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<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> 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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>





<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</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>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> = 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>4</sub></font> <a href="fsm_1.html#K2" target="_self">-admissible</a> ;<br/><b>set </b><font color="Maroon">c<sub>7</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>,<font color="Maroon">c<sub>4</sub></font> <a href="fsm_1.html#K2" target="_self">-admissible</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>3</sub></font>,<font color="Maroon">c<sub>4</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 <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>4</sub></font> <a href="fsm_1.html#K4" target="_self">-response</a> ;<br/><a NAME="E1:80_1"/><i><font color="Green">E58</font></i>: 
the <a href="fsm_1.html#U2" target="_self">InitS</a> of <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</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>
 <b>by </b><i><a class="ref" href="fsm_1.html#D18" target="_self">Def18</a></i>;<br/><i><font color="Green">E59</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>4</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>4</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 <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>4</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="80_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>10</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:80_1_2"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>10</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>4</sub></font> <a href="fsm_1.html#K4" target="_self">-response</a> </span>)</span> )
 ;<br/><a NAME="E2:80_1_2"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="ref" href="fsm_1.html#D6" target="_self">Def6</a></i>;<br/><a NAME="E3:80_1_2"/><b>then </b><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>10</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>4</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E4:80_1_2"/><b>then </b><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>10</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>4</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="E5:80_1_2"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="fsm_1.html#E3:80_1_2"><i><font color="Green">E60</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E6:80_1_2"/>
<font color="Maroon">c<sub>10</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>4</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#E4:80_1_2"><i><font color="Green">E61</font></i></a>, <a class="ref" href="fsm_1.html#D2" target="_self">Def2</a></i>;<br/><a NAME="E7:80_1_2"/><b>then </b>
<font color="Maroon">c<sub>10</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>4</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="E8:80_1_2"/><b>then </b><i><font color="Green">E63</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>4</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>10</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>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</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#E5:80_1_2"><i><font color="Green">E62</font></i></a>, <a class="ref" href="finseq_2.html#T13">FINSEQ_2:13</a></i>;<br/><a NAME="E9:80_1_2"/>
<font color="Maroon">c<sub>10</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 <span class="p3">(<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>4</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#E4:80_1_2"><i><font color="Green">E61</font></i></a>, <a class="ref" href="fsm_1.html#D2" target="_self">Def2</a></i>;<br/><a NAME="E10:80_1_2"/><b>then </b>
<font color="Maroon">c<sub>10</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 <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>4</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="E11:80_1_2"/><b>then </b><i><font color="Green">E64</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> &amp; <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>4</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>10</sub></font> is   <a href="struct_0.html#NM1">Element</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>by </b><i><a class="txt" href="fsm_1.html#E5:80_1_2"><i><font color="Green">E62</font></i></a>, <a class="ref" href="finseq_2.html#T13">FINSEQ_2:13</a></i>;<br/><a NAME="E12:80_1_2"/><i><font color="Green">E65</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>4</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>10</sub></font> <a href="hidden.html#R2">in</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>4</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>10</sub></font>
 <b>by </b><i><a class="txt" href="fsm_1.html#E1:80_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="fsm_1.html#E4:80_1_2"><i><font color="Green">E61</font></i></a>, <a class="ref" href="fsm_1.html#T57" target="_self">Th57</a></i>;<br/><b>thus </b><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>4</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>10</sub></font> = 
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#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 <span class="p3">(<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>4</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>10</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="fsm_1.html#E5:80_1_2"><i><font color="Green">E62</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>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>4</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>10</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="fsm_1.html#E8:80_1_2"><i><font color="Green">E63</font></i></a>, <a class="txt" href="fsm_1.html#E11:80_1_2"><i><font color="Green">E64</font></i></a>, <a class="txt" href="fsm_1.html#E12:80_1_2"><i><font color="Green">E65</font></i></a>, <a class="ref" href="fsm_1.html#D18" target="_self">Def18</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>4</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>10</sub></font>
<b>by </b><i><a class="txt" href="fsm_1.html#E5:80_1_2"><i><font color="Green">E62</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="E4:80_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>4</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 <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>4</sub></font> <a href="fsm_1.html#K4" target="_self">-response</a> 
 <b>by </b><i><a class="txt" href="fsm_1.html#E2:80_1"><i><font color="Green">E59</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="E2:80"/>
<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>3</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/>


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