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

<b>let </b><font color="Maroon">w</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K4">INT</a> ;<br/>

<b>set </b><font color="Maroon">p</font> = <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> <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="scmbsort.html#K1" target="_self">.--&gt;</a> <font color="Maroon">w</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">q</font> =  <a href="scmbsort.html#K3" target="_self">Bubble-Sort-Algorithm</a> ;<br/>
<a NAME="E1:78"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">s1</font>, <font color="Olive">s2</font> being  <a href="ami_1.html#NM4">State</a> of <a href="scmfsa_2.html#K1">SCM+FSA</a> <br/> for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <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> <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="scmbsort.html#K1" target="_self">.--&gt;</a> <font color="Maroon">w</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Olive">s1</font> &amp; <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> <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="scmbsort.html#K1" target="_self">.--&gt;</a> <font color="Maroon">w</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Olive">s2</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a> 10 holds <br/>( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Olive">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Olive">i</font></span>)</span> <a href="scmfsa_2.html#K20">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 0</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Olive">s2</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Olive">i</font></span>)</span> <a href="scmfsa_2.html#K20">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 0</span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Olive">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Olive">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K2">IC</a> <a href="scmfsa_2.html#K1">SCM+FSA</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Olive">s2</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Olive">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K2">IC</a> <a href="scmfsa_2.html#K1">SCM+FSA</a> </span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Olive">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Olive">i</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Olive">s2</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Olive">i</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> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">s1</font> be   <a href="ami_1.html#NM4">State</a> of <a href="scmfsa_2.html#K1">SCM+FSA</a> , <font color="Maroon">s2</font> be   <a href="ami_1.html#NM4">State</a> of <a href="scmfsa_2.html#K1">SCM+FSA</a> ;<br/><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<b>assume </b><a NAME="E1:78_1"/><i><font color="Green">E42</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> <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="scmbsort.html#K1" target="_self">.--&gt;</a> <font color="Maroon">w</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">s1</font> &amp; <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> <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="scmbsort.html#K1" target="_self">.--&gt;</a> <font color="Maroon">w</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">s2</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> 10 )
 ;<br/>

<b>set </b><font color="Maroon">Cs1</font> =  <a href="ami_1.html#K10">Computation</a> <font color="Maroon">s1</font>;<br/>
<b>set </b><font color="Maroon">Cs2</font> =  <a href="ami_1.html#K10">Computation</a> <font color="Maroon">s2</font>;<br/>
<a NAME="E2:78_1"/><i><font color="Green">E47</font></i>: 
( <span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font> &amp; <span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="ami_1.html#K11">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">s2</font> )
 
<b>by </b><i><a class="ref" href="ami_1.html#D19">AMI_1:def 19</a></i>;<br/>
<a NAME="E3:78_1"/><i><font color="Green">E48</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> <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="scmbsort.html#K1" target="_self">.--&gt;</a> <font color="Maroon">w</font></span>)</span> <a href="ami_3.html#R3">starts_at</a>  <a href="scmfsa_2.html#K5">insloc</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E4:78_1"/><b>then </b><i><font color="Green">E55</font></i>: 
<font color="Maroon">s1</font> <a href="ami_3.html#R1">starts_at</a>  <a href="scmfsa_2.html#K5">insloc</a> 0
 
<b>by </b><i><a class="txt" href="scmbsort.html#E4:78"><i><font color="Green">E79</font></i></a>, <a class="ref" href="ami_3.html#T49">AMI_3:49</a></i>;<br/>
<a NAME="E5:78_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">s2</font> <a href="ami_3.html#R1">starts_at</a>  <a href="scmfsa_2.html#K5">insloc</a> 0
 
<b>by </b><i><a class="txt" href="scmbsort.html#E4:78"><i><font color="Green">E79</font></i></a>, , <a class="ref" href="ami_3.html#T49">AMI_3:49</a></i>;<br/>
<a NAME="E6:78_1"/><i><font color="Green">E62</font></i>: 
(  <a href="scmbsort.html#K3" target="_self">Bubble-Sort-Algorithm</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">s1</font> &amp;  <a href="scmbsort.html#K3" target="_self">Bubble-Sort-Algorithm</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">s2</font> )
 
