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<div class="add">

<b>set </b><font color="Maroon">TR</font> = the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">tfsm</font>;<br/>
<b>reconsider </b><font color="Maroon">m</font> =  <a href="card_1.html#K4">card</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">tfsm</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<a NAME="E2:77_1_1"/>
 <a href="card_1.html#K4">card</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">tfsm</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="card_2.html#T59">CARD_2:59</a></i>;<br/>
<b>then consider </b><font color="Maroon">n</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:77_1_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T22">NAT_1:22</a></i>;<br/>
<b>reconsider </b><font color="Maroon">n</font> = <font color="Maroon">n</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<b>set </b><font color="Maroon">IS</font> =  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="fsm_1.html#K6" target="_self">-eq_states_EqR</a> <font color="Maroon">tfsm</font></span>)</span>,the <a href="fsm_1.html#U2" target="_self">InitS</a> of <font color="Maroon">tfsm</font>;<br/>
<b>set </b><font color="Maroon">part</font> =  <a href="fsm_1.html#K8" target="_self">final_states_partition</a> <font color="Maroon">tfsm</font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M2">Element</a> of  <a href="fsm_1.html#K8" target="_self">final_states_partition</a> <font color="Maroon">tfsm</font>,   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">IAlph</font>) <b>-&gt; </b>   <a href="subset_1.html#M2">Element</a> of  <a href="fsm_1.html#K8" target="_self">final_states_partition</a> <font color="Maroon">tfsm</font> = <font color="Maroon">a<sub>2</sub></font>,<font color="Maroon">a<sub>1</sub></font> <a href="fsm_1.html#K9" target="_self">-succ_class</a> ;<br/>
<b>consider </b><font color="Maroon">Trf</font> being   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="fsm_1.html#K8" target="_self">final_states_partition</a> <font color="Maroon">tfsm</font></span>)</span>,<font color="Maroon">IAlph</font></span><a href="zfmisc_1.html#K2">:]</a></span>, <a href="fsm_1.html#K8" target="_self">final_states_partition</a> <font color="Maroon">tfsm</font><b> such that </b><br/><a NAME="E7:77_1_1"/><i><font color="Green">E23</font></i>: 
for <font color="Olive">q</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="fsm_1.html#K8" target="_self">final_states_partition</a> <font color="Maroon">tfsm</font><br/> for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">IAlph</font> holds  <font color="Maroon">Trf</font> <a href="binop_1.html#K2">.</a> <font color="Olive">q</font>,<font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">q</font>,<font color="Olive">i</font>)
 <b>from </b><i><a class="ref" href="binop_1.html#S4">BINOP_1:sch 4</a>();<br/></i>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="subset_1.html#M2">Element</a> of  <a href="fsm_1.html#K8" target="_self">final_states_partition</a> <font color="Maroon">tfsm</font>,   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">IAlph</font>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">OAlph</font> = <font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">a<sub>2</sub></font> <a href="fsm_1.html#K10" target="_self">-class_response</a> ;<br/>
<b>consider </b><font color="Maroon">OF</font> being   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="fsm_1.html#K8" target="_self">final_states_partition</a> <font color="Maroon">tfsm</font></span>)</span>,<font color="Maroon">IAlph</font></span><a href="zfmisc_1.html#K2">:]</a></span>,<font color="Maroon">OAlph</font><b> such that </b><br/><a NAME="E9:77_1_1"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">q</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="fsm_1.html#K8" target="_self">final_states_partition</a> <font color="Maroon">tfsm</font><br/> for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">IAlph</font> holds  <font color="Maroon">OF</font> <a href="binop_1.html#K2">.</a> <font color="Olive">q</font>,<font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">q</font>,<font color="Olive">i</font>)
 <b>from </b><i><a class="ref" href="binop_1.html#S4">BINOP_1:sch 4</a>();<br/></i>
<a NAME="E10:77_1_1"/>
 <a href="fsm_1.html#K8" target="_self">final_states_partition</a> <font color="Maroon">tfsm</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="fsm_1.html#K7" target="_self">-eq_states_partition</a> <font color="Maroon">tfsm</font>
 
<b>by </b><i>, </i>;<br/>
<b>then reconsider </b><font color="Maroon">IS</font> =  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="fsm_1.html#K6" target="_self">-eq_states_EqR</a> <font color="Maroon">tfsm</font></span>)</span>,the <a href="fsm_1.html#U2" target="_self">InitS</a> of <font color="Maroon">tfsm</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="fsm_1.html#K8" target="_self">final_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/>
