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

<b>let </b><font color="Maroon">s</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">t</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">Ia</font> =  <a href="scmisort.html#K4" target="_self">Insert-Sort-Algorithm</a> ;<br/>
<b>set </b><font color="Maroon">p</font> =  <a href="scmfsa6a.html#K3">Initialized</a> <a href="scmisort.html#K4" target="_self">Insert-Sort-Algorithm</a> ;<br/>
<b>set </b><font color="Maroon">x</font> = <span class="p1">(<span class="default"><a href="scmfsa_2.html#K6">fsloc</a> 0</span>)</span> <a href="scmbsort.html#K1">.--&gt;</a> <font color="Maroon">t</font>;<br/>
<b>set </b><font color="Maroon">z</font> = <span class="p1">(<span class="default"><a href="scmfsa6b.html#K3">IExec</a> <span class="p2">(<span class="default"><a href="scmisort.html#K3" target="_self">insert-sort</a> <span class="p3">(<span class="default"><a href="scmfsa_2.html#K6">fsloc</a> 0</span>)</span></span>)</span>,<font color="Maroon">s</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>;<br/>
<b>assume </b><a NAME="E1:70"/><i><font color="Green">E35</font></i>: 
<span class="p1">(<span class="default"><a href="scmfsa6a.html#K3">Initialized</a> <a href="scmisort.html#K4" target="_self">Insert-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">.--&gt;</a> <font color="Maroon">t</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">s</font>
 ;<br/>

<a NAME="E2:70"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_2.html#K6">fsloc</a> 0</span>)</span> <a href="scmbsort.html#K1">.--&gt;</a> <font color="Maroon">t</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="scmfsa_2.html#K6">fsloc</a> 0</span>)</span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>;<br/>
<a NAME="E3:70"/><b>then </b><i><font color="Green">E41</font></i>: 
 <a href="scmfsa_2.html#K6">fsloc</a> 0 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_2.html#K6">fsloc</a> 0</span>)</span> <a href="scmbsort.html#K1">.--&gt;</a> <font color="Maroon">t</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E4:70"/><b>then </b>
 <a href="scmfsa_2.html#K6">fsloc</a> 0 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa6a.html#K3">Initialized</a> <a href="scmisort.html#K4" target="_self">Insert-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">.--&gt;</a> <font color="Maroon">t</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_4.html#T13">FUNCT_4:13</a></i>;<br/>
<b>then </b><i><font color="Green">E43</font></i>: <font color="Maroon">s</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> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa6a.html#K3">Initialized</a> <a href="scmisort.html#K4" target="_self">Insert-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">.--&gt;</a> <font color="Maroon">t</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K6">fsloc</a> 0</span>)</span>
<b>by </b><i>, <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
<br/>.= 
<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">.--&gt;</a> <font color="Maroon">t</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K6">fsloc</a> 0</span>)</span>
<b>by </b><i>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
<font color="Maroon">t</font>
<b>by </b><i><a class="ref" href="cqc_lang.html#T6">CQC_LANG:6</a></i>
;<br/>
<a NAME="E6:70"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">s</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>,<span class="p1">(<span class="default"><a href="scmfsa6b.html#K3">IExec</a> <span class="p2">(<span class="default"><a href="scmisort.html#K3" target="_self">insert-sort</a> <span class="p3">(<span class="default"><a href="scmfsa_2.html#K6">fsloc</a> 0</span>)</span></span>)</span>,<font color="Maroon">s</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="rfinseq.html#R1">are_fiberwise_equipotent</a>  &amp; ( for <font color="Olive">i</font>, <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Olive">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">s</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> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">j</font> holds <br/>for <font color="Olive">x1</font>, <font color="Olive">x2</font> being  <a href="int_1.html#NM1">Integer</a>  st <font color="Olive">x1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa6b.html#K3">IExec</a> <span class="p3">(<span class="default"><a href="scmisort.html#K3" target="_self">insert-sort</a> <span class="p4">(<span class="default"><a href="scmfsa_2.html#K6">fsloc</a> 0</span>)</span></span>)</span>,<font color="Maroon">s</font></span>)</span> <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> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font> &amp; <font color="Olive">x2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa6b.html#K3">IExec</a> <span class="p3">(<span class="default"><a href="scmisort.html#K3" target="_self">insert-sort</a> <span class="p4">(<span class="default"><a href="scmfsa_2.html#K6">fsloc</a> 0</span>)</span></span>)</span>,<font color="Maroon">s</font></span>)</span> <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> <a href="seq_1.html#K2">.</a> <font color="Olive">j</font> holds <br/><font color="Olive">x1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">x2</font> ) )
 
<b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">u</font> = <span class="p1">(<span class="default"><a href="scmfsa6b.html#K3">IExec</a> <span class="p2">(<span class="default"><a href="scmisort.html#K3" target="_self">insert-sort</a> <span class="p3">(<span class="default"><a href="scmfsa_2.html#K6">fsloc</a> 0</span>)</span></span>)</span>,<font color="Maroon">s</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> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="scmbsort.html#T38">SCMBSORT:38</a></i>;<br/>
<b>take </b>
<font color="Maroon">u</font>
;<br/>

