<?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">w</font> be    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">IAlph</font>;<br/><b>let </b><font color="Maroon">tfsm1</font> be  non <a href="struct_0.html#V3">empty</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>let </b><font color="Maroon">tfsm2</font> be  non <a href="struct_0.html#V3">empty</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>let </b><font color="Maroon">q11</font> be   <a href="fsm_1.html#NM1">State</a> of <font color="Maroon">tfsm1</font>;<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>let </b><font color="Maroon">q</font> be   <a href="fsm_1.html#NM1">State</a> of <font color="Maroon">tfsm</font>;<br/>















<b>assume </b><a NAME="E1:104"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">tfsm</font> <a href="hidden.html#R1">=</a> <font color="Maroon">tfsm1</font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">tfsm2</font> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">tfsm1</font> <a href="subset_1.html#R1">misses</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">tfsm2</font> &amp; <font color="Maroon">q11</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> )
 ;<br/>

<b>set </b><font color="Maroon">ad1</font> = <font color="Maroon">q11</font>,<font color="Maroon">w</font> <a href="fsm_1.html#K2" target="_self">-admissible</a> ;<br/>
<b>set </b><font color="Maroon">ad</font> = <font color="Maroon">q</font>,<font color="Maroon">w</font> <a href="fsm_1.html#K2" target="_self">-admissible</a> ;<br/>
<i><font color="Green">E23</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">q11</font>,<font color="Maroon">w</font> <a href="fsm_1.html#K2" target="_self">-admissible</a> </span>)</span> = 
<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/>.= 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">q</font>,<font color="Maroon">w</font> <a href="fsm_1.html#K2" target="_self">-admissible</a> </span>)</span>
<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>( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">a<sub>1</sub></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">q11</font>,<font color="Maroon">w</font> <a href="fsm_1.html#K2" target="_self">-admissible</a> </span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon">q11</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"><font color="Maroon">q</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="E3:104"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
;<br/>
<a NAME="E4:104"/><i><font color="Green">E39</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="104_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:104_2"/><i><font color="Green">E40</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</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">q11</font>,<font color="Maroon">w</font> <a href="fsm_1.html#K2" target="_self">-admissible</a> </span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon">q11</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">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">q</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">k</font> )
 ;<br/>

<b>assume </b><a NAME="E2:104_2"/><i><font color="Green">E42</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <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">q11</font>,<font color="Maroon">w</font> <a href="fsm_1.html#K2" target="_self">-admissible</a> </span>)</span> )
 ;<br/>

<a NAME="E3:104_2"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0 or ( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</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="104_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:104_2_1"/>
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0 or 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</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="104_2_1_1"><b>suppose </b></a><a NAME="E1:104_2_1_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><font color="Maroon">q11</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">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<font color="Maroon">q11</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">q</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">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i>, , </i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="104_2_1_2"><b>suppose </b></a><a NAME="E1:104_2_1_2"/><i><font color="Green">E45</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 ;<br/><div class="add"><a NAME="E2:104_2_1_2"/>
<font color="Maroon">k</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="txt" href="fsm_1.html#E7:104_2_1_2"><i><font color="Green">E79</font></i></a>, </i>;<br/><a NAME="E3:104_2_1_2"/><b>then </b><i><font color="Green">E46</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">w</font> )
 <b>by </b><i><a class="txt" href="fsm_1.html#E9:104_2_1_2"><i><font color="Green">E81</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><b>then consider </b><font color="Maroon">w1k</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">IAlph</font>, <font color="Maroon">q1k</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">tfsm1</font>, <font color="Maroon">q1k1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">tfsm1</font><b> such that </b><br/><a NAME="E5:104_2_1_2"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">w1k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> &amp; <font color="Maroon">q1k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">q11</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">k</font> &amp; <font color="Maroon">q1k1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">q11</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">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">w1k</font> <a href="fsm_1.html#K1" target="_self">-succ_of</a> <font color="Maroon">q1k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q1k1</font> )
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">wk</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">IAlph</font>, <font color="Maroon">qk</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">tfsm</font>, <font color="Maroon">qk1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">tfsm</font><b> such that </b><br/><a NAME="E7:104_2_1_2"/><i><font color="Green">E79</font></i>: 
( <font color="Maroon">wk</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> &amp; <font color="Maroon">qk</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">q</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">k</font> &amp; <font color="Maroon">qk1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">q</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">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">wk</font> <a href="fsm_1.html#K1" target="_self">-succ_of</a> <font color="Maroon">qk</font> <a href="hidden.html#R1">=</a> <font color="Maroon">qk1</font> )
 <b>by </b><i><a class="txt" href="fsm_1.html#E10:104_2_1_2"><i><font color="Green">E82</font></i></a>, </i>;<br/><a NAME="E8:104_2_1_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">w</font> <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="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E9:104_2_1_2"/><b>then </b><i><font color="Green">E81</font></i>: 
<font color="Maroon">k</font> <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="txt" href="fsm_1.html#E10:104_2_1_2"><i><font color="Green">E82</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E10:104_2_1_2"/><i><font color="Green">E82</font></i>: 
 <a href="relat_1.html#K1">dom</a> the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">tfsm2</font> <a href="hidden.html#R1">=</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">tfsm2</font>,<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="E11:104_2_1_2"/>
<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">tfsm1</font>,<font color="Maroon">IAlph</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="subset_1.html#R1">misses</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">tfsm2</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#T127">ZFMISC_1:127</a></i>;<br/><a NAME="E12:104_2_1_2"/><b>then </b><i><font color="Green">E83</font></i>: 
not <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">q1k</font>,<font color="Maroon">w1k</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</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">tfsm2</font>,<font color="Maroon">IAlph</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">q11</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">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default">the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">tfsm1</font> <a href="funct_4.html#K1">+*</a> the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">tfsm2</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">q1k</font>,<font color="Maroon">w1k</font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i>, , <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a>, <a class="txt" href="fsm_1.html#E12:104_2_1_2"><i><font color="Green">E83</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">q</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">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i>, , <a class="txt" href="fsm_1.html#E9:104_2_1_2"><i><font color="Green">E81</font></i></a>, <a class="txt" href="fsm_1.html#E12:104_2_1_2"><i><font color="Green">E83</font></i></a>, , , , </i>
;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E5:104"/>
for <font color="Olive">k</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">k</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>);<br/></i>
<b>hence </b><a NAME="E6:104"/>
<font color="Maroon">q11</font>,<font color="Maroon">w</font> <a href="fsm_1.html#K2" target="_self">-admissible</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">q</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_1.html#T18">FINSEQ_1:18</a></i>;<br/>


</div>