<b>take </b><font color="Maroon">IT</font> =  <a href="fsm_1.html#G2" target="_self">Mealy-FSM</a>(# <span class="p1">(<span class="default"><a href="fsm_1.html#K8" target="_self">final_states_partition</a> <font color="Maroon">tfsm</font></span>)</span>,<font color="Maroon">Trf</font>,<font color="Maroon">OF</font>,<font color="Maroon">IS</font> #);<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">Q</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">IT</font>;<br/><b>let </b><font color="Maroon">s</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">IAlph</font>;<br/><b>let </b><font color="Maroon">q</font> be   <a href="fsm_1.html#NM1">State</a> of <font color="Maroon">tfsm</font>;<br/><b>assume </b><a NAME="E1:77_1_1_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font>
 ;<br/><b>reconsider </b><font color="Maroon">Q'</font> = <font color="Maroon">Q</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="fsm_1.html#K8" target="_self">final_states_partition</a> <font color="Maroon">tfsm</font> ;<br/><b>reconsider </b><font color="Maroon">s'</font> = <font color="Maroon">s</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">IAlph</font> ;<br/><b>consider </b><font color="Maroon">Q1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">tfsm</font>, <font color="Maroon">n1</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E5:77_1_1_1"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">Q1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q'</font> &amp; <font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">tfsm</font> &amp; <font color="Maroon">s'</font>,<font color="Maroon">Q'</font> <a href="fsm_1.html#K9" target="_self">-succ_class</a>  <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="fsm_1.html#K6" target="_self">-eq_states_EqR</a> <font color="Maroon">tfsm</font></span>)</span>,<span class="p1">(<span class="default">the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">tfsm</font> <a href="funct_2.html#K3">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">Q1</font>,<font color="Maroon">s'</font></span><a href="domain_1.html#K1">]</a></span></span>)</span> )
 <b>by </b><i/>;<br/><a NAME="E6:77_1_1_1"/><i><font color="Green">E42</font></i>: 
the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">IT</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">Q'</font>,<font color="Maroon">s'</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="fsm_1.html#K6" target="_self">-eq_states_EqR</a> <font color="Maroon">tfsm</font></span>)</span>,<span class="p1">(<span class="default">the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">tfsm</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">Q1</font>,<font color="Maroon">s'</font></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E7:77_1_1_1"/>
 <a href="fsm_1.html#K8" target="_self">final_states_partition</a> <font color="Maroon">tfsm</font> is <a href="fsm_1.html#V4" target="_self">final</a>
 <b>by </b><i/>;<br/><a NAME="E8:77_1_1_1"/><b>then </b>
<font color="Maroon">q</font>,<font color="Maroon">Q1</font> <a href="fsm_1.html#R5" target="_self">-are_equivalent</a> 
 <b>by </b><i><a class="ref" href="fsm_1.html#T11" target="_self">Th11</a>, , </i>;<br/><a NAME="E9:77_1_1_1"/><b>then </b><i><font color="Green">E43</font></i>: 
<font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1 <a href="fsm_1.html#R6" target="_self">-equivalent</a> <font color="Maroon">q</font>,<font color="Maroon">Q1</font>
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">n1'</font> = <font color="Maroon">n1</font> as   <a href="int_1.html#NM1">Integer</a> ;<br/><b>reconsider </b><font color="Maroon">1'</font> = 1 as   <a href="int_1.html#NM1">Integer</a> ;<br/><a NAME="E12:77_1_1_1"/><i><font color="Green">E44</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n1</font> <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="E13:77_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">n1'</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">1'</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n1'</font>
 ;<br/><a NAME="E14:77_1_1_1"/><b>then </b>
<font color="Maroon">n1</font> <a href="fsm_1.html#R6" target="_self">-equivalent</a> the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">tfsm</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">q</font>,<font color="Maroon">s'</font></span><a href="domain_1.html#K1">]</a></span>,the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">tfsm</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">Q1</font>,<font color="Maroon">s'</font></span><a href="domain_1.html#K1">]</a></span>
