<?xml version="1.0"?>
<div class="add">

<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">Rtfsm</font> be  non <a href="struct_0.html#V3">empty</a> <a href="group_1.html#V6">finite</a> <a href="fsm_1.html#V5" target="_self">reduced</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>set </b><font color="Maroon">m</font> = <font color="Maroon">Rtfsm</font>;<br/>
<b>set </b><font color="Maroon">rm</font> =  <a href="fsm_1.html#K11" target="_self">the_reduction_of</a> <font color="Maroon">Rtfsm</font>;<br/>
<b>set </b><font color="Maroon">fpm</font> =  <a href="fsm_1.html#K8" target="_self">final_states_partition</a> <font color="Maroon">Rtfsm</font>;<br/>
<a NAME="E1:89"/><i><font color="Green">E22</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="fsm_1.html#K11" target="_self">the_reduction_of</a> <font color="Maroon">Rtfsm</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="fsm_1.html#K8" target="_self">final_states_partition</a> <font color="Maroon">Rtfsm</font>
 
<b>by </b><i/>;<br/>
<a NAME="E2:89"/><i><font color="Green">E23</font></i>: 
the <a href="fsm_1.html#U2" target="_self">InitS</a> of <font color="Maroon">Rtfsm</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">Rtfsm</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Rtfsm</font>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="fsm_1.html#K8" target="_self">final_states_partition</a> <font color="Maroon">Rtfsm</font> = <span class="p1">(<span class="default"><a href="borsuk_1.html#K2">proj</a> <span class="p2">(<span class="default"><a href="fsm_1.html#K8" target="_self">final_states_partition</a> <font color="Maroon">Rtfsm</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<b>consider </b><font color="Maroon">Tf</font> being   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Rtfsm</font>, <a href="fsm_1.html#K8" target="_self">final_states_partition</a> <font color="Maroon">Rtfsm</font><b> such that </b><br/><a NAME="E4:89"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">q</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Rtfsm</font> holds  <font color="Maroon">Tf</font> <a href="funct_2.html#K3">.</a> <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">q</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S4">FUNCT_2:sch 4</a>();<br/></i>
<div><i><font color="Green">E39</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="89_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:89_1"/>
not <font color="Maroon">Tf</font> is <a href="funct_1.html#V2">one-to-one</a>
 ;<br/><b>then consider </b><font color="Maroon">q1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">q2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:89_1"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">q1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Rtfsm</font> &amp; <font color="Maroon">q2</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Rtfsm</font> &amp; <font color="Maroon">Tf</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">q1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Tf</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">q2</font> &amp; <font color="Maroon">q1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q2</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#T25">FUNCT_2:25</a></i>;<br/><b>reconsider </b><font color="Maroon">q1</font> = <font color="Maroon">q1</font>, <font color="Maroon">q2</font> = <font color="Maroon">q2</font> as   <a href="fsm_1.html#NM1">State</a> of <font color="Maroon">Rtfsm</font> <b>by </b><i><a class="ref" href="fsm_1.html#T14" target="_self">Th14</a></i>;<br/><a NAME="E5:89_1"/><i><font color="Green">E42</font></i>: 
not <font color="Maroon">q1</font>,<font color="Maroon">q2</font> <a href="fsm_1.html#R5" target="_self">-are_equivalent</a> 
 <b>by </b><i><a class="ref" href="fsm_1.html#T14" target="_self">Th14</a>, <a class="ref" href="fsm_1.html#T13" target="_self">Th13</a></i>;<br/><a NAME="E6:89_1"/><i><font color="Green">E43</font></i>: 
 <a href="fsm_1.html#K8" target="_self">final_states_partition</a> <font color="Maroon">Rtfsm</font> is <a href="fsm_1.html#V4" target="_self">final</a>
 <b>by </b><i/>;<br/><a NAME="E7:89_1"/>
( <font color="Maroon">Tf</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">q1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K2">proj</a> <span class="p2">(<span class="default"><a href="fsm_1.html#K8" target="_self">final_states_partition</a> <font color="Maroon">Rtfsm</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">q1</font> &amp; <font color="Maroon">Tf</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">q2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K2">proj</a> <span class="p2">(<span class="default"><a href="fsm_1.html#K8" target="_self">final_states_partition</a> <font color="Maroon">Rtfsm</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">q2</font> )
 <b>by </b><i/>;<br/><a NAME="E8:89_1"/><b>then </b>
( <font color="Maroon">q1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Tf</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">q1</font> &amp; <font color="Maroon">q2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Tf</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">q2</font> )
 <b>by </b><i><a class="ref" href="borsuk_1.html#D1">BORSUK_1:def 1</a></i>;<br/><b>hence </b><a NAME="E9:89_1"/>
contradiction
 <b>by </b><i><a class="ref" href="fsm_1.html#T14" target="_self">Th14</a>, , , </i>;<br/></div><b>end;</b></div>
<a NAME="E6:89"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">Tf</font> <a href="hidden.html#R1">=</a>  <a href="fsm_1.html#K8" target="_self">final_states_partition</a> <font color="Maroon">Rtfsm</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:89_2"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">Tf</font> <a href="tarski.html#R1">c=</a>  <a href="fsm_1.html#K8" target="_self">final_states_partition</a> <font color="Maroon">Rtfsm</font>
 <b>by </b><i><a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<b>thus </b><a NAME="E2:89_2"/>
 <a href="fsm_1.html#K8" target="_self">final_states_partition</a> <font color="Maroon">Rtfsm</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">Tf</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:89_2_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="fsm_1.html#K8" target="_self">final_states_partition</a> <font color="Maroon">Rtfsm</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">pq</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Rtfsm</font> ;<br/>
<b>consider </b><font color="Maroon">q</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Rtfsm</font><b> such that </b><br/><a NAME="E4:89_2_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">pq</font>
 <b>by </b><i><a class="ref" href="fsm_1.html#T14" target="_self">Th14</a>, </i>;<br/>
<a NAME="E5:89_2_1"/>
<font color="Maroon">pq</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K2">proj</a> <span class="p2">(<span class="default"><a href="fsm_1.html#K8" target="_self">final_states_partition</a> <font color="Maroon">Rtfsm</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">q</font>
 
