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

<b>let </b><font color="Maroon" title="c39">s</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"> for <font color="Olive" title="b1">t</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>   st <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> <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="scmbsort.html#K1" title="SCMBSORT:func.1">.--&gt;</a> <font color="Olive" title="b1">t</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c39">s</font> holds <br/> ex <font color="Olive" title="b2">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>  st <br/>( <font color="Olive" title="b1">t</font>,<font color="Olive" title="b2">u</font> <a href="rfinseq.html#R1" title="RFINSEQ:pred.1">are_fiberwise_equipotent</a>  &amp; <font color="Olive" title="b2">u</font> is <a href="seqm_3.html#V4" title="SEQM_3:attr.4">non-increasing</a> &amp; <font color="Olive" title="b2">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="c39">s</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="Olive" title="b2">u</font> )</span><br/><b>let </b><font color="Maroon" title="c40">t</font> be    <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 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> <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="scmbsort.html#K1" title="SCMBSORT:func.1">.--&gt;</a> <font color="Maroon" title="c40">t</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c39">s</font> implies  ex <font color="Olive" title="b1">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>  st <br/>( <font color="Maroon" title="c40">t</font>,<font color="Olive" title="b1">u</font> <a href="rfinseq.html#R1" title="RFINSEQ:pred.1">are_fiberwise_equipotent</a>  &amp; <font color="Olive" title="b1">u</font> is <a href="seqm_3.html#V4" title="SEQM_3:attr.4">non-increasing</a> &amp; <font color="Olive" title="b1">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="c39">s</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="Olive" title="b1">u</font> ) )</span><br/>



<b>set </b><font color="Maroon" title="c41">Ba</font> =  <a href="scmbsort.html#K3" title="SCMBSORT:func.3">Bubble-Sort-Algorithm</a> ;<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/>
<b>set </b><font color="Maroon" title="c43">x</font> = <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">t</font>;<br/>
<b>set </b><font color="Maroon" title="c44">z</font> = <span class="p1">(<span class="default"><a href="scmfsa6b.html#K2" title="SCMFSA6B:func.2">IExec</a> <span class="p2">(<span class="default"><a href="scmbsort.html#K2" title="SCMBSORT:func.2">bubble-sort</a> <span class="p3">(<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>,<font color="Maroon" title="c39">s</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>;<br/>
<b>assume </b><a NAME="E1:67"/><i><font color="Green" title="E65">A1</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> <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="scmbsort.html#K1" title="SCMBSORT:func.1">.--&gt;</a> <font color="Maroon" title="c40">t</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c39">s</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">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>  st <br/>( <font color="Maroon" title="c40">t</font>,<font color="Olive" title="b1">u</font> <a href="rfinseq.html#R1" title="RFINSEQ:pred.1">are_fiberwise_equipotent</a>  &amp; <font color="Olive" title="b1">u</font> is <a href="seqm_3.html#V4" title="SEQM_3:attr.4">non-increasing</a> &amp; <font color="Olive" title="b1">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="c39">s</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="Olive" title="b1">u</font> )</span><br/>

