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<b>let </b><font color="Maroon" title="c39">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="ami_1.html#D29" title="AMI_1:def.29">AMI_1:def 29</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c39">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <a href="scmbsort.html#K4" title="SCMBSORT:func.4">Sorting-Function</a>  or  ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K8" title="CARD_3:func.8">sproduct</a> the <a href="ami_1.html#U3" title="AMI_1:sel.3">Object-Kind</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a>  st <br/>( <font color="Maroon" title="c39">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <span class="p1">(<span class="default"><a href="scmfsa6a.html#K4" title="SCMFSA6A:func.4">Initialized</a> <a href="scmbsort.html#K3" title="SCMBSORT:func.3">Bubble-Sort-Algorithm</a> </span>)</span> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <font color="Olive">b<sub>1</sub></font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K8" title="CARD_3:func.8">sproduct</a> the <a href="ami_1.html#U3" title="AMI_1:sel.3">Object-Kind</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a>  &amp; <a href="scmbsort.html#K4" title="SCMBSORT:func.4">Sorting-Function</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ami_1.html#K19" title="AMI_1:func.19">Result</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa6a.html#K4" title="SCMFSA6A:func.4">Initialized</a> <a href="scmbsort.html#K3" title="SCMBSORT:func.3">Bubble-Sort-Algorithm</a> </span>)</span> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> ) )</span><br/>

<b>assume </b><a NAME="E1:67"/>
<font color="Maroon" title="c39">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <a href="scmbsort.html#K4" title="SCMBSORT:func.4">Sorting-Function</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K8" title="CARD_3:func.8">sproduct</a> the <a href="ami_1.html#U3" title="AMI_1:sel.3">Object-Kind</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a>  st <br/>( <font color="Maroon" title="c39">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <span class="p1">(<span class="default"><a href="scmfsa6a.html#K4" title="SCMFSA6A:func.4">Initialized</a> <a href="scmbsort.html#K3" title="SCMBSORT:func.3">Bubble-Sort-Algorithm</a> </span>)</span> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <font color="Olive">b<sub>1</sub></font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K8" title="CARD_3:func.8">sproduct</a> the <a href="ami_1.html#U3" title="AMI_1:sel.3">Object-Kind</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a>  &amp; <a href="scmbsort.html#K4" title="SCMBSORT:func.4">Sorting-Function</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ami_1.html#K19" title="AMI_1:func.19">Result</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa6a.html#K4" title="SCMFSA6A:func.4">Initialized</a> <a href="scmbsort.html#K3" title="SCMBSORT:func.3">Bubble-Sort-Algorithm</a> </span>)</span> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> )</span><br/>

<b>then consider </b><font color="Maroon" title="c40">w</font> being    <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> <b> such that </b><br/><a NAME="E3:67"/><i><font color="Green" title="E66">A1</font></i>: 
<font color="Maroon" title="c39">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K6" title="SCMFSA_2:func.6">fsloc</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="scmbsort.html#K1" title="SCMBSORT:func.1">.--&gt;</a> <font color="Maroon" title="c40">w</font>
 <b>by </b><i><a class="ref" href="scmbsort.html#T60" target="_self" title="SCMBSORT:th.60">Th60</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c41">s</font> = <font color="Maroon" title="c39">x</font> as   <a href="ami_1.html#NM8" title="AMI_1:NM.8">FinPartState</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a>  <b>by </b><i><a class="txt" href="#E3:67"><i><font color="Green" title="E66">A1</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c42">p</font> =  <a href="scmfsa6a.html#K4" title="SCMFSA6A:func.4">Initialized</a> <a href="scmbsort.html#K3" title="SCMBSORT:func.3">Bubble-Sort-Algorithm</a> ;<br/>
<a NAME="E5:67"/><i><font color="Green" title="E67">A2</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c41">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="scmfsa_2.html#K6" title="SCMFSA_2:func.6">fsloc</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="#E3:67"><i><font color="Green" title="E66">A1</font></i></a>, <a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c41">s</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c39">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c41">s</font> &amp; <span class="p1">(<span class="default"><a href="scmfsa6a.html#K4" title="SCMFSA6A:func.4">Initialized</a> <a href="scmbsort.html#K3" title="SCMBSORT:func.3">Bubble-Sort-Algorithm</a> </span>)</span> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <font color="Maroon" title="c41">s</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K8" title="CARD_3:func.8">sproduct</a> the <a href="ami_1.html#U3" title="AMI_1:sel.3">Object-Kind</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a>  &amp; <a href="scmbsort.html#K4" title="SCMBSORT:func.4">Sorting-Function</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c41">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ami_1.html#K19" title="AMI_1:func.19">Result</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa6a.html#K4" title="SCMFSA6A:func.4">Initialized</a> <a href="scmbsort.html#K3" title="SCMBSORT:func.3">Bubble-Sort-Algorithm</a> </span>)</span> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <font color="Maroon" title="c41">s</font></span>)</span> )</span><br/>

