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

<b>let </b><font color="Maroon">OAlph</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">IAlph</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">tfsm</font> be  non <a href="struct_0.html#V3">empty</a> <a href="group_1.html#V6">finite</a>  <a href="fsm_1.html#L2" target="_self">Mealy-FSM</a> of <font color="Maroon">IAlph</font>,<font color="Maroon">OAlph</font>;<br/>





<b>set </b><font color="Maroon">cTran</font> = the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">tfsm</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="fsm_1.html#K12" target="_self">accessibleStates</a> <font color="Maroon">tfsm</font></span>)</span>,<font color="Maroon">IAlph</font></span><a href="zfmisc_1.html#K2">:]</a></span>;<br/>
<a NAME="E1:99"/><i><font color="Green">E22</font></i>: 
(  <a href="fsm_1.html#K12" target="_self">accessibleStates</a> <font color="Maroon">tfsm</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">tfsm</font> &amp; <font color="Maroon">IAlph</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">IAlph</font> )
 
<b>by </b><i><a class="ref" href="fsm_1.html#T14" target="_self">Th14</a></i>;<br/>
<a NAME="E2:99"/><b>then </b>
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="fsm_1.html#K12" target="_self">accessibleStates</a> <font color="Maroon">tfsm</font></span>)</span>,<font color="Maroon">IAlph</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">tfsm</font>,<font color="Maroon">IAlph</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/>
<a NAME="E3:99"/><b>then </b>
the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">tfsm</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="fsm_1.html#K12" target="_self">accessibleStates</a> <font color="Maroon">tfsm</font></span>)</span>,<font color="Maroon">IAlph</font></span><a href="zfmisc_1.html#K2">:]</a></span> is   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="fsm_1.html#K12" target="_self">accessibleStates</a> <font color="Maroon">tfsm</font></span>)</span>,<font color="Maroon">IAlph</font></span><a href="zfmisc_1.html#K2">:]</a></span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">tfsm</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T38">FUNCT_2:38</a></i>;<br/>
<a NAME="E4:99"/><b>then </b><i><font color="Green">E23</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">tfsm</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p3">(<span class="default"><a href="fsm_1.html#K12" target="_self">accessibleStates</a> <font color="Maroon">tfsm</font></span>)</span>,<font color="Maroon">IAlph</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="fsm_1.html#K12" target="_self">accessibleStates</a> <font color="Maroon">tfsm</font></span>)</span>,<font color="Maroon">IAlph</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E5:99"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default">the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">tfsm</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p3">(<span class="default"><a href="fsm_1.html#K12" target="_self">accessibleStates</a> <font color="Maroon">tfsm</font></span>)</span>,<font color="Maroon">IAlph</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="fsm_1.html#K12" target="_self">accessibleStates</a> <font color="Maroon">tfsm</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="99_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:99_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default">the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">tfsm</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p3">(<span class="default"><a href="fsm_1.html#K12" target="_self">accessibleStates</a> <font color="Maroon">tfsm</font></span>)</span>,<font color="Maroon">IAlph</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>
 ;<br/>

<a NAME="E2:99_1"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default">the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">tfsm</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p3">(<span class="default"><a href="fsm_1.html#K12" target="_self">accessibleStates</a> <font color="Maroon">tfsm</font></span>)</span>,<font color="Maroon">IAlph</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">tfsm</font>
 
<b>by </b><i><a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">q1</font> = <font color="Maroon">x</font> as   <a href="fsm_1.html#NM1">State</a> of <font color="Maroon">tfsm</font> <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">d</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:99_1"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">d</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">tfsm</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p3">(<span class="default"><a href="fsm_1.html#K12" target="_self">accessibleStates</a> <font color="Maroon">tfsm</font></span>)</span>,<font color="Maroon">IAlph</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">tfsm</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p3">(<span class="default"><a href="fsm_1.html#K12" target="_self">accessibleStates</a> <font color="Maroon">tfsm</font></span>)</span>,<font color="Maroon">IAlph</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">d</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E6:99_1"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">d</font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R2">in</a>  <a href="fsm_1.html#K12" target="_self">accessibleStates</a> <font color="Maroon">tfsm</font> &amp; <font color="Maroon">d</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">IAlph</font> )
 
<b>by </b><i>, , <a class="ref" href="mcart_1.html#T10">MCART_1:10</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">q</font> = <font color="Maroon">d</font> <a href="mcart_1.html#K1">`1</a>  as   <a href="fsm_1.html#NM1">State</a> of <font color="Maroon">tfsm</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">s</font> = <font color="Maroon">d</font> <a href="mcart_1.html#K2">`2</a>  as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">IAlph</font> <b>by </b><i>, , <a class="ref" href="mcart_1.html#T10">MCART_1:10</a></i>;<br/>
<b>set </b><font color="Maroon">I</font> = the <a href="fsm_1.html#U2" target="_self">InitS</a> of <font color="Maroon">tfsm</font>;<br/>
<a NAME="E9:99_1"/>
<font color="Maroon">q</font> is <a href="fsm_1.html#V6" target="_self">accessible</a>
 
