<?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="group_1.html#V6">finite</a> <a href="fsm_1.html#V7" target="_self">connected</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="E1:117"/><i><font color="Green">E87</font></i>: 
<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> 
 ;<br/>

<b>set </b><font color="Maroon">c<sub>5</sub></font> =  <a href="fsm_1.html#K11" target="_self">the_reduction_of</a> <font color="Maroon">c<sub>3</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> =  <a href="fsm_1.html#K11" target="_self">the_reduction_of</a> <font color="Maroon">c<sub>4</sub></font>;<br/>
<b>consider </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font> being  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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E3:117"/><i><font color="Green">E88</font></i>: 
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#R1">misses</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font>, <a href="fsm_1.html#K11" target="_self">the_reduction_of</a> <font color="Maroon">c<sub>3</sub></font> <a href="fsm_1.html#R7" target="_self">-are_isomorphic</a>  &amp; <font color="Maroon">c<sub>8</sub></font>, <a href="fsm_1.html#K11" target="_self">the_reduction_of</a> <font color="Maroon">c<sub>4</sub></font> <a href="fsm_1.html#R7" target="_self">-are_isomorphic</a>  )
 <b>by </b><i><a class="ref" href="fsm_1.html#T81" target="_self">Th81</a></i>;<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>7</sub></font>;<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>8</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>11</sub></font> = the <a href="fsm_1.html#U2" target="_self">InitS</a> of <span class="p1">(<span class="default"><a href="fsm_1.html#K11" target="_self">the_reduction_of</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>12</sub></font> = the <a href="fsm_1.html#U2" target="_self">InitS</a> of <span class="p1">(<span class="default"><a href="fsm_1.html#K11" target="_self">the_reduction_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>13</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>7</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>14</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>8</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>15</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="fsm_1.html#K11" target="_self">the_reduction_of</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>16</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="fsm_1.html#K11" target="_self">the_reduction_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>;<br/>
<a NAME="E4:117"/><i><font color="Green">E89</font></i>: 
 <a href="fsm_1.html#K11" target="_self">the_reduction_of</a> <font color="Maroon">c<sub>3</sub></font> is <a href="fsm_1.html#V7" target="_self">connected</a>
 
<b>by </b><i><a class="ref" href="fsm_1.html#T66" target="_self">Th66</a></i>;<br/>
<a NAME="E5:117"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">c<sub>7</sub></font> is <a href="fsm_1.html#V7" target="_self">connected</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="117_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:117_1"/>
not <font color="Maroon">c<sub>7</sub></font> is <a href="fsm_1.html#V7" target="_self">connected</a>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>17</sub></font> being    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>7</sub></font><b> such that </b><br/><a NAME="E3:117_1"/><i><font color="Green">E91</font></i>: 
not <font color="Maroon">c<sub>17</sub></font> is <a href="fsm_1.html#V6" target="_self">accessible</a>
 <b>by </b><i><a class="ref" href="fsm_1.html#D22" target="_self">Def22</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>18</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>7</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="fsm_1.html#K11" target="_self">the_reduction_of</a> <font color="Maroon">c<sub>3</sub></font></span>)</span><b> such that </b><br/><a NAME="E5:117_1"/><i><font color="Green">E92</font></i>: 
( <font color="Maroon">c<sub>18</sub></font> is <a href="funct_2.html#V3">bijective</a> &amp; <font color="Maroon">c<sub>18</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>7</sub></font> <a href="hidden.html#R1">=</a> the <a href="fsm_1.html#U2" target="_self">InitS</a> of <span class="p1">(<span class="default"><a href="fsm_1.html#K11" target="_self">the_reduction_of</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="fsm_1.html#NM1">State</a> of <font color="Maroon">c<sub>7</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>18</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>7</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 <span class="p1">(<span class="default"><a href="fsm_1.html#K11" target="_self">the_reduction_of</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</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>7</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 <span class="p1">(<span class="default"><a href="fsm_1.html#K11" target="_self">the_reduction_of</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</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="txt" href="fsm_1.html#E3:117"><i><font color="Green">E88</font></i></a>, <a class="ref" href="fsm_1.html#D19" target="_self">Def19</a></i>;<br/>
<a NAME="E6:117_1"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">c<sub>18</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="fsm_1.html#E5:117_1"><i><font color="Green">E92</font></i></a>, <a class="ref" href="funct_2.html#D4">FUNCT_2:def 4</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>17</sub></font>;<br/>
<a NAME="E7:117_1"/>
<font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>17</sub></font> is <a href="fsm_1.html#V6" target="_self">accessible</a>
 
