"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <a href="scmbsort.html#K4" target="_self">Sorting-Function</a> 
 ;<br/>

<b>then consider </b><font color="Maroon">w</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K4">INT</a> <b> such that </b><br/><a NAME="E3:79"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K6">fsloc</a> 0</span>)</span> <a href="scmbsort.html#K1" target="_self">.--&gt;</a> <font color="Maroon">w</font>
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">s</font> = <font color="Maroon">x</font> as    <a href="ami_1.html#M1">FinPartState</a> of  <a href="scmfsa_2.html#K1">SCM+FSA</a>  <b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">p</font> =  <a href="scmfsa6a.html#K3">Initialized</a> <a href="scmbsort.html#K3" target="_self">Bubble-Sort-Algorithm</a> ;<br/>
<a NAME="E5:79"/><i><font color="Green">E42</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="scmfsa_2.html#K6">fsloc</a> 0</span>)</span></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/>
<b>take </b>
<font color="Maroon">s</font>
;<br/>

<b>thus </b><a NAME="E6:79"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 ;<br/>

<a NAME="E7:79"/><i><font color="Green">E47</font></i>: 
<span class="p1">(<span class="default"><a href="scmfsa6a.html#K3">Initialized</a> <a href="scmbsort.html#K3" target="_self">Bubble-Sort-Algorithm</a> </span>)</span> <a href="ami_1.html#K17">+*</a> <font color="Maroon">s</font> is <a href="ami_1.html#V11">autonomic</a>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E8:79"/><i><font color="Green">E48</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="scmfsa6a.html#K3">Initialized</a> <a href="scmbsort.html#K3" target="_self">Bubble-Sort-Algorithm</a> </span>)</span> <a href="xboole_0.html#R1">misses</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">s</font>
 
<b>by </b><i>, </i>;<br/>
<div><i><font color="Green">E55</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="79_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">t</font> be   <a href="ami_1.html#NM4">State</a> of <a href="scmfsa_2.html#K1">SCM+FSA</a> ;<br/><b>assume </b><a NAME="E1:79_1"/><i><font color="Green">E60</font></i>: 
<span class="p1">(<span class="default"><a href="scmfsa6a.html#K3">Initialized</a> <a href="scmbsort.html#K3" target="_self">Bubble-Sort-Algorithm</a> </span>)</span> <a href="ami_1.html#K17">+*</a> <font color="Maroon">s</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">t</font>
 ;<br/><b>set </b><font color="Maroon">bf</font> =  <a href="scmbsort.html#K2" target="_self">bubble-sort</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K6">fsloc</a> 0</span>)</span>;<br/><a NAME="E2:79_1"/>
 <a href="scmfsa6a.html#K3">Initialized</a> <a href="scmbsort.html#K3" target="_self">Bubble-Sort-Algorithm</a>  <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="scmfsa6a.html#K3">Initialized</a> <a href="scmbsort.html#K3" target="_self">Bubble-Sort-Algorithm</a> </span>)</span> <a href="ami_1.html#K17">+*</a> <font color="Maroon">s</font>
 <b>by </b><i>, <a class="ref" href="funct_4.html#T33">FUNCT_4:33</a></i>;<br/><a NAME="E3:79_1"/><b>then </b>
 <a href="scmfsa6a.html#K3">Initialized</a> <a href="scmbsort.html#K3" target="_self">Bubble-Sort-Algorithm</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">t</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E4:79_1"/><b>then </b><i><font color="Green">E62</font></i>: 
 <a href="scmfsa6a.html#K3">Initialized</a> <span class="p1">(<span class="default"><a href="scmbsort.html#K2" target="_self">bubble-sort</a> <span class="p2">(<span class="default"><a href="scmfsa_2.html#K6">fsloc</a> 0</span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">t</font>
 <b>by </b><i/>;<br/><a NAME="E5:79_1"/>
 <a href="scmfsa6a.html#K3">Initialized</a> <span class="p1">(<span class="default"><a href="scmbsort.html#K2" target="_self">bubble-sort</a> <span class="p2">(<span class="default"><a href="scmfsa_2.html#K6">fsloc</a> 0</span>)</span></span>)</span> is <a href="ami_1.html#V12">halting</a>
 <b>by </b><i><a class="txt" href="scmbsort.html#E17:79"><i><font color="Green">E66</font></i></a>, <a class="ref" href="scm_halt.html#D2">SCM_HALT:def 2</a></i>;<br/><b>hence </b><a NAME="E6:79_1"/>
<font color="Maroon">t</font> is <a href="ami_1.html#V9">halting</a>
 <b>by </b><i>, <a class="ref" href="ami_1.html#D26">AMI_1:def 26</a></i>;<br/></div><b>end;</b></div>
<a NAME="E10:79"/><b>then </b><i><font color="Green">E63</font></i>: 
<span class="p1">(<span class="default"><a href="scmfsa6a.html#K3">Initialized</a> <a href="scmbsort.html#K3" target="_self">Bubble-Sort-Algorithm</a> </span>)</span> <a href="ami_1.html#K17">+*</a> <font color="Maroon">s</font> is <a href="ami_1.html#V12">halting</a>
 