<b>by </b><i><a class="txt" href="scmbsort.html#E4:78"><i><font color="Green">E79</font></i></a>, </i>;<br/>
<i><font color="Green">E63</font></i>: <font color="Maroon">s1</font> <a href="scmfsa_2.html#K20">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 0</span>)</span> = 
1
<b>by </b><i><a class="txt" href="scmbsort.html#E4:78"><i><font color="Green">E79</font></i></a>, </i>
<br/>.= 
<font color="Maroon">s2</font> <a href="scmfsa_2.html#K20">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 0</span>)</span>
<b>by </b><i><a class="txt" href="scmbsort.html#E4:78"><i><font color="Green">E79</font></i></a>, </i>
;<br/>
<i><font color="Green">E64</font></i>: <font color="Maroon">s1</font> <a href="scmfsa_2.html#K21">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K6">fsloc</a> 0</span>)</span> = 
<font color="Maroon">w</font>
<b>by </b><i><a class="txt" href="scmbsort.html#E4:78"><i><font color="Green">E79</font></i></a>, </i>
<br/>.= 
<font color="Maroon">s2</font> <a href="scmfsa_2.html#K21">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K6">fsloc</a> 0</span>)</span>
<b>by </b><i><a class="txt" href="scmbsort.html#E4:78"><i><font color="Green">E79</font></i></a>, </i>
;<br/>
<i><font color="Green">E65</font></i>:  <a href="ami_1.html#K6">IC</a> <font color="Maroon">s1</font> = 
 <a href="scmfsa_2.html#K5">insloc</a> 0
<b>by </b><i>, <a class="ref" href="ami_3.html#D14">AMI_3:def 14</a></i>
<br/>.= 
 <a href="ami_1.html#K6">IC</a> <font color="Maroon">s2</font>
<b>by </b><i>, <a class="ref" href="ami_3.html#D14">AMI_3:def 14</a></i>
;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="78_1_4"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:78_1_4"/>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 2 or <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 3 or <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 4 or <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 5 or <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 6 or <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 7 or <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 8 or <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 9 or <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 10 )
 </span><b>by </b><i><a class="txt" href="scmbsort.html#E4:78"><i><font color="Green">E79</font></i></a>, </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="78_1_4_1"><b>suppose </b></a><a NAME="E1:78_1_4_1"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:78_1_4_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</font></span>)</span> <a href="scmfsa_2.html#K20">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 0</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</font></span>)</span> <a href="scmfsa_2.html#K20">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 0</span>)</span>
 <b>by </b><i><a class="txt" href="scmbsort.html#E5:78"><i><font color="Green">E80</font></i></a>, </i>;<br/><b>thus </b><a NAME="E3:78_1_4_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K2">IC</a> <a href="scmfsa_2.html#K1">SCM+FSA</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K2">IC</a> <a href="scmfsa_2.html#K1">SCM+FSA</a> </span>)</span>
 <b>by </b><i><a class="txt" href="scmbsort.html#E5:78"><i><font color="Green">E80</font></i></a>, , </i>;<br/><b>thus </b><a NAME="E4:78_1_4_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</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>
 <b>by </b><i><a class="txt" href="scmbsort.html#E5:78"><i><font color="Green">E80</font></i></a>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="78_1_4_2"><b>suppose </b></a><a NAME="E1:78_1_4_2"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</font></span>)</span> <a href="scmfsa_2.html#K20">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 0</span>)</span> = 
<font color="Maroon">s1</font> <a href="scmfsa_2.html#K20">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 0</span>)</span>
<b>by </b><i>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</font></span>)</span> <a href="scmfsa_2.html#K20">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 0</span>)</span>
<b>by </b><i>, , , , </i>
;<br/><br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K2">IC</a> <a href="scmfsa_2.html#K1">SCM+FSA</a> </span>)</span> = 
 <a href="scmfsa_2.html#K5">insloc</a> 1
<b>by </b><i>, , , </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K2">IC</a> <a href="scmfsa_2.html#K1">SCM+FSA</a> </span>)</span>
<b>by </b><i>, , , </i>
;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</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> = 
<font color="Maroon">s1</font> <a href="scmfsa_2.html#K21">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K6">fsloc</a> 0</span>)</span>
<b>by </b><i>, , , </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</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>
<b>by </b><i>, , , , </i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="78_1_4_3"><b>suppose </b></a><a NAME="E1:78_1_4_3"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 2
 ;<br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</font></span>)</span> <a href="scmfsa_2.html#K20">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 0</span>)</span> = 
<font color="Maroon">s1</font> <a href="scmfsa_2.html#K20">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 0</span>)</span>
<b>by </b><i>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</font></span>)</span> <a href="scmfsa_2.html#K20">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 0</span>)</span>
<b>by </b><i>, , , , </i>
;<br/><br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K2">IC</a> <a href="scmfsa_2.html#K1">SCM+FSA</a> </span>)</span> = 
 <a href="scmfsa_2.html#K5">insloc</a> 2
<b>by </b><i>, , , </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K2">IC</a> <a href="scmfsa_2.html#K1">SCM+FSA</a> </span>)</span>
<b>by </b><i>, , , </i>
;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</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> = 
<font color="Maroon">s1</font> <a href="scmfsa_2.html#K21">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K6">fsloc</a> 0</span>)</span>
<b>by </b><i>, , , </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</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>
<b>by </b><i>, , , , </i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="78_1_4_4"><b>suppose </b></a><a NAME="E1:78_1_4_4"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 3
 ;<br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</font></span>)</span> <a href="scmfsa_2.html#K20">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 0</span>)</span> = 
<font color="Maroon">s1</font> <a href="scmfsa_2.html#K20">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 0</span>)</span>
<b>by </b><i>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</font></span>)</span> <a href="scmfsa_2.html#K20">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 0</span>)</span>
<b>by </b><i>, , , , </i>
;<br/><br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K2">IC</a> <a href="scmfsa_2.html#K1">SCM+FSA</a> </span>)</span> = 
 <a href="scmfsa_2.html#K5">insloc</a> 3
<b>by </b><i>, , , </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K2">IC</a> <a href="scmfsa_2.html#K1">SCM+FSA</a> </span>)</span>
<b>by </b><i>, , , </i>
;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</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> = 
<font color="Maroon">s1</font> <a href="scmfsa_2.html#K21">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K6">fsloc</a> 0</span>)</span>
<b>by </b><i>, , , </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</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>
<b>by </b><i>, , , , </i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="78_1_4_5"><b>suppose </b></a><a NAME="E1:78_1_4_5"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 4
 ;<br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</font></span>)</span> <a href="scmfsa_2.html#K20">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 0</span>)</span> = 
<font color="Maroon">s1</font> <a href="scmfsa_2.html#K20">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 0</span>)</span>
<b>by </b><i>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</font></span>)</span> <a href="scmfsa_2.html#K20">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 0</span>)</span>
<b>by </b><i>, , , , </i>
;<br/><br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K2">IC</a> <a href="scmfsa_2.html#K1">SCM+FSA</a> </span>)</span> = 
 <a href="scmfsa_2.html#K5">insloc</a> 4
<b>by </b><i>, , , </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K2">IC</a> <a href="scmfsa_2.html#K1">SCM+FSA</a> </span>)</span>
<b>by </b><i>, , , </i>
;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</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> = 
<font color="Maroon">s1</font> <a href="scmfsa_2.html#K21">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K6">fsloc</a> 0</span>)</span>
<b>by </b><i>, , , </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</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>
<b>by </b><i>, , , , </i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="78_1_4_6"><b>suppose </b></a><a NAME="E1:78_1_4_6"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 5
 ;<br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</font></span>)</span> <a href="scmfsa_2.html#K20">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 0</span>)</span> = 
<font color="Maroon">s1</font> <a href="scmfsa_2.html#K20">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 0</span>)</span>
<b>by </b><i>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</font></span>)</span> <a href="scmfsa_2.html#K20">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 0</span>)</span>
<b>by </b><i>, , , , </i>
;<br/><br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K2">IC</a> <a href="scmfsa_2.html#K1">SCM+FSA</a> </span>)</span> = 
 <a href="scmfsa_2.html#K5">insloc</a> 5
<b>by </b><i>, , , </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K2">IC</a> <a href="scmfsa_2.html#K1">SCM+FSA</a> </span>)</span>
<b>by </b><i>, , , </i>
;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</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> = 
<font color="Maroon">s1</font> <a href="scmfsa_2.html#K21">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K6">fsloc</a> 0</span>)</span>
<b>by </b><i>, , , </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</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>
<b>by </b><i>, , , , </i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="78_1_4_7"><b>suppose </b></a><a NAME="E1:78_1_4_7"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 6
 ;<br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</font></span>)</span> <a href="scmfsa_2.html#K20">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 0</span>)</span> = 
<font color="Maroon">s1</font> <a href="scmfsa_2.html#K20">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 0</span>)</span>
<b>by </b><i>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</font></span>)</span> <a href="scmfsa_2.html#K20">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 0</span>)</span>
<b>by </b><i>, , , , </i>
;<br/><br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K2">IC</a> <a href="scmfsa_2.html#K1">SCM+FSA</a> </span>)</span> = 
 <a href="scmfsa_2.html#K5">insloc</a> 6
<b>by </b><i>, , , </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K2">IC</a> <a href="scmfsa_2.html#K1">SCM+FSA</a> </span>)</span>
<b>by </b><i>, , , </i>
;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</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> = 
<font color="Maroon">s1</font> <a href="scmfsa_2.html#K21">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K6">fsloc</a> 0</span>)</span>
<b>by </b><i>, , , </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</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>
<b>by </b><i>, , , , </i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="78_1_4_8"><b>suppose </b></a><a NAME="E1:78_1_4_8"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 7
 ;<br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</font></span>)</span> <a href="scmfsa_2.html#K20">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 0</span>)</span> = 
<font color="Maroon">s1</font> <a href="scmfsa_2.html#K20">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 0</span>)</span>
<b>by </b><i>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</font></span>)</span> <a href="scmfsa_2.html#K20">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 0</span>)</span>
<b>by </b><i>, , , , </i>
;<br/><br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K2">IC</a> <a href="scmfsa_2.html#K1">SCM+FSA</a> </span>)</span> = 
 <a href="scmfsa_2.html#K5">insloc</a> 7
<b>by </b><i>, , , </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K2">IC</a> <a href="scmfsa_2.html#K1">SCM+FSA</a> </span>)</span>
<b>by </b><i>, , , </i>
;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</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> = 
<font color="Maroon">s1</font> <a href="scmfsa_2.html#K21">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K6">fsloc</a> 0</span>)</span>
<b>by </b><i>, , , </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</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>
<b>by </b><i>, , , , </i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="78_1_4_9"><b>suppose </b></a><a NAME="E1:78_1_4_9"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 8
 ;<br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</font></span>)</span> <a href="scmfsa_2.html#K20">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 0</span>)</span> = 
<font color="Maroon">s1</font> <a href="scmfsa_2.html#K20">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 0</span>)</span>
<b>by </b><i>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</font></span>)</span> <a href="scmfsa_2.html#K20">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 0</span>)</span>
<b>by </b><i>, , , , </i>
;<br/><br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K2">IC</a> <a href="scmfsa_2.html#K1">SCM+FSA</a> </span>)</span> = 
 <a href="scmfsa_2.html#K5">insloc</a> 8
<b>by </b><i>, , , </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K2">IC</a> <a href="scmfsa_2.html#K1">SCM+FSA</a> </span>)</span>
<b>by </b><i>, , , </i>
;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</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> = 
<font color="Maroon">s1</font> <a href="scmfsa_2.html#K21">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K6">fsloc</a> 0</span>)</span>
<b>by </b><i>, , , </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</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>
<b>by </b><i>, , , , </i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="78_1_4_10"><b>suppose </b></a><a NAME="E1:78_1_4_10"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 9
 ;<br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</font></span>)</span> <a href="scmfsa_2.html#K20">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 0</span>)</span> = 
<font color="Maroon">s1</font> <a href="scmfsa_2.html#K20">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 0</span>)</span>
<b>by </b><i>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</font></span>)</span> <a href="scmfsa_2.html#K20">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 0</span>)</span>
<b>by </b><i>, , , , </i>
;<br/><br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K2">IC</a> <a href="scmfsa_2.html#K1">SCM+FSA</a> </span>)</span> = 
 <a href="scmfsa_2.html#K5">insloc</a> 9
<b>by </b><i>, , , </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K2">IC</a> <a href="scmfsa_2.html#K1">SCM+FSA</a> </span>)</span>
<b>by </b><i>, , , </i>
;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</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> = 
<font color="Maroon">s1</font> <a href="scmfsa_2.html#K21">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K6">fsloc</a> 0</span>)</span>
<b>by </b><i>, , , </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</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>
<b>by </b><i>, , , , </i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="78_1_4_11"><b>suppose </b></a><a NAME="E1:78_1_4_11"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 10
 ;<br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</font></span>)</span> <a href="scmfsa_2.html#K20">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 0</span>)</span> = 
<font color="Maroon">s1</font> <a href="scmfsa_2.html#K20">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 0</span>)</span>
<b>by </b><i>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</font></span>)</span> <a href="scmfsa_2.html#K20">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 0</span>)</span>
<b>by </b><i>, , , , </i>
;<br/><br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K2">IC</a> <a href="scmfsa_2.html#K1">SCM+FSA</a> </span>)</span> = 
 <a href="scmfsa_2.html#K5">insloc</a> 10
<b>by </b><i>, , , </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K2">IC</a> <a href="scmfsa_2.html#K1">SCM+FSA</a> </span>)</span>
<b>by </b><i>, , , </i>
;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</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> = 
<font color="Maroon">s1</font> <a href="scmfsa_2.html#K21">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K6">fsloc</a> 0</span>)</span>
<b>by </b><i>, , , </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</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>
<b>by </b><i>, , , , </i>
;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>set </b><font color="Maroon">UD</font> = <span class="p1"><a href="enumset1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="scmfsa_2.html#K6">fsloc</a> 0</span>)</span>,<span class="p2">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 0</span>)</span>,<span class="p2">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 3</span>)</span>,<span class="p2">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 4</span>)</span>,<span class="p2">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 5</span>)</span>,<span class="p2">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 6</span>)</span></span><a href="enumset1.html#K6">}</a></span>;<br/>
<b>set </b><font color="Maroon">Us</font> = <span class="p1">(<span class="default"><a href="sf_mastr.html#K4">UsedInt*Loc</a> <a href="scmbsort.html#K3" target="_self">Bubble-Sort-Algorithm</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="sf_mastr.html#K2">UsedIntLoc</a> <a href="scmbsort.html#K3" target="_self">Bubble-Sort-Algorithm</a> </span>)</span>;<br/>
<i><font color="Green">E77</font></i>:  <a href="sf_mastr.html#K4">UsedInt*Loc</a> <a href="scmbsort.html#K3" target="_self">Bubble-Sort-Algorithm</a>  = 
 <a href="sf_mastr.html#K4">UsedInt*Loc</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>
<b>by </b><i/>
<br/>.= 
<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/>
;<br/>
<i><font color="Green">E78</font></i>:  <a href="sf_mastr.html#K2">UsedIntLoc</a> <a href="scmbsort.html#K3" target="_self">Bubble-Sort-Algorithm</a>  = 
 <a href="sf_mastr.html#K2">UsedIntLoc</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>
<b>by </b><i/>
<br/>.= 
<span class="p1"><a href="enumset1.html#K5">{</a><span class="default"><span class="p2">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 0</span>)</span>,<span class="p2">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 3</span>)</span>,<span class="p2">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 4</span>)</span>,<span class="p2">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 5</span>)</span>,<span class="p2">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 6</span>)</span></span><a href="enumset1.html#K5">}</a></span>
<b>by </b><i/>
;<br/>
<a NAME="E4:78"/><b>then </b><i><font color="Green">E79</font></i>: 
<span class="p1">(<span class="default"><a href="sf_mastr.html#K4">UsedInt*Loc</a> <a href="scmbsort.html#K3" target="_self">Bubble-Sort-Algorithm</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="sf_mastr.html#K2">UsedIntLoc</a> <a href="scmbsort.html#K3" target="_self">Bubble-Sort-Algorithm</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="enumset1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="scmfsa_2.html#K6">fsloc</a> 0</span>)</span>,<span class="p2">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 0</span>)</span>,<span class="p2">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 3</span>)</span>,<span class="p2">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 4</span>)</span>,<span class="p2">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 5</span>)</span>,<span class="p2">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 6</span>)</span></span><a href="enumset1.html#K6">}</a></span>
 