<b>by </b><i><a class="txt" href="fsm_1.html#E4:117"><i><font color="Green">E89</font></i></a>, <a class="ref" href="fsm_1.html#D22" target="_self">Def22</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>20</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E9:117_1"/><i><font color="Green">E94</font></i>: 
the <a href="fsm_1.html#U2" target="_self">InitS</a> of <span class="p1">(<span class="default"><a href="fsm_1.html#K11" target="_self">the_reduction_of</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>20</sub></font> <a href="fsm_1.html#R1" target="_self">-leads_to</a> <font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="ref" href="fsm_1.html#D21" target="_self">Def21</a></i>;<br/>
<a NAME="E10:117_1"/><i><font color="Green">E95</font></i>: 
( 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>20</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <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>20</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/>
<a NAME="E11:117_1"/><b>then </b><i><font color="Green">E96</font></i>: 
<font color="Maroon">c<sub>18</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>7</sub></font>,<font color="Maroon">c<sub>20</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"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></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 <span class="p2">(<span class="default"><a href="fsm_1.html#K11" target="_self">the_reduction_of</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>20</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"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="fsm_1.html#E5:117_1"><i><font color="Green">E92</font></i></a>, <a class="ref" href="fsm_1.html#T60" target="_self">Th60</a></i>;<br/>
<a NAME="E12:117_1"/><i><font color="Green">E97</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E13:117_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">c<sub>7</sub></font>,<font color="Maroon">c<sub>20</sub></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">c<sub>20</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="E14:117_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <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"><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>7</sub></font>,<font color="Maroon">c<sub>20</sub></font> <a href="fsm_1.html#K2" target="_self">-admissible</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="fsm_1.html#E10:117_1"><i><font color="Green">E95</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E15:117_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <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>7</sub></font>,<font color="Maroon">c<sub>20</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="E16:117_1"/><b>then </b><i><font color="Green">E98</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>7</sub></font>,<font color="Maroon">c<sub>20</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"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>18</sub></font>
 
<b>by </b><i><a class="txt" href="fsm_1.html#E12:117_1"><i><font color="Green">E97</font></i></a>, <a class="ref" href="finseq_2.html#T13">FINSEQ_2:13</a></i>;<br/>
<a NAME="E17:117_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</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>7</sub></font>,<font color="Maroon">c<sub>20</sub></font> <a href="fsm_1.html#K2" target="_self">-admissible</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="fsm_1.html#E9:117_1"><i><font color="Green">E94</font></i></a>, <a class="txt" href="fsm_1.html#E11:117_1"><i><font color="Green">E96</font></i></a>, <a class="ref" href="fsm_1.html#D3" target="_self">Def3</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default">the <a href="fsm_1.html#U2" target="_self">InitS</a> of <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>20</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"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="fsm_1.html#E6:117_1"><i><font color="Green">E93</font></i></a>, <a class="txt" href="fsm_1.html#E16:117_1"><i><font color="Green">E98</font></i></a>, <a class="ref" href="funct_1.html#T56">FUNCT_1:56</a></i>
<br/>.= 
<font color="Maroon">c<sub>17</sub></font>
<b>by </b><i><a class="txt" href="fsm_1.html#E6:117_1"><i><font color="Green">E93</font></i></a>, <a class="txt" href="fsm_1.html#E12:117_1"><i><font color="Green">E97</font></i></a>, <a class="ref" href="funct_1.html#T56">FUNCT_1:56</a></i>
;<br/>
<a NAME="E19:117_1"/><b>then </b>
the <a href="fsm_1.html#U2" target="_self">InitS</a> of <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>20</sub></font> <a href="fsm_1.html#R1" target="_self">-leads_to</a> <font color="Maroon">c<sub>17</sub></font>
 
