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<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">IAlph</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">OAlph</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">tfsm</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">IAlph</font>,<font color="Maroon">OAlph</font>;<br/>







<b>assume </b><a NAME="E1:70"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">k</font> <a href="fsm_1.html#K7" target="_self">-eq_states_partition</a> <font color="Maroon">tfsm</font> is <a href="fsm_1.html#V4" target="_self">final</a>
 ;<br/>

<b>set </b><font color="Maroon">S</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">tfsm</font>;<br/>
<b>set </b><font color="Maroon">keq</font> = <font color="Maroon">k</font> <a href="fsm_1.html#K6" target="_self">-eq_states_EqR</a> <font color="Maroon">tfsm</font>;<br/>
<b>set </b><font color="Maroon">k1eq</font> = <span class="p1">(<span class="default"><font color="Maroon">k</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">tfsm</font>;<br/>
<b>set </b><font color="Maroon">kpart</font> = <font color="Maroon">k</font> <a href="fsm_1.html#K7" target="_self">-eq_states_partition</a> <font color="Maroon">tfsm</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">x</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">E23</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">k</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">tfsm</font>
 ;<br/><b>then consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:70_1_1"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">tfsm</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">tfsm</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">a</font> = <font color="Maroon">a</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">tfsm</font> <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">b</font> = <font color="Maroon">b</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">tfsm</font> <b>by </b><i/>;<br/><a NAME="E6:70_1_1"/>
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="fsm_1.html#R6" target="_self">-equivalent</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
 <b>by </b><i>, , </i>;<br/><a NAME="E7:70_1_1"/><b>then </b>
<font color="Maroon">k</font> <a href="fsm_1.html#R6" target="_self">-equivalent</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
 <b>by </b><i><a class="txt" href="fsm_1.html#E2:70_1"><i><font color="Green">E39</font></i></a></i>;<br/><b>hence </b><a NAME="E8:70_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">k</font> <a href="fsm_1.html#K6" target="_self">-eq_states_EqR</a> <font color="Maroon">tfsm</font>
 <b>by </b><i>, </i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:70_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">k</font> <a href="fsm_1.html#K6" target="_self">-eq_states_EqR</a> <font color="Maroon">tfsm</font>
 ;<br/><b>then consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:70_1"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">tfsm</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">tfsm</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">a</font> = <font color="Maroon">a</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">tfsm</font> <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">b</font> = <font color="Maroon">b</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">tfsm</font> <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">cl</font> =  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="fsm_1.html#K6" target="_self">-eq_states_EqR</a> <font color="Maroon">tfsm</font></span>)</span>,<font color="Maroon">b</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">k</font> <a href="fsm_1.html#K7" target="_self">-eq_states_partition</a> <font color="Maroon">tfsm</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">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">cl</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">cl</font> )
 <b>by </b><i>, , <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">a</font>,<font color="Maroon">b</font> <a href="fsm_1.html#R5" target="_self">-are_equivalent</a> 
 <b>by </b><i><a class="ref" href="fsm_1.html#T10" target="_self">Th10</a>, </i>;<br/><a NAME="E10:70_1"/><b>then </b>
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="fsm_1.html#R6" target="_self">-equivalent</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E11:70_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">k</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">tfsm</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:70"/>
<span class="p1">(<span class="default"><font color="Maroon">k</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">tfsm</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="fsm_1.html#K6" target="_self">-eq_states_EqR</a> <font color="Maroon">tfsm</font>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


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