<a NAME="E2:67"/>
 <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="scmbsort.html#K1" title="SCMBSORT:func.1">.--&gt;</a> <font color="Maroon" title="c40">t</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="E3:67"/><b>then </b><i><font color="Green" title="E66">A2</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"><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="scmbsort.html#K1" title="SCMBSORT:func.1">.--&gt;</a> <font color="Maroon" title="c40">t</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E4:67"/><b>then </b>
 <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"><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> <span class="p2">(<span class="default"><span class="p3">(<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">t</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_4.html#T13" title="FUNCT_4:th.13">FUNCT_4:13</a></i>;<br/>
<b>then </b><i><font color="Green" title="E67">A3</font></i>: <font color="Maroon" title="c39">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#K6" title="SCMFSA_2:func.6">fsloc</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> = 
<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> <span class="p2">(<span class="default"><span class="p3">(<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">t</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.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>
<b>by </b><i><a class="txt" href="scmbsort.html#E1:67"><i><font color="Green" title="E65">A1</font></i></a>, <a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</a></i>
<br/>.= 
<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="scmbsort.html#K1" title="SCMBSORT:func.1">.--&gt;</a> <font color="Maroon" title="c40">t</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.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>
<b>by </b><i><a class="txt" href="scmbsort.html#E3:67"><i><font color="Green" title="E66">A2</font></i></a>, <a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></i>
<br/>.= 
<font color="Maroon" title="c40">t</font>
<b>by </b><i><a class="ref" href="funcop_1.html#T87" title="FUNCOP_1:th.87">FUNCOP_1:87</a></i>
;<br/>
<a NAME="E6:67"/><i><font color="Green" title="E68">A4</font></i>: 
( <font color="Maroon" title="c39">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#K6" title="SCMFSA_2:func.6">fsloc</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p1">(<span class="default"><a href="scmfsa6b.html#K2" title="SCMFSA6B:func.2">IExec</a> <span class="p2">(<span class="default"><a href="scmbsort.html#K2" title="SCMBSORT:func.2">bubble-sort</a> <span class="p3">(<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>,<font color="Maroon" title="c39">s</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="rfinseq.html#R1" title="RFINSEQ:pred.1">are_fiberwise_equipotent</a>  &amp; ( for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</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>   st <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c39">s</font> <a href="scmfsa_2.html#K20" title="SCMFSA_2:func.20">.</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> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">j</font> holds <br/>for <font color="Olive" title="b3">x1</font>, <font color="Olive" title="b4">x2</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st <font color="Olive" title="b3">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa6b.html#K2" title="SCMFSA6B:func.2">IExec</a> <span class="p3">(<span class="default"><a href="scmbsort.html#K2" title="SCMBSORT:func.2">bubble-sort</a> <span class="p4">(<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>,<font color="Maroon" title="c39">s</font></span>)</span> <a href="scmfsa_2.html#K20" title="SCMFSA_2:func.20">.</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> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b4">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa6b.html#K2" title="SCMFSA6B:func.2">IExec</a> <span class="p3">(<span class="default"><a href="scmbsort.html#K2" title="SCMBSORT:func.2">bubble-sort</a> <span class="p4">(<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>,<font color="Maroon" title="c39">s</font></span>)</span> <a href="scmfsa_2.html#K20" title="SCMFSA_2:func.20">.</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> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">j</font> holds <br/><font color="Olive" title="b3">x1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b4">x2</font> ) )
 
<b>by </b><i><a class="ref" href="scmbsort.html#T65" target="_self" title="SCMBSORT:th.65">Th65</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c45">u</font> = <span class="p1">(<span class="default"><a href="scmfsa6b.html#K2" title="SCMFSA6B:func.2">IExec</a> <span class="p2">(<span class="default"><a href="scmbsort.html#K2" title="SCMBSORT:func.2">bubble-sort</a> <span class="p3">(<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>,<font color="Maroon" title="c39">s</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> as    <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>by </b><i><a class="ref" href="finseq_3.html#T126" title="FINSEQ_3:th.126">FINSEQ_3:126</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c45">u</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c40">t</font>,<font color="Maroon" title="c45">u</font> <a href="rfinseq.html#R1" title="RFINSEQ:pred.1">are_fiberwise_equipotent</a>  &amp; <font color="Maroon" title="c45">u</font> is <a href="seqm_3.html#V4" title="SEQM_3:attr.4">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="c39">s</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> )</span><br/>

<b>thus </b><a NAME="E8:67"/>
<font color="Maroon" title="c40">t</font>,<font color="Maroon" title="c45">u</font> <a href="rfinseq.html#R1" title="RFINSEQ:pred.1">are_fiberwise_equipotent</a> 
 <b>by </b><i><a class="txt" href="scmbsort.html#E5:67"><i><font color="Green" title="E67">A3</font></i></a>, <a class="ref" href="scmbsort.html#T65" target="_self" title="SCMBSORT:th.65">Th65</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c45">u</font> is <a href="seqm_3.html#V4" title="SEQM_3:attr.4">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="c39">s</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> )</span><br/>

<a NAME="E9:67"/><i><font color="Green" title="E69">A5</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c39">s</font> <a href="scmfsa_2.html#K20" title="SCMFSA_2:func.20">.</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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c45">u</font>
 