<b>by </b><i><a class="txt" href="scmbsort.html#E4:78"><i><font color="Green">E79</font></i></a>, <a class="ref" href="enumset1.html#T62">ENUMSET1:62</a></i>;<br/>
<a NAME="E5:78"/><i><font color="Green">E80</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">s1</font>, <font color="Olive">s2</font> being  <a href="ami_1.html#NM4">State</a> of <a href="scmfsa_2.html#K1">SCM+FSA</a>   st 11 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <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> <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="scmbsort.html#K1" target="_self">.--&gt;</a> <font color="Maroon">w</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Olive">s1</font> &amp; <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> <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="scmbsort.html#K1" target="_self">.--&gt;</a> <font color="Maroon">w</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Olive">s2</font> holds <br/>( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Olive">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Olive">i</font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sf_mastr.html#K4">UsedInt*Loc</a> <a href="scmbsort.html#K3" target="_self">Bubble-Sort-Algorithm</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><a href="sf_mastr.html#K2">UsedIntLoc</a> <a href="scmbsort.html#K3" target="_self">Bubble-Sort-Algorithm</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Olive">s2</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Olive">i</font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sf_mastr.html#K4">UsedInt*Loc</a> <a href="scmbsort.html#K3" target="_self">Bubble-Sort-Algorithm</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><a href="sf_mastr.html#K2">UsedIntLoc</a> <a href="scmbsort.html#K3" target="_self">Bubble-Sort-Algorithm</a> </span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Olive">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Olive">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K2">IC</a> <a href="scmfsa_2.html#K1">SCM+FSA</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Olive">s2</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Olive">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K2">IC</a> <a href="scmfsa_2.html#K1">SCM+FSA</a> </span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">s1</font> be   <a href="ami_1.html#NM4">State</a> of <a href="scmfsa_2.html#K1">SCM+FSA</a> , <font color="Maroon">s2</font> be   <a href="ami_1.html#NM4">State</a> of <a href="scmfsa_2.html#K1">SCM+FSA</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:78_4"/><i><font color="Green">E81</font></i>: 
11 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font>
 <b>and </b><br/><a NAME="E2:78_4"/><i><font color="Green">E82</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> <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="scmbsort.html#K1" target="_self">.--&gt;</a> <font color="Maroon">w</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">s1</font>
 <b>and </b><br/><a NAME="E3:78_4"/><i><font color="Green">E83</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> <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="scmbsort.html#K1" target="_self">.--&gt;</a> <font color="Maroon">w</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">s2</font>
 ;<br/>

