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

<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">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">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">M</font> = <font color="Maroon">tfsm</font>;<br/>
<b>let </b><font color="Maroon">cTran</font> be   <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>;<br/><b>let </b><font color="Maroon">cOFun</font> be   <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>,<font color="Maroon">OAlph</font>;<br/><b>let </b><font color="Maroon">cInitS</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="fsm_1.html#K12" target="_self">accessibleStates</a> <font color="Maroon">tfsm</font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:100"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">cTran</font> <a href="hidden.html#R1">=</a> 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>
 <b>and </b><br/><a NAME="E2:100"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">cOFun</font> <a href="hidden.html#R1">=</a> the <a href="fsm_1.html#U3" target="_self">OFun</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>
 <b>and </b><br/><a NAME="E3:100"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">cInitS</font> <a href="hidden.html#R1">=</a> the <a href="fsm_1.html#U2" target="_self">InitS</a> of <font color="Maroon">tfsm</font>
 ;<br/>

<b>set </b><font color="Maroon">cm</font> =  <a href="fsm_1.html#G2" target="_self">Mealy-FSM</a>(# <span class="p1">(<span class="default"><a href="fsm_1.html#K12" target="_self">accessibleStates</a> <font color="Maroon">tfsm</font></span>)</span>,<font color="Maroon">cTran</font>,<font color="Maroon">cOFun</font>,<font color="Maroon">cInitS</font> #);<br/>
<b>let </b><font color="Maroon">w</font> be    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">IAlph</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="fsm_1.html#D9" target="_self">FSM_1:def 9</a><br/>

<b>set </b><font color="Maroon">ma</font> = 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#K2" target="_self">-admissible</a> ;<br/>
<b>set </b><font color="Maroon">cma</font> = the <a href="fsm_1.html#U2" target="_self">InitS</a> of <a href="fsm_1.html#G2" target="_self">Mealy-FSM</a>(# <span class="p1">(<span class="default"><a href="fsm_1.html#K12" target="_self">accessibleStates</a> <font color="Maroon">tfsm</font></span>)</span>,<font color="Maroon">cTran</font>,<font color="Maroon">cOFun</font>,<font color="Maroon">cInitS</font> #),<font color="Maroon">w</font> <a href="fsm_1.html#K2" target="_self">-admissible</a> ;<br/>
<div><i><font color="Green">E39</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="100_1"><b>now </b></a><div class="add"><b>thus </b><a NAME="E1:100_1"/>
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default">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#K2" target="_self">-admissible</a> </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> 1 &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default">the <a href="fsm_1.html#U2" target="_self">InitS</a> of <a href="fsm_1.html#G2" target="_self">Mealy-FSM</a>(# <span class="p1">(<span class="default"><a href="fsm_1.html#K12" target="_self">accessibleStates</a> <font color="Maroon">tfsm</font></span>)</span>,<font color="Maroon">cTran</font>,<font color="Maroon">cOFun</font>,<font color="Maroon">cInitS</font> #),<font color="Maroon">w</font> <a href="fsm_1.html#K2" target="_self">-admissible</a> </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> 1 )
 <b>by </b><i/>;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <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> implies <span class="p1">(<span class="default">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#K2" target="_self">-admissible</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">the <a href="fsm_1.html#U2" target="_self">InitS</a> of <a href="fsm_1.html#G2" target="_self">Mealy-FSM</a>(# <span class="p1">(<span class="default"><a href="fsm_1.html#K12" target="_self">accessibleStates</a> <font color="Maroon">tfsm</font></span>)</span>,<font color="Maroon">cTran</font>,<font color="Maroon">cOFun</font>,<font color="Maroon">cInitS</font> #),<font color="Maroon">w</font> <a href="fsm_1.html#K2" target="_self">-admissible</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> );<br/><a NAME="E2:100_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E3:100_1"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="100_1_1"><b>proof </b></a><div class="add">

<b>let </b><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="E1:100_1_1"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <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> implies <span class="p1">(<span class="default">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#K2" target="_self">-admissible</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">the <a href="fsm_1.html#U2" target="_self">InitS</a> of <a href="fsm_1.html#G2" target="_self">Mealy-FSM</a>(# <span class="p1">(<span class="default"><a href="fsm_1.html#K12" target="_self">accessibleStates</a> <font color="Maroon">tfsm</font></span>)</span>,<font color="Maroon">cTran</font>,<font color="Maroon">cOFun</font>,<font color="Maroon">cInitS</font> #),<font color="Maroon">w</font> <a href="fsm_1.html#K2" target="_self">-admissible</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> )
 ;<br/>