<b>by </b><i><a class="ref" href="fsm_1.html#D3" target="_self">Def3</a></i>;<br/>
<b>hence </b><a NAME="E20:117_1"/>
contradiction
 <b>by </b><i><a class="txt" href="fsm_1.html#E3:117_1"><i><font color="Green">E91</font></i></a>, <a class="ref" href="fsm_1.html#D21" target="_self">Def21</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:117"/><i><font color="Green">E91</font></i>: 
 <a href="fsm_1.html#K11" target="_self">the_reduction_of</a> <font color="Maroon">c<sub>4</sub></font> is <a href="fsm_1.html#V7" target="_self">connected</a>
 
<b>by </b><i><a class="ref" href="fsm_1.html#T66" target="_self">Th66</a></i>;<br/>
<a NAME="E7:117"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">c<sub>8</sub></font> is <a href="fsm_1.html#V7" target="_self">connected</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="117_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:117_2"/>
not <font color="Maroon">c<sub>8</sub></font> is <a href="fsm_1.html#V7" target="_self">connected</a>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>17</sub></font> being    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>8</sub></font><b> such that </b><br/><a NAME="E3:117_2"/><i><font color="Green">E93</font></i>: 
not <font color="Maroon">c<sub>17</sub></font> is <a href="fsm_1.html#V6" target="_self">accessible</a>
 <b>by </b><i><a class="ref" href="fsm_1.html#D22" target="_self">Def22</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>18</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>8</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="fsm_1.html#K11" target="_self">the_reduction_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span><b> such that </b><br/><a NAME="E5:117_2"/><i><font color="Green">E94</font></i>: 
( <font color="Maroon">c<sub>18</sub></font> is <a href="funct_2.html#V3">bijective</a> &amp; <font color="Maroon">c<sub>18</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>8</sub></font> <a href="hidden.html#R1">=</a> the <a href="fsm_1.html#U2" target="_self">InitS</a> of <span class="p1">(<span class="default"><a href="fsm_1.html#K11" target="_self">the_reduction_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="fsm_1.html#NM1">State</a> of <font color="Maroon">c<sub>8</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>18</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>8</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 <span class="p1">(<span class="default"><a href="fsm_1.html#K11" target="_self">the_reduction_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</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>8</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 <span class="p1">(<span class="default"><a href="fsm_1.html#K11" target="_self">the_reduction_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</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="txt" href="fsm_1.html#E3:117"><i><font color="Green">E88</font></i></a>, <a class="ref" href="fsm_1.html#D19" target="_self">Def19</a></i>;<br/>
<a NAME="E6:117_2"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">c<sub>18</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="fsm_1.html#E5:117_2"><i><font color="Green">E94</font></i></a>, <a class="ref" href="funct_2.html#D4">FUNCT_2:def 4</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>17</sub></font>;<br/>
<a NAME="E7:117_2"/>
<font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>17</sub></font> is <a href="fsm_1.html#V6" target="_self">accessible</a>
 