<b>by </b><i><a class="txt" href="scmbsort.html#E6:67"><i><font color="Green" title="E68">A4</font></i></a>, <a class="ref" href="rfinseq.html#T16" title="RFINSEQ:th.16">RFINSEQ:16</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c46">i</font>, <font color="Maroon" title="c47">j</font> be    <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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c46">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c45">u</font> &amp; <font color="Maroon" title="c47">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c45">u</font> &amp; <font color="Maroon" title="c46">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c47">j</font> implies <font color="Maroon" title="c45">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c46">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c45">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c47">j</font> )</span><br/><b>assume </b><a NAME="E1:67_2"/><i><font color="Green" title="E70">A6</font></i>: 
( <font color="Maroon" title="c46">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c45">u</font> &amp; <font color="Maroon" title="c47">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c45">u</font> &amp; <font color="Maroon" title="c46">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c47">j</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c45">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c46">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c45">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c47">j</font></span><br/><a NAME="E2:67_2"/><b>then </b><i><font color="Green" title="E71">A7</font></i>: 
<font color="Maroon" title="c46">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E3:67_2"/><i><font color="Green" title="E72">A8</font></i>: 
<font color="Maroon" title="c47">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c39">s</font> <a href="scmfsa_2.html#K20" title="SCMFSA_2:func.20">.</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>
 <b>by </b><i><a class="txt" href="scmbsort.html#E9:67"><i><font color="Green" title="E69">A5</font></i></a>, <a class="txt" href="scmbsort.html#E1:67_2"><i><font color="Green" title="E70">A6</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c48">y1</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa6b.html#K2" title="SCMFSA6B:func.2">IExec</a> <span class="p3">(<span class="default"><a href="scmbsort.html#K2" title="SCMBSORT:func.2">bubble-sort</a> <span class="p4">(<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>,<font color="Maroon" title="c39">s</font></span>)</span> <a href="scmfsa_2.html#K20" title="SCMFSA_2:func.20">.</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> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c46">i</font> as   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> ;<br/><b>reconsider </b><font color="Maroon" title="c49">y2</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa6b.html#K2" title="SCMFSA6B:func.2">IExec</a> <span class="p3">(<span class="default"><a href="scmbsort.html#K2" title="SCMBSORT:func.2">bubble-sort</a> <span class="p4">(<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>,<font color="Maroon" title="c39">s</font></span>)</span> <a href="scmfsa_2.html#K20" title="SCMFSA_2:func.20">.</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> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c47">j</font> as   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> ;<br/><a NAME="E6:67_2"/>
<font color="Maroon" title="c48">y1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c49">y2</font>
 <b>by </b><i><a class="txt" href="scmbsort.html#E1:67_2"><i><font color="Green" title="E70">A6</font></i></a>, <a class="txt" href="scmbsort.html#E2:67_2"><i><font color="Green" title="E71">A7</font></i></a>, <a class="txt" href="scmbsort.html#E3:67_2"><i><font color="Green" title="E72">A8</font></i></a>, <a class="ref" href="scmbsort.html#T65" target="_self" title="SCMBSORT:th.65">Th65</a></i>;<br/><b>hence </b><a NAME="E7:67_2"/>
<font color="Maroon" title="c45">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c46">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c45">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c47">j</font>
 ; <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>
<b>hence </b><a NAME="E11:67"/>
<font color="Maroon" title="c45">u</font> is <a href="seqm_3.html#V4" title="SEQM_3:attr.4">non-increasing</a>
 <b>by </b><i><a class="ref" href="rfinseq.html#T32" title="RFINSEQ:th.32">RFINSEQ:32</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <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="c39">s</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> )</span><br/>

<b>thus </b><a NAME="E12:67"/>
<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> 
 ; <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="ami_1.html#K12" title="AMI_1:func.12">Result</a> <font color="Maroon" title="c39">s</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></span><br/>

<a NAME="E13:67"/>
 <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> <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="scmbsort.html#K1" title="SCMBSORT:func.1">.--&gt;</a> <font color="Maroon" title="c40">t</font></span>)</span>
 
<b>by </b><i><a class="ref" href="scmbsort.html#T46" target="_self" title="SCMBSORT:th.46">Th46</a></i>;<br/>
<a NAME="E14:67"/><b>then </b>
 <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> <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="scmbsort.html#K1" title="SCMBSORT:func.1">.--&gt;</a> <font color="Maroon" title="c40">t</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_4.html#T33" title="FUNCT_4:th.33">FUNCT_4:33</a></i>;<br/>
<a NAME="E15:67"/><b>then </b>
 <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="c39">s</font>
 
<b>by </b><i><a class="txt" href="scmbsort.html#E1:67"><i><font color="Green" title="E65">A1</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E16:67"/><b>then </b>
<font color="Maroon" title="c39">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c39">s</font> <a href="ami_1.html#K36" title="AMI_1:func.36">+*</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>
 
<b>by </b><i><a class="ref" href="funct_4.html#T79" title="FUNCT_4:th.79">FUNCT_4:79</a></i>;<br/>
<b>hence </b><a NAME="E17:67"/>
<span class="p1">(<span class="default"><a href="ami_1.html#K12" title="AMI_1:func.12">Result</a> <font color="Maroon" title="c39">s</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="ref" href="scmbsort.html#T57" target="_self" title="SCMBSORT:th.57">Th57</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>