<b>assume </b><a NAME="E2:100_1_1"/>
<font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <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>
 ;<br/>

<a NAME="E3:100_1_1"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</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> 1 )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E4:100_1_1"/><b>then </b><i><font color="Green">E44</font></i>: 
( <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">w</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</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> 1 )
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E5:100_1_1"/><i><font color="Green">E45</font></i>: 
( 0 <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> or ( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font> &amp; 0 <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> 1 ) )
 
;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="100_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:100_1_1_1"/>
( 0 <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> or 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> )
 </span><b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="100_1_1_1_1"><b>suppose </b></a><a NAME="E1:100_1_1_1_1"/><i><font color="Green">E46</font></i>: 
0 <a href="hidden.html#R1">=</a> <font color="Maroon">j</font>
 ;<br/><div class="add"><b>hence </b><span class="p1">(<span class="default">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#K2" target="_self">-admissible</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
the <a href="fsm_1.html#U2" target="_self">InitS</a> of <font color="Maroon">tfsm</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default">the <a href="fsm_1.html#U2" target="_self">InitS</a> of <a href="fsm_1.html#G2" target="_self">Mealy-FSM</a>(# <span class="p1">(<span class="default"><a href="fsm_1.html#K12" target="_self">accessibleStates</a> <font color="Maroon">tfsm</font></span>)</span>,<font color="Maroon">cTran</font>,<font color="Maroon">cOFun</font>,<font color="Maroon">cInitS</font> #),<font color="Maroon">w</font> <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"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i>, <a class="txt" href="fsm_1.html#E5:100"><i><font color="Green">E81</font></i></a>, </i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="100_1_1_1_2"><b>suppose </b></a><a NAME="E1:100_1_1_1_2"/><i><font color="Green">E47</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">mwj</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">IAlph</font>, <font color="Maroon">mqj</font> being   <a href="fsm_1.html#NM1">State</a> of <font color="Maroon">tfsm</font>, <font color="Maroon">mqj1</font> being   <a href="fsm_1.html#NM1">State</a> of <font color="Maroon">tfsm</font><b> such that </b><br/><a NAME="E3:100_1_1_1_2"/><i><font color="Green">E79</font></i>: 
( <font color="Maroon">mwj</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> &amp; <font color="Maroon">mqj</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">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#K2" target="_self">-admissible</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> &amp; <font color="Maroon">mqj1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">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#K2" target="_self">-admissible</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">mwj</font> <a href="fsm_1.html#K1" target="_self">-succ_of</a> <font color="Maroon">mqj</font> <a href="hidden.html#R1">=</a> <font color="Maroon">mqj1</font> )
 <b>by </b><i><a class="txt" href="fsm_1.html#E3:100_1_1_1_2"><i><font color="Green">E79</font></i></a>, </i>;<br/><a NAME="E4:100_1_1_1_2"/>
