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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</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>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>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>;<br/>







<b>set </b><font color="Maroon">c<sub>5</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>1</sub></font> <a href="fsm_1.html#K7" target="_self">-eq_states_partition</a> <font color="Maroon">c<sub>4</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="fsm_1.html#K7" target="_self">-eq_states_partition</a> <font color="Maroon">c<sub>4</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>1</sub></font> <a href="fsm_1.html#K6" target="_self">-eq_states_EqR</a> <font color="Maroon">c<sub>4</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>9</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="fsm_1.html#K6" target="_self">-eq_states_EqR</a> <font color="Maroon">c<sub>4</sub></font>;<br/>
<b>assume </b><a NAME="E1:71"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="fsm_1.html#K7" target="_self">-eq_states_partition</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="fsm_1.html#K7" target="_self">-eq_states_partition</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>11</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>4</sub></font>;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:71_1_1"/>
<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="fsm_1.html#R5" target="_self">-are_equivalent</a> 
 ;<br/><a NAME="E2:71_1_1"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="fsm_1.html#R6" target="_self">-equivalent</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="ref" href="fsm_1.html#T41" target="_self">Th41</a></i>;<br/><a NAME="E3:71_1_1"/><b>then </b><i><font color="Green">E52</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="fsm_1.html#K6" target="_self">-eq_states_EqR</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="fsm_1.html#D12" target="_self">Def12</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>12</sub></font> =  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="fsm_1.html#K6" target="_self">-eq_states_EqR</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>11</sub></font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>1</sub></font> <a href="fsm_1.html#K7" target="_self">-eq_states_partition</a> <font color="Maroon">c<sub>4</sub></font> <b>by </b><i><a class="ref" href="eqrel_1.html#D5">EQREL_1:def 5</a></i>;<br/><b>take </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>12</sub></font>;<br/><b>thus </b><a NAME="E5:71_1_1"/>
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font> )
 <b>by </b><i><a class="txt" href="fsm_1.html#E3:71_1_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="eqrel_1.html#T27">EQREL_1:27</a>, <a class="ref" href="eqrel_1.html#T28">EQREL_1:28</a></i>;<br/></div>
<b>end;</b></div><b>given </b><font color="Maroon">c<sub>12</sub></font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>1</sub></font> <a href="fsm_1.html#K7" target="_self">-eq_states_partition</a> <font color="Maroon">c<sub>4</sub></font><b> such that </b><a NAME="E2:71_1"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font> )
 ;<br/><b>consider </b><font color="Maroon">c<sub>13</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:71_1"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="fsm_1.html#K6" target="_self">-eq_states_EqR</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>13</sub></font> )
 <b>by </b><i><a class="ref" href="eqrel_1.html#D5">EQREL_1:def 5</a></i>;<br/><a NAME="E5:71_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="fsm_1.html#K6" target="_self">-eq_states_EqR</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="fsm_1.html#E2:71_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="fsm_1.html#E4:71_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="eqrel_1.html#T27">EQREL_1:27</a></i>;<br/><a NAME="E6:71_1"/><b>then </b><i><font color="Green">E54</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="fsm_1.html#K6" target="_self">-eq_states_EqR</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="eqrel_1.html#T12">EQREL_1:12</a></i>;<br/><a NAME="E7:71_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="fsm_1.html#K6" target="_self">-eq_states_EqR</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="fsm_1.html#E2:71_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="fsm_1.html#E4:71_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="eqrel_1.html#T27">EQREL_1:27</a></i>;<br/><a NAME="E8:71_1"/><b>then </b><i><font color="Green">E55</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="fsm_1.html#K6" target="_self">-eq_states_EqR</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="fsm_1.html#E6:71_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="eqrel_1.html#T13">EQREL_1:13</a></i>;<br/><a NAME="E9:71_1"/><b>then </b><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="fsm_1.html#R6" target="_self">-equivalent</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="ref" href="fsm_1.html#D12" target="_self">Def12</a></i>;<br/><b>thus </b><a NAME="E10:71_1"/>
<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="fsm_1.html#R5" target="_self">-are_equivalent</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>14</sub></font> be    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">c<sub>2</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="fsm_1.html#D10" target="_self">FSM_1:def 10</a><br/>

<b>consider </b><font color="Maroon">c<sub>15</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E2:71_1_2"/><i><font color="Green">E57</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D2">FINSEQ_1:def 2</a></i>;<br/>
<a NAME="E3:71_1_2"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="txt" href="fsm_1.html#E2:71_1_2"><i><font color="Green">E57</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="71_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:71_1_2_1"/>
( <font color="Maroon">c<sub>15</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> or <font color="Maroon">c<sub>15</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>1</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="71_1_2_1_1"><b>suppose </b></a><a NAME="E1:71_1_2_1_1"/>
<font color="Maroon">c<sub>15</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:71_1_2_1_1"/>
<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="fsm_1.html#K4" target="_self">-response</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="fsm_1.html#K4" target="_self">-response</a> 
 <b>by </b><i><a class="txt" href="fsm_1.html#E9:71_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="fsm_1.html#E3:71_1_2"><i><font color="Green">E58</font></i></a>, <a class="ref" href="fsm_1.html#D11" target="_self">Def11</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="71_1_2_1_2"><b>suppose </b></a><a NAME="E1:71_1_2_1_2"/>
<font color="Maroon">c<sub>15</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>16</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:71_1_2_1_2"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>16</sub></font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>16</sub></font> )
 <b>by </b><i><a class="ref" href="fsm_1.html#T1" target="_self">Th1</a></i>;<br/><a NAME="E4:71_1_2_1_2"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="fsm_1.html#K7" target="_self">-eq_states_partition</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="fsm_1.html#K7" target="_self">-eq_states_partition</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="fsm_1.html#E1:71"><i><font color="Green">E51</font></i></a>, <a class="txt" href="fsm_1.html#E3:71_1_2_1_2"><i><font color="Green">E59</font></i></a>, <a class="ref" href="fsm_1.html#T46" target="_self">Th46</a></i>;<br/><a NAME="E5:71_1_2_1_2"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>15</sub></font> <a href="fsm_1.html#K6" target="_self">-eq_states_EqR</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="fsm_1.html#E8:71_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="fsm_1.html#T8" target="_self">Th8</a>, <a class="txt" href="fsm_1.html#E4:71_1_2_1_2"><i><font color="Green">E60</font></i></a></i>;<br/><a NAME="E6:71_1_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>15</sub></font> <a href="fsm_1.html#R6" target="_self">-equivalent</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="ref" href="fsm_1.html#D12" target="_self">Def12</a></i>;<br/><b>hence </b><a NAME="E7:71_1_2_1_2"/>
<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="fsm_1.html#K4" target="_self">-response</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="fsm_1.html#K4" target="_self">-response</a> 
 <b>by </b><i><a class="txt" href="fsm_1.html#E3:71_1_2"><i><font color="Green">E58</font></i></a>, <a class="ref" href="fsm_1.html#D11" target="_self">Def11</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:71"/>
<font color="Maroon">c<sub>1</sub></font> <a href="fsm_1.html#K7" target="_self">-eq_states_partition</a> <font color="Maroon">c<sub>4</sub></font> is <a href="fsm_1.html#V4" target="_self">final</a>
 <b>by </b><i><a class="ref" href="fsm_1.html#D14" target="_self">Def14</a></i>;<br/>


</div>