<b>by </b><i><a class="ref" href="ami_1.html#D26">AMI_1:def 26</a></i>;<br/>
<b>thus </b><a NAME="E11:79"/>
<span class="p1">(<span class="default"><a href="scmfsa6a.html#K3">Initialized</a> <a href="scmbsort.html#K3" target="_self">Bubble-Sort-Algorithm</a> </span>)</span> <a href="ami_1.html#K17">+*</a> <font color="Maroon">s</font> is   <a href="ami_1.html#NM6">pre-program</a> of  <a href="scmfsa_2.html#K1">SCM+FSA</a> 
 <b>by </b><i>, , , <a class="ref" href="ami_1.html#D26">AMI_1:def 26</a></i>;<br/>

<b>consider </b><font color="Maroon">z</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E13:79"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">w</font>,<font color="Maroon">z</font> <a href="rfinseq.html#R1">are_fiberwise_equipotent</a>  &amp; <font color="Maroon">z</font> is <a href="rfinseq.html#V1">non-increasing</a> &amp; <font color="Maroon">z</font> is    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K4">INT</a>  &amp; <a href="scmbsort.html#K4" target="_self">Sorting-Function</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K6">fsloc</a> 0</span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">z</font> )
 <b>by </b><i>, <a class="txt" href="scmbsort.html#E9:79"><i><font color="Green">E55</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">t</font> being   <a href="ami_1.html#NM4">State</a> of <a href="scmfsa_2.html#K1">SCM+FSA</a> <b> such that </b><br/><a NAME="E15:79"/><i><font color="Green">E65</font></i>: 
<span class="p1">(<span class="default"><a href="scmfsa6a.html#K3">Initialized</a> <a href="scmbsort.html#K3" target="_self">Bubble-Sort-Algorithm</a> </span>)</span> <a href="ami_1.html#K17">+*</a> <font color="Maroon">s</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">t</font>
 <b>by </b><i><a class="ref" href="ami_3.html#T39">AMI_3:39</a></i>;<br/>
<b>consider </b><font color="Maroon">u</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E17:79"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">w</font>,<font color="Maroon">u</font> <a href="rfinseq.html#R1">are_fiberwise_equipotent</a>  &amp; <font color="Maroon">u</font> is <a href="rfinseq.html#V1">non-increasing</a> &amp; <font color="Maroon">u</font> is    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K4">INT</a>  &amp; <span class="p1">(<span class="default"><a href="ami_1.html#K12">Result</a> <font color="Maroon">t</font></span>)</span> <a href="scmfsa_2.html#K21">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K6">fsloc</a> 0</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> )
 <b>by </b><i>, , </i>;<br/>
<a NAME="E18:79"/>
<font color="Maroon">u</font>,<font color="Maroon">z</font> <a href="rfinseq.html#R1">are_fiberwise_equipotent</a> 
 