<b>by </b><i><a class="txt" href="fsm_1.html#E6:117"><i><font color="Green">E91</font></i></a>, <a class="ref" href="fsm_1.html#D22" target="_self">Def22</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>20</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E9:117_2"/><i><font color="Green">E96</font></i>: 
the <a href="fsm_1.html#U2" target="_self">InitS</a> of <span class="p1">(<span class="default"><a href="fsm_1.html#K11" target="_self">the_reduction_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>20</sub></font> <a href="fsm_1.html#R1" target="_self">-leads_to</a> <font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="ref" href="fsm_1.html#D21" target="_self">Def21</a></i>;<br/>
<a NAME="E10:117_2"/><i><font color="Green">E97</font></i>: 
( 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>20</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <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>20</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/>
<a NAME="E11:117_2"/><b>then </b><i><font color="Green">E98</font></i>: 
<font color="Maroon">c<sub>18</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>8</sub></font>,<font color="Maroon">c<sub>20</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"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></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 <span class="p2">(<span class="default"><a href="fsm_1.html#K11" target="_self">the_reduction_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>20</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"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="fsm_1.html#E5:117_2"><i><font color="Green">E94</font></i></a>, <a class="ref" href="fsm_1.html#T60" target="_self">Th60</a></i>;<br/>
<a NAME="E12:117_2"/><i><font color="Green">E99</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E13:117_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">c<sub>8</sub></font>,<font color="Maroon">c<sub>20</sub></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">c<sub>20</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="E14:117_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <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"><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>8</sub></font>,<font color="Maroon">c<sub>20</sub></font> <a href="fsm_1.html#K2" target="_self">-admissible</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="fsm_1.html#E10:117_2"><i><font color="Green">E97</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E15:117_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <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>8</sub></font>,<font color="Maroon">c<sub>20</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="E16:117_2"/><b>then </b><i><font color="Green">E100</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>8</sub></font>,<font color="Maroon">c<sub>20</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"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>18</sub></font>
 
<b>by </b><i><a class="txt" href="fsm_1.html#E12:117_2"><i><font color="Green">E99</font></i></a>, <a class="ref" href="finseq_2.html#T13">FINSEQ_2:13</a></i>;<br/>
<a NAME="E17:117_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</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>8</sub></font>,<font color="Maroon">c<sub>20</sub></font> <a href="fsm_1.html#K2" target="_self">-admissible</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="fsm_1.html#E9:117_2"><i><font color="Green">E96</font></i></a>, <a class="txt" href="fsm_1.html#E11:117_2"><i><font color="Green">E98</font></i></a>, <a class="ref" href="fsm_1.html#D3" target="_self">Def3</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default">the <a href="fsm_1.html#U2" target="_self">InitS</a> of <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>20</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"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="fsm_1.html#E6:117_2"><i><font color="Green">E95</font></i></a>, <a class="txt" href="fsm_1.html#E16:117_2"><i><font color="Green">E100</font></i></a>, <a class="ref" href="funct_1.html#T56">FUNCT_1:56</a></i>
<br/>.= 
<font color="Maroon">c<sub>17</sub></font>
<b>by </b><i><a class="txt" href="fsm_1.html#E6:117_2"><i><font color="Green">E95</font></i></a>, <a class="txt" href="fsm_1.html#E12:117_2"><i><font color="Green">E99</font></i></a>, <a class="ref" href="funct_1.html#T56">FUNCT_1:56</a></i>
;<br/>
<a NAME="E19:117_2"/><b>then </b>
the <a href="fsm_1.html#U2" target="_self">InitS</a> of <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>20</sub></font> <a href="fsm_1.html#R1" target="_self">-leads_to</a> <font color="Maroon">c<sub>17</sub></font>
 