<b>by </b><i>, <a class="ref" href="fsm_1.html#T14" target="_self">Th14</a></i>;<br/>
<b>then consider </b><font color="Maroon">w</font> being    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">IAlph</font><b> such that </b><br/><a NAME="E11:99_1"/><i><font color="Green">E42</font></i>: 
the <a href="fsm_1.html#U2" target="_self">InitS</a> of <font color="Maroon">tfsm</font>,<font color="Maroon">w</font> <a href="fsm_1.html#R1" target="_self">-leads_to</a> <font color="Maroon">q</font>
 <b>by </b><i/>;<br/>
<a NAME="E12:99_1"/><i><font color="Green">E43</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">s</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">w</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">s</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/>
<b>set </b><font color="Maroon">qsa</font> = <font color="Maroon">q</font>,<span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">s</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="fsm_1.html#K2" target="_self">-admissible</a> ;<br/>
<a NAME="E13:99_1"/><i><font color="Green">E44</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">q</font>,<span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">s</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="fsm_1.html#K2" target="_self">-admissible</a> </span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">q</font>
 
<b>by </b><i/>;<br/>
<a NAME="E14:99_1"/><i><font color="Green">E45</font></i>: 
<span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">s</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/>
<a NAME="E15:99_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">s</font></span><a href="finseq_1.html#K12">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/>
<a NAME="E16:99_1"/><b>then </b><i><font color="Green">E46</font></i>: 
 ex <font color="Olive">wi</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">IAlph</font> ex <font color="Olive">qi</font>, <font color="Olive">qi1</font> being  <a href="fsm_1.html#NM1">State</a> of <font color="Maroon">tfsm</font> st <br/>( <font color="Olive">wi</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">s</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 1 &amp; <font color="Olive">qi</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">q</font>,<span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">s</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="fsm_1.html#K2" target="_self">-admissible</a> </span>)</span> <a href="funct_1.html#K1">.</a> 1 &amp; <font color="Olive">qi1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">q</font>,<span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">s</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="fsm_1.html#K2" target="_self">-admissible</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Olive">wi</font> <a href="fsm_1.html#K1" target="_self">-succ_of</a> <font color="Olive">qi</font> <a href="hidden.html#R1">=</a> <font color="Olive">qi1</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E17:99_1"/>
the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">tfsm</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q1</font>
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>
<a NAME="E18:99_1"/><b>then </b><i><font color="Green">E47</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">q</font>,<span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">s</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="fsm_1.html#K2" target="_self">-admissible</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">q1</font>
 
<b>by </b><i><a class="txt" href="fsm_1.html#E19:99_1"><i><font color="Green">E79</font></i></a>, , , , , <a class="ref" href="mcart_1.html#T23">MCART_1:23</a></i>;<br/>
<i><font color="Green">E79</font></i>: <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">w</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">s</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">w</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i>, <a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">w</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span>

;<br/>
<a NAME="E20:99_1"/>
1 <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> 2
 
;<br/>
<a NAME="E21:99_1"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> 2 &amp; 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">s</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/>
<a NAME="E22:99_1"/><b>then </b>
<span class="p1">(<span class="default">the <a href="fsm_1.html#U2" target="_self">InitS</a> of <font color="Maroon">tfsm</font>,<span class="p2">(<span class="default"><font color="Maroon">w</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">s</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="fsm_1.html#K2" target="_self">-admissible</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">w</font> <a href="finseq_1.html#K8">^</a> <span class="p4"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">s</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">q1</font>
 
<b>by </b><i><a class="ref" href="fsm_1.html#T15" target="_self">Th15</a>, , , </i>;<br/>
<a NAME="E23:99_1"/><b>then </b>
the <a href="fsm_1.html#U2" target="_self">InitS</a> of <font color="Maroon">tfsm</font>,<font color="Maroon">w</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">s</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="fsm_1.html#R1" target="_self">-leads_to</a> <font color="Maroon">q1</font>
 
<b>by </b><i/>;<br/>
<a NAME="E24:99_1"/><b>then </b>
<font color="Maroon">q1</font> is <a href="fsm_1.html#V6" target="_self">accessible</a>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E25:99_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="fsm_1.html#K12" target="_self">accessibleStates</a> <font color="Maroon">tfsm</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:99"/>
the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">tfsm</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="fsm_1.html#K12" target="_self">accessibleStates</a> <font color="Maroon">tfsm</font></span>)</span>,<font color="Maroon">IAlph</font></span><a href="zfmisc_1.html#K2">:]</a></span> is   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="fsm_1.html#K12" target="_self">accessibleStates</a> <font color="Maroon">tfsm</font></span>)</span>,<font color="Maroon">IAlph</font></span><a href="zfmisc_1.html#K2">:]</a></span>, <a href="fsm_1.html#K12" target="_self">accessibleStates</a> <font color="Maroon">tfsm</font>
 <b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>


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