<b>by </b><i>, , <a class="ref" href="rfinseq.html#T2">RFINSEQ:2</a></i>;<br/>
<a NAME="E19:79"/><b>then </b><i><font color="Green">E67</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font>
 
<b>by </b><i>, , <a class="ref" href="rfinseq.html#T36">RFINSEQ:36</a></i>;<br/>
<a NAME="E20:79"/>
 <a href="scmfsa_2.html#K6">fsloc</a> 0 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <a href="scmfsa_2.html#K1">SCM+FSA</a> 
 
;<br/>
<a NAME="E21:79"/><b>then </b>
 <a href="scmfsa_2.html#K6">fsloc</a> 0 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="ami_1.html#K12">Result</a> <font color="Maroon">t</font></span>)</span>
 
<b>by </b><i><a class="ref" href="ami_3.html#T36">AMI_3:36</a></i>;<br/>
<a NAME="E22:79"/><b>then </b><i><font color="Green">E68</font></i>: 
<a href="scmbsort.html#K4" target="_self">Sorting-Function</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> <a href="tarski.html#R1">c=</a>  <a href="ami_1.html#K12">Result</a> <font color="Maroon">t</font>
 
<b>by </b><i>, , , <a class="ref" href="ami_3.html#T27">AMI_3:27</a></i>;<br/>
<a NAME="E23:79"/>
<font color="Maroon">s</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="scmfsa6a.html#K3">Initialized</a> <a href="scmbsort.html#K3" target="_self">Bubble-Sort-Algorithm</a> </span>)</span> <a href="ami_1.html#K17">+*</a> <font color="Maroon">s</font>
 
<b>by </b><i><a class="ref" href="funct_4.html#T26">FUNCT_4:26</a></i>;<br/>
<a NAME="E24:79"/><b>then </b><i><font color="Green">E69</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">s</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa6a.html#K3">Initialized</a> <a href="scmbsort.html#K3" target="_self">Bubble-Sort-Algorithm</a> </span>)</span> <a href="ami_1.html#K17">+*</a> <font color="Maroon">s</font></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T25">RELAT_1:25</a></i>;<br/>
<a NAME="E25:79"/><i><font color="Green">E70</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_2.html#K6">fsloc</a> 0</span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="scmfsa_2.html#K6">fsloc</a> 0</span>)</span></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="E26:79"/>
 <a href="ami_1.html#K18">Result</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa6a.html#K3">Initialized</a> <a href="scmbsort.html#K3" target="_self">Bubble-Sort-Algorithm</a> </span>)</span> <a href="ami_1.html#K17">+*</a> <font color="Maroon">s</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K12">Result</a> <font color="Maroon">t</font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmfsa6a.html#K3">Initialized</a> <a href="scmbsort.html#K3" target="_self">Bubble-Sort-Algorithm</a> </span>)</span> <a href="ami_1.html#K17">+*</a> <font color="Maroon">s</font></span>)</span></span>)</span>
 
<b>by </b><i>, , , <a class="ref" href="ami_1.html#D28">AMI_1:def 28</a></i>;<br/>
<b>hence </b><a NAME="E27:79"/>
<a href="scmbsort.html#K4" target="_self">Sorting-Function</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> <a href="tarski.html#R1">c=</a>  <a href="ami_1.html#K18">Result</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa6a.html#K3">Initialized</a> <a href="scmbsort.html#K3" target="_self">Bubble-Sort-Algorithm</a> </span>)</span> <a href="ami_1.html#K17">+*</a> <font color="Maroon">s</font></span>)</span>
 <b>by </b><i>, , , , , <a class="ref" href="relat_1.html#T186">RELAT_1:186</a></i>;<br/>


</div>