<b>by </b><i><a class="ref" href="fsm_1.html#D3" target="_self">Def3</a></i>;<br/>
<b>hence </b><a NAME="E20:117_2"/>
contradiction
 <b>by </b><i><a class="txt" href="fsm_1.html#E3:117_2"><i><font color="Green">E93</font></i></a>, <a class="ref" href="fsm_1.html#D21" target="_self">Def21</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:117"/><i><font color="Green">E93</font></i>: 
(  <a href="fsm_1.html#K11" target="_self">the_reduction_of</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="fsm_1.html#R4" target="_self">-are_equivalent</a>  &amp;  <a href="fsm_1.html#K11" target="_self">the_reduction_of</a> <font color="Maroon">c<sub>4</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#T58" target="_self">Th58</a></i>;<br/>
<a NAME="E9:117"/>
( <font color="Maroon">c<sub>7</sub></font>, <a href="fsm_1.html#K11" target="_self">the_reduction_of</a> <font color="Maroon">c<sub>3</sub></font> <a href="fsm_1.html#R4" target="_self">-are_equivalent</a>  &amp; <font color="Maroon">c<sub>8</sub></font>, <a href="fsm_1.html#K11" target="_self">the_reduction_of</a> <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="txt" href="fsm_1.html#E3:117"><i><font color="Green">E88</font></i></a>, <a class="ref" href="fsm_1.html#T82" target="_self">Th82</a></i>;<br/>
<a NAME="E10:117"/><b>then </b><i><font color="Green">E94</font></i>: 
( <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="fsm_1.html#R4" target="_self">-are_equivalent</a>  &amp; <font color="Maroon">c<sub>8</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="txt" href="fsm_1.html#E8:117"><i><font color="Green">E93</font></i></a>, <a class="ref" href="fsm_1.html#T30" target="_self">Th30</a></i>;<br/>
<a NAME="E11:117"/><b>then </b>
<font color="Maroon">c<sub>7</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="txt" href="fsm_1.html#E1:117"><i><font color="Green">E87</font></i></a>, <a class="ref" href="fsm_1.html#T30" target="_self">Th30</a></i>;<br/>
<a NAME="E12:117"/><b>then </b><i><font color="Green">E95</font></i>: 
<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="fsm_1.html#R4" target="_self">-are_equivalent</a> 
 
<b>by </b><i><a class="txt" href="fsm_1.html#E10:117"><i><font color="Green">E94</font></i></a>, <a class="ref" href="fsm_1.html#T30" target="_self">Th30</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>8</sub></font> as  non <a href="struct_0.html#V3">empty</a> <a href="group_1.html#V6">finite</a> <a href="fsm_1.html#V5" target="_self">reduced</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> <b>by </b><i><a class="txt" href="fsm_1.html#E3:117"><i><font color="Green">E88</font></i></a>, <a class="ref" href="fsm_1.html#T65" target="_self">Th65</a></i>;<br/>
<a NAME="E14:117"/>
<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>18</sub></font> <a href="fsm_1.html#R7" target="_self">-are_isomorphic</a> 
 
<b>by </b><i><a class="txt" href="fsm_1.html#E3:117"><i><font color="Green">E88</font></i></a>, <a class="txt" href="fsm_1.html#E5:117"><i><font color="Green">E90</font></i></a>, <a class="txt" href="fsm_1.html#E7:117"><i><font color="Green">E92</font></i></a>, <a class="txt" href="fsm_1.html#E12:117"><i><font color="Green">E95</font></i></a>, <a class="ref" href="fsm_1.html#T83" target="_self">Th83</a></i>;<br/>
<a NAME="E15:117"/><b>then </b>
<font color="Maroon">c<sub>17</sub></font>, <a href="fsm_1.html#K11" target="_self">the_reduction_of</a> <font color="Maroon">c<sub>4</sub></font> <a href="fsm_1.html#R7" target="_self">-are_isomorphic</a> 
 
<b>by </b><i><a class="txt" href="fsm_1.html#E3:117"><i><font color="Green">E88</font></i></a>, <a class="ref" href="fsm_1.html#T59" target="_self">Th59</a></i>;<br/>
<b>hence </b><a NAME="E16:117"/>
 <a href="fsm_1.html#K11" target="_self">the_reduction_of</a> <font color="Maroon">c<sub>3</sub></font>, <a href="fsm_1.html#K11" target="_self">the_reduction_of</a> <font color="Maroon">c<sub>4</sub></font> <a href="fsm_1.html#R7" target="_self">-are_isomorphic</a> 
 <b>by </b><i><a class="txt" href="fsm_1.html#E3:117"><i><font color="Green">E88</font></i></a>, <a class="ref" href="fsm_1.html#T59" target="_self">Th59</a></i>;<br/>


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