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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>







<b>assume </b><a NAME="E1:115"/>
<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="fsm_1.html#R7" target="_self">-are_isomorphic</a> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font><b> such that </b><br/><a NAME="E3:115"/><i><font color="Green">E85</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> is <a href="funct_2.html#V3">bijective</a> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> the <a href="fsm_1.html#U2" target="_self">InitS</a> of <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> the <a href="fsm_1.html#U2" target="_self">InitS</a> of <font color="Maroon">c<sub>4</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>3</sub></font><br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>( <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">c<sub>4</sub></font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<font color="Olive">b<sub>2</sub></font> &amp; the <a href="fsm_1.html#U3" target="_self">OFun</a> of <font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> the <a href="fsm_1.html#U3" target="_self">OFun</a> of <font color="Maroon">c<sub>4</sub></font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<font color="Olive">b<sub>2</sub></font> ) ) )
 <b>by </b><i><a class="ref" href="fsm_1.html#D19" target="_self">Def19</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>7</sub></font> = the <a href="fsm_1.html#U2" target="_self">InitS</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="fsm_1.html#K4" target="_self">-response</a> ;<br/><b>set </b><font color="Maroon">c<sub>8</sub></font> = the <a href="fsm_1.html#U2" target="_self">InitS</a> of <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="fsm_1.html#K4" target="_self">-response</a> ;<br/><b>set </b><font color="Maroon">c<sub>9</sub></font> = the <a href="fsm_1.html#U2" target="_self">InitS</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="fsm_1.html#K2" target="_self">-admissible</a> ;<br/><b>set </b><font color="Maroon">c<sub>10</sub></font> = the <a href="fsm_1.html#U2" target="_self">InitS</a> of <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="fsm_1.html#K2" target="_self">-admissible</a> ;<br/><a NAME="E1:115_1"/><i><font color="Green">E86</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></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">c<sub>6</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 holds <br/><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="fsm_1.html#U2" target="_self">InitS</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="fsm_1.html#K2" target="_self">-admissible</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></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 <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="fsm_1.html#K2" target="_self">-admissible</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_1_1"><b>proof </b></a><div class="add">

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means ( 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> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 implies <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="fsm_1.html#U2" target="_self">InitS</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></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></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 <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></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="E1:115_1_1"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
;<br/>
<a NAME="E2:115_1_1"/><i><font color="Green">E88</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>11</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:115_1_1_1"/><i><font color="Green">E89</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>11</sub></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">c<sub>6</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 implies <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="fsm_1.html#U2" target="_self">InitS</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="fsm_1.html#K2" target="_self">-admissible</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font></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 <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="fsm_1.html#K2" target="_self">-admissible</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> )
 ;<br/>

<b>assume </b><a NAME="E2:115_1_1_1"/><i><font color="Green">E90</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>11</sub></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">c<sub>6</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/>

<a NAME="E3:115_1_1_1"/><i><font color="Green">E91</font></i>: 
( 0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> or ( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>11</sub></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="115_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="E1:115_1_1_1_1"/>
( 0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> or 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>11</sub></font> )
 </span><b>by </b><i><a class="txt" href="fsm_1.html#E3:115_1_1_1"><i><font color="Green">E91</font></i></a>, <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="115_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:115_1_1_1_1_1"/><i><font color="Green">E92</font></i>: 
0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font>
 ;<br/><div class="add"><b>hence </b><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="fsm_1.html#U2" target="_self">InitS</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="fsm_1.html#K2" target="_self">-admissible</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> = 
the <a href="fsm_1.html#U2" target="_self">InitS</a> of <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="fsm_1.html#E3:115"><i><font color="Green">E85</font></i></a>, <a class="ref" href="fsm_1.html#D2" target="_self">Def2</a></i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="fsm_1.html#U2" target="_self">InitS</a> of <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></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">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="fsm_1.html#E1:115_1_1_1_1_1"><i><font color="Green">E92</font></i></a>, <a class="ref" href="fsm_1.html#D2" target="_self">Def2</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="115_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:115_1_1_1_1_2"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>11</sub></font>
 ;<br/><div class="add"><a NAME="E2:115_1_1_1_1_2"/><b>then </b><i><font color="Green">E92</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="fsm_1.html#E2:115_1_1_1"><i><font color="Green">E90</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>12</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>13</sub></font>, <font color="Maroon">c<sub>14</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E4:115_1_1_1_1_2"/><i><font color="Green">E93</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>13</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 <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="fsm_1.html#K2" target="_self">-admissible</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>14</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 <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></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">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="fsm_1.html#K1" target="_self">-succ_of</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> )
 <b>by </b><i><a class="ref" href="fsm_1.html#D2" target="_self">Def2</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>15</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>16</sub></font>, <font color="Maroon">c<sub>17</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>4</sub></font><b> such that </b><br/><a NAME="E6:115_1_1_1_1_2"/><i><font color="Green">E94</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>16</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 <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="fsm_1.html#K2" target="_self">-admissible</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>17</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 <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></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">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="fsm_1.html#K1" target="_self">-succ_of</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> )
 <b>by </b><i><a class="txt" href="fsm_1.html#E2:115_1_1_1_1_2"><i><font color="Green">E92</font></i></a>, <a class="ref" href="fsm_1.html#D2" target="_self">Def2</a></i>;<br/><a NAME="E7:115_1_1_1_1_2"/><i><font color="Green">E95</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></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">c<sub>6</sub></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/><b>thus </b><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="fsm_1.html#U2" target="_self">InitS</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="fsm_1.html#K2" target="_self">-admissible</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> = 
<font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="fsm_1.html#E4:115_1_1_1_1_2"><i><font color="Green">E93</font></i></a></i>
<br/>.= 
the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">c<sub>4</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font>
<b>by </b><i><a class="txt" href="fsm_1.html#E3:115"><i><font color="Green">E85</font></i></a>, <a class="txt" href="fsm_1.html#E1:115_1_1_1"><i><font color="Green">E89</font></i></a>, <a class="txt" href="fsm_1.html#E2:115_1_1_1_1_2"><i><font color="Green">E92</font></i></a>, <a class="txt" href="fsm_1.html#E4:115_1_1_1_1_2"><i><font color="Green">E93</font></i></a>, <a class="txt" href="fsm_1.html#E6:115_1_1_1_1_2"><i><font color="Green">E94</font></i></a>, <a class="txt" href="fsm_1.html#E7:115_1_1_1_1_2"><i><font color="Green">E95</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="fsm_1.html#U2" target="_self">InitS</a> of <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></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">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="fsm_1.html#E6:115_1_1_1_1_2"><i><font color="Green">E94</font></i></a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>thus </b><a NAME="E3:115_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="fsm_1.html#E1:115_1_1"><i><font color="Green">E87</font></i></a>, <a class="txt" href="fsm_1.html#E2:115_1_1"><i><font color="Green">E88</font></i></a>);</i><br/>