 <b>by </b><i><a class="ref" href="fsm_1.html#T12" target="_self">Th12</a>, , <a class="txt" href="fsm_1.html#E5:77_1_1_1"><i><font color="Green">E40</font></i></a></i>;<br/><a NAME="E15:77_1_1_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default">the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">tfsm</font> <a href="funct_2.html#K3">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">q</font>,<font color="Maroon">s'</font></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<span class="p2">(<span class="default">the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">tfsm</font> <a href="funct_2.html#K3">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">Q1</font>,<font color="Maroon">s'</font></span><a href="domain_1.html#K1">]</a></span></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">n1</font> <a href="fsm_1.html#K6" target="_self">-eq_states_EqR</a> <font color="Maroon">tfsm</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E16:77_1_1_1"/>
the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">tfsm</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">q</font>,<font color="Maroon">s</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">IT</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">Q</font>,<font color="Maroon">s</font>
 <b>by </b><i>, <a class="ref" href="eqrel_1.html#T27">EQREL_1:27</a></i>;<br/><a NAME="E17:77_1_1_1"/><i><font color="Green">E45</font></i>: 
the <a href="fsm_1.html#U3" target="_self">OFun</a> of <font color="Maroon">IT</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">Q'</font>,<font color="Maroon">s'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Q'</font>,<font color="Maroon">s'</font> <a href="fsm_1.html#K10" target="_self">-class_response</a> 
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">Q1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">tfsm</font><b> such that </b><br/><a NAME="E19:77_1_1_1"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">Q1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q'</font> &amp; <font color="Maroon">Q'</font>,<font color="Maroon">s'</font> <a href="fsm_1.html#K10" target="_self">-class_response</a>  <a href="hidden.html#R1">=</a> the <a href="fsm_1.html#U3" target="_self">OFun</a> of <font color="Maroon">tfsm</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">Q1</font>,<font color="Maroon">s'</font></span><a href="domain_1.html#K1">]</a></span> )
 <b>by </b><i/>;<br/><a NAME="E20:77_1_1_1"/>
 <a href="fsm_1.html#K8" target="_self">final_states_partition</a> <font color="Maroon">tfsm</font> is <a href="fsm_1.html#V4" target="_self">final</a>
 <b>by </b><i/>;<br/><a NAME="E21:77_1_1_1"/><b>then </b>
<font color="Maroon">q</font>,<font color="Maroon">Q1</font> <a href="fsm_1.html#R5" target="_self">-are_equivalent</a> 
 <b>by </b><i><a class="ref" href="fsm_1.html#T11" target="_self">Th11</a>, , </i>;<br/><b>hence </b><a NAME="E22:77_1_1_1"/>
the <a href="fsm_1.html#U3" target="_self">OFun</a> of <font color="Maroon">tfsm</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">q</font>,<font color="Maroon">s</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> the <a href="fsm_1.html#U3" target="_self">OFun</a> of <font color="Maroon">IT</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">Q</font>,<font color="Maroon">s</font></span><a href="domain_1.html#K1">]</a></span>
 <b>by </b><i><a class="ref" href="fsm_1.html#T13" target="_self">Th13</a>, , </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E13:77_1_1"/>
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">IT</font> <a href="hidden.html#R1">=</a>  <a href="fsm_1.html#K8" target="_self">final_states_partition</a> <font color="Maroon">tfsm</font> &amp; ( for <font color="Olive">Q</font> being  <a href="fsm_1.html#NM1">State</a> of <font color="Maroon">IT</font><br/> for <font color="Olive">s</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">IAlph</font><br/> for <font color="Olive">q</font> being  <a href="fsm_1.html#NM1">State</a> of <font color="Maroon">tfsm</font>  st <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Q</font> holds <br/>( the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">tfsm</font> <a href="binop_1.html#K2">.</a> <font color="Olive">q</font>,<font color="Olive">s</font> <a href="hidden.html#R2">in</a> the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">IT</font> <a href="binop_1.html#K1">.</a> <font color="Olive">Q</font>,<font color="Olive">s</font> &amp; the <a href="fsm_1.html#U3" target="_self">OFun</a> of <font color="Maroon">tfsm</font> <a href="binop_1.html#K2">.</a> <font color="Olive">q</font>,<font color="Olive">s</font> <a href="hidden.html#R1">=</a> the <a href="fsm_1.html#U3" target="_self">OFun</a> of <font color="Maroon">IT</font> <a href="binop_1.html#K1">.</a> <font color="Olive">Q</font>,<font color="Olive">s</font> ) ) &amp; the <a href="fsm_1.html#U2" target="_self">InitS</a> of <font color="Maroon">tfsm</font> <a href="hidden.html#R2">in</a> the <a href="fsm_1.html#U2" target="_self">InitS</a> of <font color="Maroon">IT</font> )
 <b>by </b><i><a class="ref" href="eqrel_1.html#T28">EQREL_1:28</a></i>;<br/>


</div>