 ex <font color="Olive">cmwj</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">IAlph</font> ex <font color="Olive">cmqj</font>, <font color="Olive">cmqj1</font> being  <a href="fsm_1.html#NM1">State</a> of <a href="fsm_1.html#G2" target="_self">Mealy-FSM</a>(# <span class="p1">(<span class="default"><a href="fsm_1.html#K12" target="_self">accessibleStates</a> <font color="Maroon">tfsm</font></span>)</span>,<font color="Maroon">cTran</font>,<font color="Maroon">cOFun</font>,<font color="Maroon">cInitS</font> #) st <br/>( <font color="Olive">cmwj</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> &amp; <font color="Olive">cmqj</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">the <a href="fsm_1.html#U2" target="_self">InitS</a> of <a href="fsm_1.html#G2" target="_self">Mealy-FSM</a>(# <span class="p1">(<span class="default"><a href="fsm_1.html#K12" target="_self">accessibleStates</a> <font color="Maroon">tfsm</font></span>)</span>,<font color="Maroon">cTran</font>,<font color="Maroon">cOFun</font>,<font color="Maroon">cInitS</font> #),<font color="Maroon">w</font> <a href="fsm_1.html#K2" target="_self">-admissible</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> &amp; <font color="Olive">cmqj1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">the <a href="fsm_1.html#U2" target="_self">InitS</a> of <a href="fsm_1.html#G2" target="_self">Mealy-FSM</a>(# <span class="p1">(<span class="default"><a href="fsm_1.html#K12" target="_self">accessibleStates</a> <font color="Maroon">tfsm</font></span>)</span>,<font color="Maroon">cTran</font>,<font color="Maroon">cOFun</font>,<font color="Maroon">cInitS</font> #),<font color="Maroon">w</font> <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"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Olive">cmwj</font> <a href="fsm_1.html#K1" target="_self">-succ_of</a> <font color="Olive">cmqj</font> <a href="hidden.html#R1">=</a> <font color="Olive">cmqj1</font> )
 <b>by </b><i><a class="txt" href="fsm_1.html#E3:100_1_1_1_2"><i><font color="Green">E79</font></i></a>, <a class="txt" href="fsm_1.html#E5:100"><i><font color="Green">E81</font></i></a>, </i>;<br/><b>hence </b><a NAME="E5:100_1_1_1_2"/>
<span class="p1">(<span class="default">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#K2" target="_self">-admissible</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">the <a href="fsm_1.html#U2" target="_self">InitS</a> of <a href="fsm_1.html#G2" target="_self">Mealy-FSM</a>(# <span class="p1">(<span class="default"><a href="fsm_1.html#K12" target="_self">accessibleStates</a> <font color="Maroon">tfsm</font></span>)</span>,<font color="Maroon">cTran</font>,<font color="Maroon">cOFun</font>,<font color="Maroon">cInitS</font> #),<font color="Maroon">w</font> <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"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, , <a class="txt" href="fsm_1.html#E3:100_1_1_1_2"><i><font color="Green">E79</font></i></a>, <a class="txt" href="fsm_1.html#E5:100"><i><font color="Green">E81</font></i></a>, , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>thus </b><a NAME="E4:100_1"/>
for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">j</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, <a class="ref" href="fsm_1.html#T15" target="_self">Th15</a>);</i><br/></div><b>end;</b></div>
<a NAME="E5:100"/><b>then </b><i><font color="Green">E81</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#K2" target="_self">-admissible</a>  <a href="hidden.html#R1">=</a> the <a href="fsm_1.html#U2" target="_self">InitS</a> of <a href="fsm_1.html#G2" target="_self">Mealy-FSM</a>(# <span class="p1">(<span class="default"><a href="fsm_1.html#K12" target="_self">accessibleStates</a> <font color="Maroon">tfsm</font></span>)</span>,<font color="Maroon">cTran</font>,<font color="Maroon">cOFun</font>,<font color="Maroon">cInitS</font> #),<font color="Maroon">w</font> <a href="fsm_1.html#K2" target="_self">-admissible</a> 
 