<b>thus </b><a NAME="E6:67"/>
<font color="Maroon" title="c39">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c41">s</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="scmfsa6a.html#K4" title="SCMFSA6A:func.4">Initialized</a> <a href="scmbsort.html#K3" title="SCMBSORT:func.3">Bubble-Sort-Algorithm</a> </span>)</span> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <font color="Maroon" title="c41">s</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K8" title="CARD_3:func.8">sproduct</a> the <a href="ami_1.html#U3" title="AMI_1:sel.3">Object-Kind</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a>  &amp; <a href="scmbsort.html#K4" title="SCMBSORT:func.4">Sorting-Function</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c41">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ami_1.html#K19" title="AMI_1:func.19">Result</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa6a.html#K4" title="SCMFSA6A:func.4">Initialized</a> <a href="scmbsort.html#K3" title="SCMBSORT:func.3">Bubble-Sort-Algorithm</a> </span>)</span> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <font color="Maroon" title="c41">s</font></span>)</span> )</span><br/>

<a NAME="E7:67"/><i><font color="Green" title="E68">A3</font></i>: 
<span class="p1">(<span class="default"><a href="scmfsa6a.html#K4" title="SCMFSA6A:func.4">Initialized</a> <a href="scmbsort.html#K3" title="SCMBSORT:func.3">Bubble-Sort-Algorithm</a> </span>)</span> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <font color="Maroon" title="c41">s</font> is <a href="ami_1.html#V10" title="AMI_1:attr.10">autonomic</a>
 
<b>by </b><i><a class="txt" href="#E3:67"><i><font color="Green" title="E66">A1</font></i></a>, <a class="ref" href="scmbsort.html#T68" target="_self" title="SCMBSORT:th.68">Th68</a></i>;<br/>
<a NAME="E8:67"/><i><font color="Green" title="E69">A4</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="scmfsa6a.html#K4" title="SCMFSA6A:func.4">Initialized</a> <a href="scmbsort.html#K3" title="SCMBSORT:func.3">Bubble-Sort-Algorithm</a> </span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c41">s</font>
 