<b>set </b><font color="Maroon">Cs11</font> = <span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> 11;<br/>
<b>set </b><font color="Maroon">Cs21</font> = <span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="ami_1.html#K11">.</a> 11;<br/>
<a NAME="E4:78_4"/><i><font color="Green">E85</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> <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="scmbsort.html#K1" target="_self">.--&gt;</a> <font color="Maroon">w</font></span>)</span> <a href="ami_3.html#R3">starts_at</a>  <a href="scmfsa_2.html#K5">insloc</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E5:78_4"/><b>then </b><i><font color="Green">E86</font></i>: 
<font color="Maroon">s1</font> <a href="ami_3.html#R1">starts_at</a>  <a href="scmfsa_2.html#K5">insloc</a> 0
 
<b>by </b><i>, <a class="ref" href="ami_3.html#T49">AMI_3:49</a></i>;<br/>
<a NAME="E6:78_4"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">s2</font> <a href="ami_3.html#R1">starts_at</a>  <a href="scmfsa_2.html#K5">insloc</a> 0
 
<b>by </b><i><a class="txt" href="scmbsort.html#E4:78_4"><i><font color="Green">E85</font></i></a>, <a class="txt" href="scmbsort.html#E5:78_4"><i><font color="Green">E86</font></i></a>, <a class="ref" href="ami_3.html#T49">AMI_3:49</a></i>;<br/>
<a NAME="E7:78_4"/><i><font color="Green">E88</font></i>: 
 <a href="scmbsort.html#K3" target="_self">Bubble-Sort-Algorithm</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">s1</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E8:78_4"/><i><font color="Green">E89</font></i>: 
 <a href="scmbsort.html#K3" target="_self">Bubble-Sort-Algorithm</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">s2</font>
 