<b>thus </b><a NAME="E8:70"/>
<font color="Maroon">t</font>,<font color="Maroon">u</font> <a href="rfinseq.html#R1">are_fiberwise_equipotent</a> 
 <b>by </b><i>, </i>;<br/>

<a NAME="E9:70"/><i><font color="Green">E47</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">s</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> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">u</font>
 
<b>by </b><i>, <a class="ref" href="rfinseq.html#T16">RFINSEQ:16</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_2"><b>now </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> , <font color="Maroon">j</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:70_2"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">u</font> &amp; <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">u</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font> )
 ;<br/><a NAME="E2:70_2"/><b>then </b><i><font color="Green">E50</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E3:70_2"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">s</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="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E4:70_2"/><i><font color="Green">E53</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa6b.html#K3">IExec</a> <span class="p3">(<span class="default"><a href="scmisort.html#K3" target="_self">insert-sort</a> <span class="p4">(<span class="default"><a href="scmfsa_2.html#K6">fsloc</a> 0</span>)</span></span>)</span>,<font color="Maroon">s</font></span>)</span> <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> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K4">INT</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa6b.html#K3">IExec</a> <span class="p3">(<span class="default"><a href="scmisort.html#K3" target="_self">insert-sort</a> <span class="p4">(<span class="default"><a href="scmfsa_2.html#K6">fsloc</a> 0</span>)</span></span>)</span>,<font color="Maroon">s</font></span>)</span> <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> <a href="seq_1.html#K2">.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K4">INT</a>  )
 <b>by </b><i>, <a class="ref" href="finseq_2.html#T13">FINSEQ_2:13</a></i>;<br/><b>then reconsider </b><font color="Maroon">y1</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa6b.html#K3">IExec</a> <span class="p3">(<span class="default"><a href="scmisort.html#K3" target="_self">insert-sort</a> <span class="p4">(<span class="default"><a href="scmfsa_2.html#K6">fsloc</a> 0</span>)</span></span>)</span>,<font color="Maroon">s</font></span>)</span> <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> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> as   <a href="int_1.html#NM1">Integer</a> <b>by </b><i><a class="ref" href="int_1.html#D2">INT_1:def 2</a></i>;<br/><b>reconsider </b><font color="Maroon">y2</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa6b.html#K3">IExec</a> <span class="p3">(<span class="default"><a href="scmisort.html#K3" target="_self">insert-sort</a> <span class="p4">(<span class="default"><a href="scmfsa_2.html#K6">fsloc</a> 0</span>)</span></span>)</span>,<font color="Maroon">s</font></span>)</span> <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> <a href="seq_1.html#K2">.</a> <font color="Maroon">j</font> as   <a href="int_1.html#NM1">Integer</a> <b>by </b><i>, <a class="ref" href="int_1.html#D2">INT_1:def 2</a></i>;<br/><a NAME="E7:70_2"/>
<font color="Maroon">y1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">y2</font>
 <b>by </b><i>, , , </i>;<br/><b>hence </b><a NAME="E8:70_2"/>
<font color="Maroon">u</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">u</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">j</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E11:70"/>
<font color="Maroon">u</font> is <a href="rfinseq.html#V1">non-increasing</a>
 <b>by </b><i><a class="ref" href="rfinseq.html#T32">RFINSEQ:32</a></i>;<br/>

<b>thus </b><a NAME="E12:70"/>
<font color="Maroon">u</font> is    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K4">INT</a> 
 ;<br/>

<a NAME="E13:70"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="scmfsa6a.html#K3">Initialized</a> <a href="scmisort.html#K4" target="_self">Insert-Sort-Algorithm</a> </span>)</span> <a href="xboole_0.html#R1">misses</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_2.html#K6">fsloc</a> 0</span>)</span> <a href="scmbsort.html#K1">.--&gt;</a> <font color="Maroon">t</font></span>)</span>
 
<b>by </b><i><a class="ref" href="scmbsort.html#T46">SCMBSORT:46</a></i>;<br/>
<a NAME="E14:70"/><b>then </b>
 <a href="scmfsa6a.html#K3">Initialized</a> <a href="scmisort.html#K4" target="_self">Insert-Sort-Algorithm</a>  <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="scmfsa6a.html#K3">Initialized</a> <a href="scmisort.html#K4" target="_self">Insert-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">.--&gt;</a> <font color="Maroon">t</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_4.html#T33">FUNCT_4:33</a></i>;<br/>
<a NAME="E15:70"/><b>then </b>
 <a href="scmfsa6a.html#K3">Initialized</a> <a href="scmisort.html#K4" target="_self">Insert-Sort-Algorithm</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">s</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E16:70"/><b>then </b>
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="ami_5.html#K8">+*</a> <span class="p1">(<span class="default"><a href="scmfsa6a.html#K3">Initialized</a> <a href="scmisort.html#K4" target="_self">Insert-Sort-Algorithm</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="funct_4.html#T79">FUNCT_4:79</a></i>;<br/>
<b>hence </b><a NAME="E17:70"/>
<span class="p1">(<span class="default"><a href="ami_1.html#K12">Result</a> <font color="Maroon">s</font></span>)</span> <a href="scmfsa_2.html#K21">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K6">fsloc</a> 0</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font>
 <b>by </b><i><a class="ref" href="scmbsort.html#T57">SCMBSORT:57</a></i>;<br/>


</div>