<b>by </b><i><a class="ref" href="fsm_1.html#T14" target="_self">Th14</a>, , <a class="ref" href="borsuk_1.html#T29">BORSUK_1:29</a></i>;<br/>
<a NAME="E6:89_2_1"/><b>then </b>
<font color="Maroon">Tf</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">pq</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E7:89_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">Tf</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T6">FUNCT_2:6</a></i>;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div>
<a NAME="E7:89"/><b>then </b>
<font color="Maroon">Tf</font> is <a href="funct_2.html#V2">onto</a>
 
<b>by </b><i><a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a></i>;<br/>
<a NAME="E8:89"/><b>then </b><i><font color="Green">E46</font></i>: 
<font color="Maroon">Tf</font> is <a href="funct_2.html#V3">bijective</a>
 
<b>by </b><i>, <a class="ref" href="funct_2.html#D4">FUNCT_2:def 4</a></i>;<br/>
<b>set </b><font color="Maroon">Im</font> = the <a href="fsm_1.html#U2" target="_self">InitS</a> of <font color="Maroon">Rtfsm</font>;<br/>
<a NAME="E9:89"/>
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">Rtfsm</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K2">proj</a> <span class="p2">(<span class="default"><a href="fsm_1.html#K8" target="_self">final_states_partition</a> <font color="Maroon">Rtfsm</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> the <a href="fsm_1.html#U2" target="_self">InitS</a> of <font color="Maroon">Rtfsm</font>
 