<b>by </b><i><a class="txt" href="scmbsort.html#E4:78_4"><i><font color="Green">E85</font></i></a>, </i>;<br/>
<i><font color="Green">E90</font></i>: <font color="Maroon">s1</font> <a href="scmfsa_2.html#K20">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 0</span>)</span> = 
1
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">s2</font> <a href="scmfsa_2.html#K20">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 0</span>)</span>
<b>by </b><i><a class="txt" href="scmbsort.html#E4:78_4"><i><font color="Green">E85</font></i></a>, </i>
;<br/>
<i><font color="Green">E127</font></i>: <font color="Maroon">s1</font> <a href="scmfsa_2.html#K21">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K6">fsloc</a> 0</span>)</span> = 
<font color="Maroon">w</font>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">s2</font> <a href="scmfsa_2.html#K21">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K6">fsloc</a> 0</span>)</span>
<b>by </b><i><a class="txt" href="scmbsort.html#E4:78_4"><i><font color="Green">E85</font></i></a>, </i>
;<br/>
<a NAME="E11:78_4"/><i><font color="Green">E128</font></i>: 
<span class="p1">(<span class="default"><a href="sf_mastr.html#K4">UsedInt*Loc</a> <a href="scmbsort.html#K3" target="_self">Bubble-Sort-Algorithm</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="sf_mastr.html#K2">UsedIntLoc</a> <a href="scmbsort.html#K3" target="_self">Bubble-Sort-Algorithm</a> </span>)</span> <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="ami_1.html#K10">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> 11</span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E12:78_4"/><i><font color="Green">E129</font></i>: 
<span class="p1">(<span class="default"><a href="sf_mastr.html#K4">UsedInt*Loc</a> <a href="scmbsort.html#K3" target="_self">Bubble-Sort-Algorithm</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="sf_mastr.html#K2">UsedIntLoc</a> <a href="scmbsort.html#K3" target="_self">Bubble-Sort-Algorithm</a> </span>)</span> <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="ami_1.html#K10">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="ami_1.html#K11">.</a> 11</span>)</span>
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_4_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:78_4_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="sf_mastr.html#K4">UsedInt*Loc</a> <a href="scmbsort.html#K3" target="_self">Bubble-Sort-Algorithm</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="sf_mastr.html#K2">UsedIntLoc</a> <a href="scmbsort.html#K3" target="_self">Bubble-Sort-Algorithm</a> </span>)</span>
 ;<br/><a NAME="E2:78_4_3"/><b>then </b><i><font color="Green">E130</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="enumset1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="scmfsa_2.html#K6">fsloc</a> 0</span>)</span>,<span class="p2">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 0</span>)</span>,<span class="p2">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 3</span>)</span>,<span class="p2">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 4</span>)</span>,<span class="p2">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 5</span>)</span>,<span class="p2">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 6</span>)</span></span><a href="enumset1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="scmbsort.html#E4:78"><i><font color="Green">E79</font></i></a>, <a class="txt" href="scmbsort.html#E5:78"><i><font color="Green">E80</font></i></a>, <a class="ref" href="enumset1.html#T62">ENUMSET1:62</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="78_4_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:78_4_3_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="scmfsa_2.html#K6">fsloc</a> 0 or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="scmfsa_2.html#K4">intloc</a> 0 or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="scmfsa_2.html#K4">intloc</a> 1 or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="scmfsa_2.html#K4">intloc</a> 2 or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="scmfsa_2.html#K4">intloc</a> 3 or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="scmfsa_2.html#K4">intloc</a> 4 or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="scmfsa_2.html#K4">intloc</a> 5 or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="scmfsa_2.html#K4">intloc</a> 6 )
 </span><b>by </b><i><a class="ref" href="scmbsort.html#T11" target="_self">Th11</a>, <a class="ref" href="enumset1.html#D6">ENUMSET1:def 6</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="78_4_3_1_1"><b>suppose </b></a><a NAME="E1:78_4_3_1_1"/><i><font color="Green">E131</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="scmfsa_2.html#K6">fsloc</a> 0
 ;<br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> 11</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<font color="Maroon">s1</font> <a href="scmfsa_2.html#K21">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K6">fsloc</a> 0</span>)</span>
<b>by </b><i><a class="txt" href="scmbsort.html#E6:78_4"><i><font color="Green">E87</font></i></a>, <a class="txt" href="scmbsort.html#E8:78_4"><i><font color="Green">E89</font></i></a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="ami_1.html#K11">.</a> 11</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="txt" href="scmbsort.html#E7:78_4"><i><font color="Green">E88</font></i></a>, <a class="txt" href="scmbsort.html#E9:78_4"><i><font color="Green">E90</font></i></a>, , <a class="ref" href="scmbsort.html#T12" target="_self">Th12</a>, </i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="78_4_3_1_2"><b>suppose </b></a><a NAME="E1:78_4_3_1_2"/><i><font color="Green">E132</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="scmfsa_2.html#K4">intloc</a> 0
 ;<br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> 11</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<font color="Maroon">s1</font> <a href="scmfsa_2.html#K20">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 0</span>)</span>
<b>by </b><i><a class="txt" href="scmbsort.html#E6:78_4"><i><font color="Green">E87</font></i></a>, <a class="txt" href="scmbsort.html#E8:78_4"><i><font color="Green">E89</font></i></a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="ami_1.html#K11">.</a> 11</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="txt" href="scmbsort.html#E7:78_4"><i><font color="Green">E88</font></i></a>, <a class="txt" href="scmbsort.html#E9:78_4"><i><font color="Green">E90</font></i></a>, <a class="ref" href="scmbsort.html#T8" target="_self">Th8</a>, <a class="ref" href="scmbsort.html#T12" target="_self">Th12</a>, </i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="78_4_3_1_3"><b>suppose </b></a><a NAME="E1:78_4_3_1_3"/><i><font color="Green">E133</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="scmfsa_2.html#K4">intloc</a> 1
 ;<br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> 11</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="scmfsa_2.html#K21">.</a> <span class="p2">(<span class="default"><a href="scmfsa_2.html#K6">fsloc</a> 0</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="scmbsort.html#E6:78_4"><i><font color="Green">E87</font></i></a>, <a class="txt" href="scmbsort.html#E8:78_4"><i><font color="Green">E89</font></i></a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="ami_1.html#K11">.</a> 11</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="txt" href="scmbsort.html#E7:78_4"><i><font color="Green">E88</font></i></a>, <a class="txt" href="scmbsort.html#E9:78_4"><i><font color="Green">E90</font></i></a>, , <a class="ref" href="scmbsort.html#T12" target="_self">Th12</a>, </i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="78_4_3_1_4"><b>suppose </b></a><a NAME="E1:78_4_3_1_4"/><i><font color="Green">E134</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="scmfsa_2.html#K4">intloc</a> 2
 ;<br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> 11</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<font color="Maroon">s1</font> <a href="scmfsa_2.html#K20">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 0</span>)</span>
<b>by </b><i><a class="txt" href="scmbsort.html#E6:78_4"><i><font color="Green">E87</font></i></a>, <a class="txt" href="scmbsort.html#E8:78_4"><i><font color="Green">E89</font></i></a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="ami_1.html#K11">.</a> 11</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="txt" href="scmbsort.html#E7:78_4"><i><font color="Green">E88</font></i></a>, <a class="txt" href="scmbsort.html#E9:78_4"><i><font color="Green">E90</font></i></a>, <a class="ref" href="scmbsort.html#T8" target="_self">Th8</a>, <a class="ref" href="scmbsort.html#T12" target="_self">Th12</a>, </i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="78_4_3_1_5"><b>suppose </b></a><a NAME="E1:78_4_3_1_5"/><i><font color="Green">E135</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="scmfsa_2.html#K4">intloc</a> 3
 ;<br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> 11</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<font color="Maroon">s1</font> <a href="scmfsa_2.html#K20">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 0</span>)</span>
<b>by </b><i><a class="txt" href="scmbsort.html#E6:78_4"><i><font color="Green">E87</font></i></a>, <a class="txt" href="scmbsort.html#E8:78_4"><i><font color="Green">E89</font></i></a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="ami_1.html#K11">.</a> 11</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="txt" href="scmbsort.html#E7:78_4"><i><font color="Green">E88</font></i></a>, <a class="txt" href="scmbsort.html#E9:78_4"><i><font color="Green">E90</font></i></a>, <a class="ref" href="scmbsort.html#T8" target="_self">Th8</a>, <a class="ref" href="scmbsort.html#T12" target="_self">Th12</a>, </i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="78_4_3_1_6"><b>suppose </b></a><a NAME="E1:78_4_3_1_6"/><i><font color="Green">E136</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="scmfsa_2.html#K4">intloc</a> 4
 ;<br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> 11</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<font color="Maroon">s1</font> <a href="scmfsa_2.html#K20">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 0</span>)</span>
<b>by </b><i><a class="txt" href="scmbsort.html#E6:78_4"><i><font color="Green">E87</font></i></a>, <a class="txt" href="scmbsort.html#E8:78_4"><i><font color="Green">E89</font></i></a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="ami_1.html#K11">.</a> 11</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="txt" href="scmbsort.html#E7:78_4"><i><font color="Green">E88</font></i></a>, <a class="txt" href="scmbsort.html#E9:78_4"><i><font color="Green">E90</font></i></a>, <a class="ref" href="scmbsort.html#T8" target="_self">Th8</a>, <a class="ref" href="scmbsort.html#T12" target="_self">Th12</a>, </i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="78_4_3_1_7"><b>suppose </b></a><a NAME="E1:78_4_3_1_7"/><i><font color="Green">E137</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="scmfsa_2.html#K4">intloc</a> 5
 ;<br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> 11</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<font color="Maroon">s1</font> <a href="scmfsa_2.html#K20">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 0</span>)</span>
<b>by </b><i><a class="txt" href="scmbsort.html#E6:78_4"><i><font color="Green">E87</font></i></a>, <a class="txt" href="scmbsort.html#E8:78_4"><i><font color="Green">E89</font></i></a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="ami_1.html#K11">.</a> 11</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="txt" href="scmbsort.html#E7:78_4"><i><font color="Green">E88</font></i></a>, <a class="txt" href="scmbsort.html#E9:78_4"><i><font color="Green">E90</font></i></a>, <a class="ref" href="scmbsort.html#T8" target="_self">Th8</a>, <a class="ref" href="scmbsort.html#T12" target="_self">Th12</a>, </i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="78_4_3_1_8"><b>suppose </b></a><a NAME="E1:78_4_3_1_8"/><i><font color="Green">E138</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="scmfsa_2.html#K4">intloc</a> 6
 ;<br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> 11</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<font color="Maroon">s1</font> <a href="scmfsa_2.html#K20">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 0</span>)</span>
<b>by </b><i><a class="txt" href="scmbsort.html#E6:78_4"><i><font color="Green">E87</font></i></a>, <a class="txt" href="scmbsort.html#E8:78_4"><i><font color="Green">E89</font></i></a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="ami_1.html#K11">.</a> 11</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="txt" href="scmbsort.html#E7:78_4"><i><font color="Green">E88</font></i></a>, <a class="txt" href="scmbsort.html#E9:78_4"><i><font color="Green">E90</font></i></a>, <a class="ref" href="scmbsort.html#T8" target="_self">Th8</a>, <a class="ref" href="scmbsort.html#T12" target="_self">Th12</a>, </i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E14:78_4"/><b>then </b><i><font color="Green">E139</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> 11</span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sf_mastr.html#K4">UsedInt*Loc</a> <a href="scmbsort.html#K3" target="_self">Bubble-Sort-Algorithm</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><a href="sf_mastr.html#K2">UsedIntLoc</a> <a href="scmbsort.html#K3" target="_self">Bubble-Sort-Algorithm</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="ami_1.html#K11">.</a> 11</span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sf_mastr.html#K4">UsedInt*Loc</a> <a href="scmbsort.html#K3" target="_self">Bubble-Sort-Algorithm</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><a href="sf_mastr.html#K2">UsedIntLoc</a> <a href="scmbsort.html#K3" target="_self">Bubble-Sort-Algorithm</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="scmbsort.html#T9" target="_self">Th9</a>, <a class="ref" href="scmbsort.html#T10" target="_self">Th10</a>, <a class="ref" href="scmfsa6a.html#T9">SCMFSA6A:9</a></i>;<br/>
<i><font color="Green">E140</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> 11</span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K2">IC</a> <a href="scmfsa_2.html#K1">SCM+FSA</a> </span>)</span> = 
 <a href="scmfsa_2.html#K5">insloc</a> 11
<b>by </b><i><a class="txt" href="scmbsort.html#E6:78_4"><i><font color="Green">E87</font></i></a>, <a class="txt" href="scmbsort.html#E8:78_4"><i><font color="Green">E89</font></i></a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="ami_1.html#K11">.</a> 11</span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K2">IC</a> <a href="scmfsa_2.html#K1">SCM+FSA</a> </span>)</span>
<b>by </b><i><a class="txt" href="scmbsort.html#E7:78_4"><i><font color="Green">E88</font></i></a>, <a class="txt" href="scmbsort.html#E9:78_4"><i><font color="Green">E90</font></i></a>, </i>
;<br/>
<a NAME="E16:78_4"/><i><font color="Green">E141</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds   <a href="ami_1.html#K6">IC</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Olive">i</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <a href="scmbsort.html#K3" target="_self">Bubble-Sort-Algorithm</a> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E17:78_4"/>
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds   <a href="ami_1.html#K6">IC</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Olive">i</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <a href="scmbsort.html#K3" target="_self">Bubble-Sort-Algorithm</a> 
 