</div><b>end;</b></div><i><font color="Green">E87</font></i>:  <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">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="fsm_1.html#K4" target="_self">-response</a> </span>)</span> = 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="ref" href="fsm_1.html#D6" target="_self">Def6</a></i>
<br/>.= 
 <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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="fsm_1.html#K4" target="_self">-response</a> </span>)</span>
<b>by </b><i><a class="ref" href="fsm_1.html#D6" target="_self">Def6</a></i>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>11</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:115_1_3"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <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">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="fsm_1.html#K4" target="_self">-response</a> </span>)</span> )
 ;<br/><a NAME="E2:115_1_3"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="ref" href="fsm_1.html#D6" target="_self">Def6</a></i>;<br/><a NAME="E3:115_1_3"/><b>then </b><i><font color="Green">E88</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E4:115_1_3"/><b>then </b>
<font color="Maroon">c<sub>11</sub></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">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E5:115_1_3"/><b>then </b>
<font color="Maroon">c<sub>11</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">c<sub>6</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="finseq_2.html#T9">FINSEQ_2:9</a></i>;<br/><a NAME="E6:115_1_3"/><b>then </b>
<font color="Maroon">c<sub>11</sub></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> <span class="p2">(<span class="default">the <a href="fsm_1.html#U2" target="_self">InitS</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="fsm_1.html#K2" target="_self">-admissible</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="fsm_1.html#D2" target="_self">Def2</a></i>;<br/><a NAME="E7:115_1_3"/><b>then </b><i><font color="Green">E89</font></i>: 
<font color="Maroon">c<sub>11</sub></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">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></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="E8:115_1_3"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <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">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="fsm_1.html#K2" target="_self">-admissible</a> </span>)</span> )
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E9:115_1_3"/><b>then </b><i><font color="Green">E90</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>11</sub></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">c<sub>6</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="ref" href="fsm_1.html#D2" target="_self">Def2</a></i>;<br/><a NAME="E10:115_1_3"/><i><font color="Green">E91</font></i>: 
( <span class="p1">(<span class="default">the <a href="fsm_1.html#U2" target="_self">InitS</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="fsm_1.html#K2" target="_self">-admissible</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="fsm_1.html#E3:115_1_3"><i><font color="Green">E88</font></i></a>, <a class="txt" href="fsm_1.html#E7:115_1_3"><i><font color="Green">E89</font></i></a>, <a class="ref" href="finseq_2.html#T13">FINSEQ_2:13</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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="fsm_1.html#K4" target="_self">-response</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> = 
the <a href="fsm_1.html#U3" target="_self">OFun</a> of <font color="Maroon">c<sub>4</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="fsm_1.html#K2" target="_self">-admissible</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="fsm_1.html#E3:115_1_3"><i><font color="Green">E88</font></i></a>, <a class="ref" href="fsm_1.html#D6" target="_self">Def6</a></i>
<br/>.= 
the <a href="fsm_1.html#U3" target="_self">OFun</a> of <font color="Maroon">c<sub>4</sub></font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <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">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="fsm_1.html#K2" target="_self">-admissible</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="fsm_1.html#E1:115_1"><i><font color="Green">E86</font></i></a>, <a class="txt" href="fsm_1.html#E9:115_1_3"><i><font color="Green">E90</font></i></a></i>
<br/>.= 
the <a href="fsm_1.html#U3" target="_self">OFun</a> of <font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="fsm_1.html#U2" target="_self">InitS</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="fsm_1.html#K2" target="_self">-admissible</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="fsm_1.html#E3:115"><i><font color="Green">E85</font></i></a>, <a class="txt" href="fsm_1.html#E10:115_1_3"><i><font color="Green">E91</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="fsm_1.html#U2" target="_self">InitS</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="fsm_1.html#K4" target="_self">-response</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font>
<b>by </b><i><a class="txt" href="fsm_1.html#E3:115_1_3"><i><font color="Green">E88</font></i></a>, <a class="ref" href="fsm_1.html#D6" target="_self">Def6</a></i>
;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:115_1"/>
the <a href="fsm_1.html#U2" target="_self">InitS</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></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 <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="fsm_1.html#K4" target="_self">-response</a> 
 <b>by </b><i><a class="txt" href="fsm_1.html#E2:115_1"><i><font color="Green">E87</font></i></a>, <a class="ref" href="finseq_1.html#T18">FINSEQ_1:18</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:115"/>
<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="fsm_1.html#R4" target="_self">-are_equivalent</a> 
 <b>by </b><i><a class="ref" href="fsm_1.html#D9" target="_self">Def9</a></i>;<br/>


</div>