<b>by </b><i><a class="txt" href="#E3:67"><i><font color="Green" title="E66">A1</font></i></a>, <a class="ref" href="scmbsort.html#T46" target="_self" title="SCMBSORT:th.46">Th46</a></i>;<br/>
<div><i><font color="Green" title="E70">A5</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c43">t</font> be   <a href="ami_1.html#NM5" title="AMI_1:NM.5">State</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="scmfsa6a.html#K4" title="SCMFSA6A:func.4">Initialized</a> <a href="scmbsort.html#K3" title="SCMBSORT:func.3">Bubble-Sort-Algorithm</a> </span>)</span> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <font color="Maroon" title="c41">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c43">t</font> implies <font color="Maroon" title="c43">t</font> is <a href="ami_1.html#V8" title="AMI_1:attr.8">halting</a> )</span><br/><b>assume </b><a NAME="E1:67_1"/><i><font color="Green" title="E71">A6</font></i>: 
<span class="p1">(<span class="default"><a href="scmfsa6a.html#K4" title="SCMFSA6A:func.4">Initialized</a> <a href="scmbsort.html#K3" title="SCMBSORT:func.3">Bubble-Sort-Algorithm</a> </span>)</span> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <font color="Maroon" title="c41">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c43">t</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c43">t</font> is <a href="ami_1.html#V8" title="AMI_1:attr.8">halting</a></span><br/><b>set </b><font color="Maroon" title="c44">bf</font> =  <a href="scmbsort.html#K2" title="SCMBSORT:func.2">bubble-sort</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K6" title="SCMFSA_2:func.6">fsloc</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>;<br/><a NAME="E2:67_1"/>
 <a href="scmfsa6a.html#K4" title="SCMFSA6A:func.4">Initialized</a> <a href="scmbsort.html#K3" title="SCMBSORT:func.3">Bubble-Sort-Algorithm</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="scmfsa6a.html#K4" title="SCMFSA6A:func.4">Initialized</a> <a href="scmbsort.html#K3" title="SCMBSORT:func.3">Bubble-Sort-Algorithm</a> </span>)</span> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <font color="Maroon" title="c41">s</font>
 <b>by </b><i><a class="txt" href="#E8:67"><i><font color="Green" title="E69">A4</font></i></a>, <a class="ref" href="funct_4.html#T33" title="FUNCT_4:th.33">FUNCT_4:33</a></i>;<br/><a NAME="E3:67_1"/><b>then </b><i><font color="Green" title="E72">A7</font></i>: 
 <a href="scmfsa6a.html#K4" title="SCMFSA6A:func.4">Initialized</a> <a href="scmbsort.html#K3" title="SCMBSORT:func.3">Bubble-Sort-Algorithm</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c43">t</font>
 <b>by </b><i><a class="txt" href="#E1:67_1"><i><font color="Green" title="E71">A6</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E4:67_1"/>
 <a href="scmfsa6a.html#K4" title="SCMFSA6A:func.4">Initialized</a> <span class="p1">(<span class="default"><a href="scmbsort.html#K2" title="SCMBSORT:func.2">bubble-sort</a> <span class="p2">(<span class="default"><a href="scmfsa_2.html#K6" title="SCMFSA_2:func.6">fsloc</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> is <a href="ami_1.html#V11" title="AMI_1:attr.11">halting</a>
 <b>by </b><i><a class="ref" href="scmbsort.html#T64" target="_self" title="SCMBSORT:th.64">Th64</a></i>;<br/><b>hence </b><a NAME="E5:67_1"/>
<font color="Maroon" title="c43">t</font> is <a href="ami_1.html#V8" title="AMI_1:attr.8">halting</a>
 <b>by </b><i><a class="txt" href="#E3:67_1"><i><font color="Green" title="E72">A7</font></i></a>, <a class="ref" href="ami_1.html#D26" title="AMI_1:def.26">AMI_1:def 26</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E10:67"/><b>then </b><i><font color="Green" title="E71">A8</font></i>: 
<span class="p1">(<span class="default"><a href="scmfsa6a.html#K4" title="SCMFSA6A:func.4">Initialized</a> <a href="scmbsort.html#K3" title="SCMBSORT:func.3">Bubble-Sort-Algorithm</a> </span>)</span> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <font color="Maroon" title="c41">s</font> is <a href="ami_1.html#V11" title="AMI_1:attr.11">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/>
<b>thus </b><a NAME="E11:67"/>
<span class="p1">(<span class="default"><a href="scmfsa6a.html#K4" title="SCMFSA6A:func.4">Initialized</a> <a href="scmbsort.html#K3" title="SCMBSORT:func.3">Bubble-Sort-Algorithm</a> </span>)</span> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <font color="Maroon" title="c41">s</font> is   <a href="ami_1.html#NM9" title="AMI_1:NM.9">pre-program</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a> 
 <b>by </b><i><a class="txt" href="#E3:67"><i><font color="Green" title="E66">A1</font></i></a>, <a class="txt" href="#E9:67"><i><font color="Green" title="E70">A5</font></i></a>, <a class="ref" href="scmbsort.html#T68" target="_self" title="SCMBSORT:th.68">Th68</a>, <a class="ref" href="ami_1.html#D26" title="AMI_1:def.26">AMI_1:def 26</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <a href="scmbsort.html#K4" title="SCMBSORT:func.4">Sorting-Function</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c41">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ami_1.html#K19" title="AMI_1:func.19">Result</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa6a.html#K4" title="SCMFSA6A:func.4">Initialized</a> <a href="scmbsort.html#K3" title="SCMBSORT:func.3">Bubble-Sort-Algorithm</a> </span>)</span> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <font color="Maroon" title="c41">s</font></span>)</span></span><br/>