<b>by </b><i><a class="txt" href="scmbsort.html#E4:78_4"><i><font color="Green">E85</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E18:78_4"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sf_mastr.html#K4">UsedInt*Loc</a> <a href="scmbsort.html#K3" target="_self">Bubble-Sort-Algorithm</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><a href="sf_mastr.html#K2">UsedIntLoc</a> <a href="scmbsort.html#K3" target="_self">Bubble-Sort-Algorithm</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sf_mastr.html#K4">UsedInt*Loc</a> <a href="scmbsort.html#K3" target="_self">Bubble-Sort-Algorithm</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><a href="sf_mastr.html#K2">UsedIntLoc</a> <a href="scmbsort.html#K3" target="_self">Bubble-Sort-Algorithm</a> </span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K2">IC</a> <a href="scmfsa_2.html#K1">SCM+FSA</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K2">IC</a> <a href="scmfsa_2.html#K1">SCM+FSA</a> </span>)</span> )
 <b>by </b><i><a class="txt" href="scmbsort.html#E3:78_4"><i><font color="Green">E83</font></i></a>, <a class="txt" href="scmbsort.html#E8:78_4"><i><font color="Green">E89</font></i></a>, <a class="txt" href="scmbsort.html#E9:78_4"><i><font color="Green">E90</font></i></a>, <a class="ref" href="scmbsort.html#T11" target="_self">Th11</a>, <a class="ref" href="scmbsort.html#T12" target="_self">Th12</a>, , </i>;<br/>


