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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>assume </b><a NAME="E1:70"/><i><font color="Green">E50</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> is <a href="fsm_1.html#V4" target="_self">final</a>
 ;<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#K6" target="_self">-eq_states_EqR</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#K6" target="_self">-eq_states_EqR</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#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="70_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>9</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:70_1_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</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#K6" target="_self">-eq_states_EqR</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>11</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:70_1_1"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <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>11</sub></font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">c<sub>10</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>11</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> )
 <b>by </b><i><a class="ref" href="relset_1.html#T6">RELSET_1:6</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>10</sub></font> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font> <b>by </b><i><a class="txt" href="fsm_1.html#E3:70_1_1"><i><font color="Green">E52</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>11</sub></font> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font> <b>by </b><i><a class="txt" href="fsm_1.html#E3:70_1_1"><i><font color="Green">E52</font></i></a></i>;<br/><a NAME="E6:70_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="fsm_1.html#R6" target="_self">-equivalent</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="fsm_1.html#E1:70_1_1"><i><font color="Green">E51</font></i></a>, <a class="txt" href="fsm_1.html#E3:70_1_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="fsm_1.html#D12" target="_self">Def12</a></i>;<br/><a NAME="E7:70_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>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="ref" href="fsm_1.html#T43" target="_self">Th43</a></i>;<br/><b>hence </b><a NAME="E8:70_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <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#E3:70_1_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="fsm_1.html#D12" target="_self">Def12</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:70_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <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>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>11</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:70_1"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <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>11</sub></font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">c<sub>10</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>11</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> )
 <b>by </b><i><a class="ref" href="relset_1.html#T6">RELSET_1:6</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>10</sub></font> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font> <b>by </b><i><a class="txt" href="fsm_1.html#E4:70_1"><i><font color="Green">E52</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>11</sub></font> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font> <b>by </b><i><a class="txt" href="fsm_1.html#E4:70_1"><i><font color="Green">E52</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>14</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>13</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/><a NAME="E8:70_1"/>
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font> )
 <b>by </b><i><a class="txt" href="fsm_1.html#E2:70_1"><i><font color="Green">E51</font></i></a>, <a class="txt" href="fsm_1.html#E4:70_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/><a NAME="E9:70_1"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font> <a href="fsm_1.html#R5" target="_self">-are_equivalent</a> 
 <b>by </b><i><a class="txt" href="fsm_1.html#E1:70"><i><font color="Green">E50</font></i></a>, <a class="ref" href="fsm_1.html#D14" target="_self">Def14</a></i>;<br/><a NAME="E10:70_1"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="fsm_1.html#R6" target="_self">-equivalent</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="ref" href="fsm_1.html#T41" target="_self">Th41</a></i>;<br/><b>hence </b><a NAME="E11:70_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</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#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#E4:70_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="fsm_1.html#D12" target="_self">Def12</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:70"/>
<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> <a href="hidden.html#R1">=</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="tarski.html#T2">TARSKI:2</a></i>;<br/>


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