<b>consider </b><font color="Maroon" title="c43">z</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <b> such that </b><br/><a NAME="E13:67"/><i><font color="Green" title="E72">A9</font></i>: 
( <font color="Maroon" title="c40">w</font>,<font color="Maroon" title="c43">z</font> <a href="classes1.html#R2" title="CLASSES1:pred.2">are_fiberwise_equipotent</a>  &amp; <font color="Maroon" title="c43">z</font> is <a href="valued_0.html#V10" title="VALUED_0:attr.10">non-increasing</a> &amp; <font color="Maroon" title="c43">z</font> is    <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>  &amp; <a href="scmbsort.html#K4" title="SCMBSORT:func.4">Sorting-Function</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c41">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K6" title="SCMFSA_2:func.6">fsloc</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <font color="Maroon" title="c43">z</font> )
 <b>by </b><i><a class="txt" href="#E3:67"><i><font color="Green" title="E66">A1</font></i></a>, <a class="ref" href="scmbsort.html#T61" target="_self" title="SCMBSORT:th.61">Th61</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c44">t</font> being   <a href="ami_1.html#NM5" title="AMI_1:NM.5">State</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a> <b> such that </b><br/><a NAME="E15:67"/><i><font color="Green" title="E73">A10</font></i>: 
<span class="p1">(<span class="default"><a href="scmfsa6a.html#K4" title="SCMFSA6A:func.4">Initialized</a> <a href="scmbsort.html#K3" title="SCMBSORT:func.3">Bubble-Sort-Algorithm</a> </span>)</span> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <font color="Maroon" title="c41">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c44">t</font>
 <b>by </b><i><a class="ref" href="card_3.html#T97" title="CARD_3:th.97">CARD_3:97</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c45">u</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <b> such that </b><br/><a NAME="E17:67"/><i><font color="Green" title="E74">A11</font></i>: 
( <font color="Maroon" title="c40">w</font>,<font color="Maroon" title="c45">u</font> <a href="classes1.html#R2" title="CLASSES1:pred.2">are_fiberwise_equipotent</a>  &amp; <font color="Maroon" title="c45">u</font> is <a href="valued_0.html#V10" title="VALUED_0:attr.10">non-increasing</a> &amp; <font color="Maroon" title="c45">u</font> is    <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>  &amp; <span class="p1">(<span class="default"><a href="ami_1.html#K12" title="AMI_1:func.12">Result</a> <font color="Maroon" title="c44">t</font></span>)</span> <a href="scmfsa_2.html#K20" title="SCMFSA_2:func.20">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K6" title="SCMFSA_2:func.6">fsloc</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c45">u</font> )
 <b>by </b><i><a class="txt" href="#E3:67"><i><font color="Green" title="E66">A1</font></i></a>, <a class="txt" href="#E15:67"><i><font color="Green" title="E73">A10</font></i></a>, <a class="ref" href="scmbsort.html#T67" target="_self" title="SCMBSORT:th.67">Th67</a></i>;<br/>
<a NAME="E18:67"/><i><font color="Green" title="E75">A12</font></i>: 
<font color="Maroon" title="c45">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c43">z</font>
 
<b>by </b><i><a class="txt" href="#E13:67"><i><font color="Green" title="E72">A9</font></i></a>, <a class="txt" href="#E17:67"><i><font color="Green" title="E74">A11</font></i></a>, <a class="ref" href="classes1.html#T84" title="CLASSES1:th.84">CLASSES1:84</a>, <a class="ref" href="rfinseq.html#T36" title="RFINSEQ:th.36">RFINSEQ:36</a></i>;<br/>
<a NAME="E19:67"/>
 <a href="scmfsa_2.html#K6" title="SCMFSA_2:func.6">fsloc</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a> 
 