</div><b>end;</b></div>
<b>set </b><font color="Maroon">DD</font> = <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 0</span>)</span>,<span class="p2">(<span class="default"><a href="ami_1.html#K2">IC</a> <a href="scmfsa_2.html#K1">SCM+FSA</a> </span>)</span>,<span class="p2">(<span class="default"><a href="scmfsa_2.html#K6">fsloc</a> 0</span>)</span></span><a href="enumset1.html#K1">}</a></span>;<br/>
<a NAME="E6:78"/><i><font color="Green">E142</font></i>: 
 <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> <span class="p2">(<span class="default"><span class="p3">(<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></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <a href="scmbsort.html#K3" target="_self">Bubble-Sort-Algorithm</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 0</span>)</span>,<span class="p2">(<span class="default"><a href="ami_1.html#K2">IC</a> <a href="scmfsa_2.html#K1">SCM+FSA</a> </span>)</span>,<span class="p2">(<span class="default"><a href="scmfsa_2.html#K6">fsloc</a> 0</span>)</span></span><a href="enumset1.html#K1">}</a></span>
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">s1</font> be   <a href="ami_1.html#NM4">State</a> of <a href="scmfsa_2.html#K1">SCM+FSA</a> , <font color="Maroon">s2</font> be   <a href="ami_1.html#NM4">State</a> of <a href="scmfsa_2.html#K1">SCM+FSA</a> ;<br/><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:78_5"/><i><font color="Green">E143</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> <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="scmbsort.html#K1" target="_self">.--&gt;</a> <font color="Maroon">w</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">s1</font> &amp; <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> <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="scmbsort.html#K1" target="_self">.--&gt;</a> <font color="Maroon">w</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">s2</font> )
 ;<br/><b>set </b><font color="Maroon">Cs1i</font> = <span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</font>;<br/><b>set </b><font color="Maroon">Cs2i</font> = <span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</font>;<br/><a NAME="E2:78_5"/>
(  <a href="scmbsort.html#K3" target="_self">Bubble-Sort-Algorithm</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">s1</font> &amp;  <a href="scmbsort.html#K3" target="_self">Bubble-Sort-Algorithm</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">s2</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E3:78_5"/><b>then </b><i><font color="Green">E144</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <a href="scmbsort.html#K3" target="_self">Bubble-Sort-Algorithm</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <a href="scmbsort.html#K3" target="_self">Bubble-Sort-Algorithm</a> </span>)</span>
 <b>by </b><i/>;<br/><a NAME="E4:78_5"/><i><font color="Green">E145</font></i>: 
<span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 0</span>)</span>,<span class="p2">(<span class="default"><a href="ami_1.html#K2">IC</a> <a href="scmfsa_2.html#K1">SCM+FSA</a> </span>)</span>,<span class="p2">(<span class="default"><a href="scmfsa_2.html#K6">fsloc</a> 0</span>)</span></span><a href="enumset1.html#K1">}</a></span> <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="ami_1.html#K10">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E5:78_5"/><i><font color="Green">E146</font></i>: 
<span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 0</span>)</span>,<span class="p2">(<span class="default"><a href="ami_1.html#K2">IC</a> <a href="scmfsa_2.html#K1">SCM+FSA</a> </span>)</span>,<span class="p2">(<span class="default"><a href="scmfsa_2.html#K6">fsloc</a> 0</span>)</span></span><a href="enumset1.html#K1">}</a></span> <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="ami_1.html#K10">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E6:78_5"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 0</span>)</span>,<span class="p2">(<span class="default"><a href="ami_1.html#K2">IC</a> <a href="scmfsa_2.html#K1">SCM+FSA</a> </span>)</span>,<span class="p2">(<span class="default"><a href="scmfsa_2.html#K6">fsloc</a> 0</span>)</span></span><a href="enumset1.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 0</span>)</span>,<span class="p2">(<span class="default"><a href="ami_1.html#K2">IC</a> <a href="scmfsa_2.html#K1">SCM+FSA</a> </span>)</span>,<span class="p2">(<span class="default"><a href="scmfsa_2.html#K6">fsloc</a> 0</span>)</span></span><a href="enumset1.html#K1">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_5_1"><b>proof </b></a><div class="add">

<a NAME="E1:78_5_1"/><i><font color="Green">E147</font></i>: 
 <a href="scmfsa_2.html#K4">intloc</a> 0 <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="sf_mastr.html#K4">UsedInt*Loc</a> <a href="scmbsort.html#K3" target="_self">Bubble-Sort-Algorithm</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="sf_mastr.html#K2">UsedIntLoc</a> <a href="scmbsort.html#K3" target="_self">Bubble-Sort-Algorithm</a> </span>)</span>
 
<b>by </b><i>, <a class="ref" href="enumset1.html#D6">ENUMSET1:def 6</a></i>;<br/>
<a NAME="E2:78_5_1"/><i><font color="Green">E148</font></i>: 
 <a href="scmfsa_2.html#K6">fsloc</a> 0 <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="sf_mastr.html#K4">UsedInt*Loc</a> <a href="scmbsort.html#K3" target="_self">Bubble-Sort-Algorithm</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="sf_mastr.html#K2">UsedIntLoc</a> <a href="scmbsort.html#K3" target="_self">Bubble-Sort-Algorithm</a> </span>)</span>
 