<b>by </b><i><a class="ref" href="finseq_2.html#T10">FINSEQ_2:10</a></i>;<br/>
<b>set </b><font color="Maroon">mr</font> = 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#K4" target="_self">-response</a> ;<br/>
<b>set </b><font color="Maroon">cmr</font> = the <a href="fsm_1.html#U2" target="_self">InitS</a> of <a href="fsm_1.html#G2" target="_self">Mealy-FSM</a>(# <span class="p1">(<span class="default"><a href="fsm_1.html#K12" target="_self">accessibleStates</a> <font color="Maroon">tfsm</font></span>)</span>,<font color="Maroon">cTran</font>,<font color="Maroon">cOFun</font>,<font color="Maroon">cInitS</font> #),<font color="Maroon">w</font> <a href="fsm_1.html#K4" target="_self">-response</a> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="100_2"><b>now </b></a><div class="add"><b>thus </b><a NAME="E1:100_2"/>
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default">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#K4" target="_self">-response</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">w</font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default">the <a href="fsm_1.html#U2" target="_self">InitS</a> of <a href="fsm_1.html#G2" target="_self">Mealy-FSM</a>(# <span class="p1">(<span class="default"><a href="fsm_1.html#K12" target="_self">accessibleStates</a> <font color="Maroon">tfsm</font></span>)</span>,<font color="Maroon">cTran</font>,<font color="Maroon">cOFun</font>,<font color="Maroon">cInitS</font> #),<font color="Maroon">w</font> <a href="fsm_1.html#K4" target="_self">-response</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">w</font> )
 <b>by </b><i/>;<br/><b>let </b><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="E2:100_2"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">w</font></span>)</span>
 ;<br/><a NAME="E3:100_2"/><b>then </b><i><font color="Green">E83</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">w</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E4:100_2"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <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>
 <b>by </b><i><a class="ref" href="fsm_1.html#T15" target="_self">Th15</a>, <a class="ref" href="finseq_2.html#T9">FINSEQ_2:9</a></i>;<br/><a NAME="E5:100_2"/><b>then </b>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default">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#K2" target="_self">-admissible</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E6:100_2"/><b>then </b><i><font color="Green">E86</font></i>: 
( <span class="p1">(<span class="default">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#K2" target="_self">-admissible</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">tfsm</font> &amp; <font color="Maroon">w</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">IAlph</font> )
 <b>by </b><i>, <a class="ref" href="finseq_2.html#T13">FINSEQ_2:13</a></i>;<br/><a NAME="E7:100_2"/>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default">the <a href="fsm_1.html#U2" target="_self">InitS</a> of <a href="fsm_1.html#G2" target="_self">Mealy-FSM</a>(# <span class="p1">(<span class="default"><a href="fsm_1.html#K12" target="_self">accessibleStates</a> <font color="Maroon">tfsm</font></span>)</span>,<font color="Maroon">cTran</font>,<font color="Maroon">cOFun</font>,<font color="Maroon">cInitS</font> #),<font color="Maroon">w</font> <a href="fsm_1.html#K2" target="_self">-admissible</a> </span>)</span>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E8:100_2"/><b>then </b>
<span class="p1">(<span class="default">the <a href="fsm_1.html#U2" target="_self">InitS</a> of <a href="fsm_1.html#G2" target="_self">Mealy-FSM</a>(# <span class="p1">(<span class="default"><a href="fsm_1.html#K12" target="_self">accessibleStates</a> <font color="Maroon">tfsm</font></span>)</span>,<font color="Maroon">cTran</font>,<font color="Maroon">cOFun</font>,<font color="Maroon">cInitS</font> #),<font color="Maroon">w</font> <a href="fsm_1.html#K2" target="_self">-admissible</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="fsm_1.html#K12" target="_self">accessibleStates</a> <font color="Maroon">tfsm</font>
 <b>by </b><i><a class="ref" href="finseq_2.html#T13">FINSEQ_2:13</a></i>;<br/><a NAME="E9:100_2"/><b>then </b><i><font color="Green">E90</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">the <a href="fsm_1.html#U2" target="_self">InitS</a> of <a href="fsm_1.html#G2" target="_self">Mealy-FSM</a>(# <span class="p1">(<span class="default"><a href="fsm_1.html#K12" target="_self">accessibleStates</a> <font color="Maroon">tfsm</font></span>)</span>,<font color="Maroon">cTran</font>,<font color="Maroon">cOFun</font>,<font color="Maroon">cInitS</font> #),<font color="Maroon">w</font> <a href="fsm_1.html#K2" target="_self">-admissible</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">w</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</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="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>thus </b><span class="p1">(<span class="default">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#K4" target="_self">-response</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> = 
the <a href="fsm_1.html#U3" target="_self">OFun</a> of <font color="Maroon">tfsm</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">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#K2" target="_self">-admissible</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">w</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">cOFun</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">the <a href="fsm_1.html#U2" target="_self">InitS</a> of <a href="fsm_1.html#G2" target="_self">Mealy-FSM</a>(# <span class="p1">(<span class="default"><a href="fsm_1.html#K12" target="_self">accessibleStates</a> <font color="Maroon">tfsm</font></span>)</span>,<font color="Maroon">cTran</font>,<font color="Maroon">cOFun</font>,<font color="Maroon">cInitS</font> #),<font color="Maroon">w</font> <a href="fsm_1.html#K2" target="_self">-admissible</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">w</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i>, , <a class="txt" href="fsm_1.html#E5:100"><i><font color="Green">E81</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="fsm_1.html#U2" target="_self">InitS</a> of <a href="fsm_1.html#G2" target="_self">Mealy-FSM</a>(# <span class="p1">(<span class="default"><a href="fsm_1.html#K12" target="_self">accessibleStates</a> <font color="Maroon">tfsm</font></span>)</span>,<font color="Maroon">cTran</font>,<font color="Maroon">cOFun</font>,<font color="Maroon">cInitS</font> #),<font color="Maroon">w</font> <a href="fsm_1.html#K4" target="_self">-response</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
<b>by </b><i>, </i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:100"/>
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#K4" target="_self">-response</a>  <a href="hidden.html#R1">=</a> the <a href="fsm_1.html#U2" target="_self">InitS</a> of <a href="fsm_1.html#G2" target="_self">Mealy-FSM</a>(# <span class="p1">(<span class="default"><a href="fsm_1.html#K12" target="_self">accessibleStates</a> <font color="Maroon">tfsm</font></span>)</span>,<font color="Maroon">cTran</font>,<font color="Maroon">cOFun</font>,<font color="Maroon">cInitS</font> #),<font color="Maroon">w</font> <a href="fsm_1.html#K4" target="_self">-response</a> 
 <b>by </b><i><a class="ref" href="finseq_2.html#T10">FINSEQ_2:10</a></i>;<br/>


</div>