;<br/>
<a NAME="E20:67"/><b>then </b><i><font color="Green" title="E76">A13</font></i>: 
 <a href="scmfsa_2.html#K6" title="SCMFSA_2:func.6">fsloc</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="ami_1.html#K12" title="AMI_1:func.12">Result</a> <font color="Maroon" title="c44">t</font></span>)</span>
 
<b>by </b><i><a class="ref" href="ami_1.html#T79" title="AMI_1:th.79">AMI_1:79</a></i>;<br/>
<a NAME="E21:67"/>
<font color="Maroon" title="c41">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="scmfsa6a.html#K4" title="SCMFSA6A:func.4">Initialized</a> <a href="scmbsort.html#K3" title="SCMBSORT:func.3">Bubble-Sort-Algorithm</a> </span>)</span> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <font color="Maroon" title="c41">s</font>
 
<b>by </b><i><a class="ref" href="funct_4.html#T26" title="FUNCT_4:th.26">FUNCT_4:26</a></i>;<br/>
<a NAME="E22:67"/><b>then </b><i><font color="Green" title="E77">A14</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c41">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa6a.html#K4" title="SCMFSA6A:func.4">Initialized</a> <a href="scmbsort.html#K3" title="SCMBSORT:func.3">Bubble-Sort-Algorithm</a> </span>)</span> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <font color="Maroon" title="c41">s</font></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T25" title="RELAT_1:th.25">RELAT_1:25</a></i>;<br/>
<a NAME="E23:67"/><i><font color="Green" title="E78">A15</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_2.html#K6" title="SCMFSA_2:func.6">fsloc</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <font color="Maroon" title="c43">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="scmfsa_2.html#K6" title="SCMFSA_2:func.6">fsloc</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></i>;<br/>
<a NAME="E24:67"/>
 <a href="ami_1.html#K19" title="AMI_1:func.19">Result</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa6a.html#K4" title="SCMFSA6A:func.4">Initialized</a> <a href="scmbsort.html#K3" title="SCMBSORT:func.3">Bubble-Sort-Algorithm</a> </span>)</span> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <font color="Maroon" title="c41">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K12" title="AMI_1:func.12">Result</a> <font color="Maroon" title="c44">t</font></span>)</span> <a href="card_3.html#K11" title="CARD_3:func.11">|</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmfsa6a.html#K4" title="SCMFSA6A:func.4">Initialized</a> <a href="scmbsort.html#K3" title="SCMBSORT:func.3">Bubble-Sort-Algorithm</a> </span>)</span> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <font color="Maroon" title="c41">s</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E7:67"><i><font color="Green" title="E68">A3</font></i></a>, <a class="txt" href="#E10:67"><i><font color="Green" title="E71">A8</font></i></a>, <a class="txt" href="#E15:67"><i><font color="Green" title="E73">A10</font></i></a>, <a class="ref" href="ami_1.html#D28" title="AMI_1:def.28">AMI_1:def 28</a></i>;<br/>
<b>hence </b><a NAME="E25:67"/>
<a href="scmbsort.html#K4" title="SCMBSORT:func.4">Sorting-Function</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c41">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ami_1.html#K19" title="AMI_1:func.19">Result</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa6a.html#K4" title="SCMFSA6A:func.4">Initialized</a> <a href="scmbsort.html#K3" title="SCMBSORT:func.3">Bubble-Sort-Algorithm</a> </span>)</span> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <font color="Maroon" title="c41">s</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E5:67"><i><font color="Green" title="E67">A2</font></i></a>, <a class="txt" href="#E13:67"><i><font color="Green" title="E72">A9</font></i></a>, <a class="txt" href="#E17:67"><i><font color="Green" title="E74">A11</font></i></a>, <a class="txt" href="#E18:67"><i><font color="Green" title="E75">A12</font></i></a>, <a class="txt" href="#E20:67"><i><font color="Green" title="E76">A13</font></i></a>, <a class="txt" href="#E22:67"><i><font color="Green" title="E77">A14</font></i></a>, <a class="txt" href="#E23:67"><i><font color="Green" title="E78">A15</font></i></a>, <a class="ref" href="funct_4.html#T90" title="FUNCT_4:th.90">FUNCT_4:90</a>, <a class="ref" href="relat_1.html#T186" title="RELAT_1:th.186">RELAT_1:186</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