<b>by </b><i>, <a class="ref" href="enumset1.html#D6">ENUMSET1:def 6</a></i>;<br/>
<a NAME="E3:78_5_1"/><i><font color="Green">E149</font></i>: 
<span class="p1">(<span class="default"><a href="sf_mastr.html#K4">UsedInt*Loc</a> <a href="scmbsort.html#K3" target="_self">Bubble-Sort-Algorithm</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="sf_mastr.html#K2">UsedIntLoc</a> <a href="scmbsort.html#K3" target="_self">Bubble-Sort-Algorithm</a> </span>)</span> <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="ami_1.html#K10">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E4:78_5_1"/><i><font color="Green">E150</font></i>: 
<span class="p1">(<span class="default"><a href="sf_mastr.html#K4">UsedInt*Loc</a> <a href="scmbsort.html#K3" target="_self">Bubble-Sort-Algorithm</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="sf_mastr.html#K2">UsedIntLoc</a> <a href="scmbsort.html#K3" target="_self">Bubble-Sort-Algorithm</a> </span>)</span> <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="ami_1.html#K10">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E5:78_5_1"/><i><font color="Green">E151</font></i>: 
( <font color="Maroon">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 10 implies 10 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_5_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:78_5_1_1"/><i><font color="Green">E152</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 0</span>)</span>,<span class="p2">(<span class="default"><a href="ami_1.html#K2">IC</a> <a href="scmfsa_2.html#K1">SCM+FSA</a> </span>)</span>,<span class="p2">(<span class="default"><a href="scmfsa_2.html#K6">fsloc</a> 0</span>)</span></span><a href="enumset1.html#K1">}</a></span>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="78_5_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:78_5_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="scmfsa_2.html#K4">intloc</a> 0 or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="ami_1.html#K2">IC</a> <a href="scmfsa_2.html#K1">SCM+FSA</a>  or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="scmfsa_2.html#K6">fsloc</a> 0 )
 </span><b>by </b><i><a class="ref" href="scmbsort.html#T9" target="_self">Th9</a>, <a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="78_5_1_1_1_1"><b>suppose </b></a><a NAME="E1:78_5_1_1_1_1"/><i><font color="Green">E153</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="scmfsa_2.html#K4">intloc</a> 0
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_5_1_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="78_5_1_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:78_5_1_1_1_1_1_1"/>
( <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> 10 or <font color="Maroon">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 10 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="78_5_1_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:78_5_1_1_1_1_1_1_1"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> 10
 ;<br/><div class="add"><b>hence </b><a NAME="E2:78_5_1_1_1_1_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="txt" href="scmbsort.html#E3:78"><i><font color="Green">E78</font></i></a>, , <a class="ref" href="scmbsort.html#T10" target="_self">Th10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="78_5_1_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:78_5_1_1_1_1_1_1_2"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 10
 ;<br/><div class="add"><a NAME="E2:78_5_1_1_1_1_1_1_2"/><b>then </b>
11 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:78_5_1_1_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sf_mastr.html#K4">UsedInt*Loc</a> <a href="scmbsort.html#K3" target="_self">Bubble-Sort-Algorithm</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><a href="sf_mastr.html#K2">UsedIntLoc</a> <a href="scmbsort.html#K3" target="_self">Bubble-Sort-Algorithm</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sf_mastr.html#K4">UsedInt*Loc</a> <a href="scmbsort.html#K3" target="_self">Bubble-Sort-Algorithm</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><a href="sf_mastr.html#K2">UsedIntLoc</a> <a href="scmbsort.html#K3" target="_self">Bubble-Sort-Algorithm</a> </span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E4:78_5_1_1_1_1_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, , <a class="ref" href="scmbsort.html#T8" target="_self">Th8</a>, <a class="ref" href="scmbsort.html#T10" target="_self">Th10</a>, <a class="ref" href="scmfsa6a.html#T9">SCMFSA6A:9</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:78_5_1_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="78_5_1_1_1_2"><b>suppose </b></a><a NAME="E1:78_5_1_1_1_2"/><i><font color="Green">E154</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="ami_1.html#K2">IC</a> <a href="scmfsa_2.html#K1">SCM+FSA</a> 
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_5_1_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="78_5_1_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:78_5_1_1_1_2_1_1"/>
( <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> 10 or <font color="Maroon">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 10 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="78_5_1_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:78_5_1_1_1_2_1_1_1"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> 10
 ;<br/><div class="add"><b>hence </b><a NAME="E2:78_5_1_1_1_2_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="txt" href="scmbsort.html#E3:78"><i><font color="Green">E78</font></i></a>, , <a class="ref" href="scmbsort.html#T10" target="_self">Th10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="78_5_1_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:78_5_1_1_1_2_1_1_2"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 10
 ;<br/><div class="add"><a NAME="E2:78_5_1_1_1_2_1_1_2"/><b>then </b>
11 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>hence </b><a NAME="E3:78_5_1_1_1_2_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, , <a class="ref" href="scmbsort.html#T10" target="_self">Th10</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:78_5_1_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="78_5_1_1_1_3"><b>suppose </b></a><a NAME="E1:78_5_1_1_1_3"/><i><font color="Green">E155</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="scmfsa_2.html#K6">fsloc</a> 0
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_5_1_1_1_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="78_5_1_1_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:78_5_1_1_1_3_1_1"/>
( <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> 10 or <font color="Maroon">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 10 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="78_5_1_1_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:78_5_1_1_1_3_1_1_1"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> 10
 ;<br/><div class="add"><b>hence </b><a NAME="E2:78_5_1_1_1_3_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="txt" href="scmbsort.html#E3:78"><i><font color="Green">E78</font></i></a>, , <a class="ref" href="scmbsort.html#T10" target="_self">Th10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="78_5_1_1_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:78_5_1_1_1_3_1_1_2"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 10
 ;<br/><div class="add"><a NAME="E2:78_5_1_1_1_3_1_1_2"/><b>then </b>
11 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:78_5_1_1_1_3_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sf_mastr.html#K4">UsedInt*Loc</a> <a href="scmbsort.html#K3" target="_self">Bubble-Sort-Algorithm</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><a href="sf_mastr.html#K2">UsedIntLoc</a> <a href="scmbsort.html#K3" target="_self">Bubble-Sort-Algorithm</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sf_mastr.html#K4">UsedInt*Loc</a> <a href="scmbsort.html#K3" target="_self">Bubble-Sort-Algorithm</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><a href="sf_mastr.html#K2">UsedIntLoc</a> <a href="scmbsort.html#K3" target="_self">Bubble-Sort-Algorithm</a> </span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E4:78_5_1_1_1_3_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, , <a class="ref" href="scmbsort.html#T8" target="_self">Th8</a>, <a class="ref" href="scmbsort.html#T10" target="_self">Th10</a>, <a class="ref" href="scmfsa6a.html#T9">SCMFSA6A:9</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:78_5_1_1_1_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E7:78_5_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 0</span>)</span>,<span class="p2">(<span class="default"><a href="ami_1.html#K2">IC</a> <a href="scmfsa_2.html#K1">SCM+FSA</a> </span>)</span>,<span class="p2">(<span class="default"><a href="scmfsa_2.html#K6">fsloc</a> 0</span>)</span></span><a href="enumset1.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 0</span>)</span>,<span class="p2">(<span class="default"><a href="ami_1.html#K2">IC</a> <a href="scmfsa_2.html#K1">SCM+FSA</a> </span>)</span>,<span class="p2">(<span class="default"><a href="scmfsa_2.html#K6">fsloc</a> 0</span>)</span></span><a href="enumset1.html#K1">}</a></span>
 <b>by </b><i>, , <a class="ref" href="scmfsa6a.html#T9">SCMFSA6A:9</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E7:78_5"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</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> <span class="p3">(<span class="default"><span class="p4">(<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></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</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> <span class="p3">(<span class="default"><span class="p4">(<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></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="relat_1.html#T185">RELAT_1:185</a></i>;<br/></div><b>end;</b></div>
<a NAME="E8:78"/><b>then </b>
for <font color="Olive">s1</font>, <font color="Olive">s2</font> being  <a href="ami_1.html#NM4">State</a> of <a href="scmfsa_2.html#K1">SCM+FSA</a>   st <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> <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="scmbsort.html#K1" target="_self">.--&gt;</a> <font color="Maroon">w</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Olive">s1</font> &amp; <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> <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="scmbsort.html#K1" target="_self">.--&gt;</a> <font color="Maroon">w</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Olive">s2</font> holds <br/>for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Olive">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Olive">i</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> <span class="p3">(<span class="default"><span class="p4">(<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></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Olive">s2</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Olive">i</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> <span class="p3">(<span class="default"><span class="p4">(<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></span>)</span></span>)</span></span>)</span>
 
;<br/>
<b>hence </b><a NAME="E9:78"/>
<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> <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="scmbsort.html#K1" target="_self">.--&gt;</a> <font color="Maroon">w</font></span>)</span> is <a href="ami_1.html#V11">autonomic</a>
 <b>by </b><i><a class="ref" href="ami_1.html#D25">AMI_1:def 25</a></i>;<br/>


</div>