<b>by </b><i>, , <a class="ref" href="borsuk_1.html#T29">BORSUK_1:29</a></i>;<br/>
<a NAME="E10:89"/><b>then </b><i><font color="Green">E47</font></i>: 
<font color="Maroon">Tf</font> <a href="funct_2.html#K3">.</a> the <a href="fsm_1.html#U2" target="_self">InitS</a> of <font color="Maroon">Rtfsm</font> <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">Rtfsm</font></span>)</span>
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">q</font> be   <a href="fsm_1.html#NM1">State</a> of <font color="Maroon">Rtfsm</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>reconsider </b><font color="Maroon">Tfq</font> = <font color="Maroon">Tf</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">q</font> as   <a href="fsm_1.html#NM1">State</a> of <span class="p1">(<span class="default"><a href="fsm_1.html#K11" target="_self">the_reduction_of</a> <font color="Maroon">Rtfsm</font></span>)</span> <b>by </b><i/>;<br/><a NAME="E2:89_3"/>
<font color="Maroon">Tf</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K2">proj</a> <span class="p2">(<span class="default"><a href="fsm_1.html#K8" target="_self">final_states_partition</a> <font color="Maroon">Rtfsm</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">q</font>
 <b>by </b><i/>;<br/><a NAME="E3:89_3"/><b>then </b><i><font color="Green">E79</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Tf</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">q</font>
 <b>by </b><i><a class="ref" href="borsuk_1.html#D1">BORSUK_1:def 1</a></i>;<br/><a NAME="E4:89_3"/><b>then </b><i><font color="Green">E81</font></i>: 
the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">Rtfsm</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">q</font>,<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> the <a href="fsm_1.html#U1" target="_self">Tran</a> of <span class="p1">(<span class="default"><a href="fsm_1.html#K11" target="_self">the_reduction_of</a> <font color="Maroon">Rtfsm</font></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">Tf</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">q</font></span>)</span>,<font color="Maroon">s</font>
 <b>by </b><i>, </i>;<br/><a NAME="E5:89_3"/>
the <a href="fsm_1.html#U1" target="_self">Tran</a> of <span class="p1">(<span class="default"><a href="fsm_1.html#K11" target="_self">the_reduction_of</a> <font color="Maroon">Rtfsm</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">Tfq</font>,<font color="Maroon">s</font></span><a href="domain_1.html#K1">]</a></span> is   <a href="fsm_1.html#NM1">State</a> of <span class="p1">(<span class="default"><a href="fsm_1.html#K11" target="_self">the_reduction_of</a> <font color="Maroon">Rtfsm</font></span>)</span>
 ;<br/><a NAME="E6:89_3"/><b>then </b>
the <a href="fsm_1.html#U1" target="_self">Tran</a> of <span class="p1">(<span class="default"><a href="fsm_1.html#K11" target="_self">the_reduction_of</a> <font color="Maroon">Rtfsm</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Tf</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">q</font></span>)</span>,<font color="Maroon">s</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K2">proj</a> <span class="p2">(<span class="default"><a href="fsm_1.html#K8" target="_self">final_states_partition</a> <font color="Maroon">Rtfsm</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">Rtfsm</font> <a href="funct_2.html#K3">.</a> <span class="p2"><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>
 <b>by </b><i>, , <a class="ref" href="borsuk_1.html#T29">BORSUK_1:29</a></i>;<br/><b>hence </b><a NAME="E7:89_3"/>
<font color="Maroon">Tf</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">Rtfsm</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">q</font>,<font color="Maroon">s</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="fsm_1.html#U1" target="_self">Tran</a> of <span class="p1">(<span class="default"><a href="fsm_1.html#K11" target="_self">the_reduction_of</a> <font color="Maroon">Rtfsm</font></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">Tf</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">q</font></span>)</span>,<font color="Maroon">s</font>
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E8:89_3"/>
the <a href="fsm_1.html#U3" target="_self">OFun</a> of <font color="Maroon">Rtfsm</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">q</font>,<font color="Maroon">s</font> <a href="hidden.html#R1">=</a> 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">Rtfsm</font></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">Tf</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">q</font></span>)</span>,<font color="Maroon">s</font>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E12:89"/>
<font color="Maroon">Rtfsm</font>, <a href="fsm_1.html#K11" target="_self">the_reduction_of</a> <font color="Maroon">Rtfsm</font> <a href="fsm_1.html#R7" target="_self">-are_isomorphic</a> 
 <b>by </b><i>, <a class="ref" href="fsm_1.html#T14" target="_self">Th14</a>, , <a class="ref" href="fsm_1.html#T11" target="_self">Th11</a></i>;<br/>


</div>
