<?xml version="1.0"?>
<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>let </b><font color="Maroon">it1</font> be  <a href="fsm_1.html#V2" target="_self">strict</a>  <a href="fsm_1.html#L2" target="_self">Mealy-FSM</a> of <font color="Maroon">IAlph</font>,<font color="Maroon">OAlph</font>, <font color="Maroon">it2</font> be  <a href="fsm_1.html#V2" target="_self">strict</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="E1:77_1_2"/><i><font color="Green">E47</font></i>: 
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">it1</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="struct_0.html#NM1">Element</a> of <font color="Maroon">it1</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">it1</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">it1</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">it1</font> )
 ;<br/>

<b>assume </b><a NAME="E2:77_1_2"/><i><font color="Green">E79</font></i>: 
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">it2</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="struct_0.html#NM1">Element</a> of <font color="Maroon">it2</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">it2</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">it2</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">it2</font> )
 ;<br/>

<a NAME="E3:77_1_2"/><i><font color="Green">E81</font></i>: 
the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">it1</font> <a href="hidden.html#R1">=</a> the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">it2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1_2_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">T1</font> = the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">it1</font> as   <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>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">T2</font> = the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">it2</font> as   <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>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">Q</font> be    <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>let </b><font color="Maroon">s</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">IAlph</font>;<br/><b>consider </b><font color="Maroon">q</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">tfsm</font><b> such that </b><br/><a NAME="E2:77_1_2_1_1"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font>
 <b>by </b><i/>;<br/><a NAME="E3:77_1_2_1_1"/><i><font color="Green">E83</font></i>: 
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">q</font>,<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T1</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="fsm_1.html#T11" target="_self">Th11</a></i>;<br/><a NAME="E4:77_1_2_1_1"/><i><font color="Green">E84</font></i>: 
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">q</font>,<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T2</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="fsm_1.html#T11" target="_self">Th11</a></i>;<br/><b>reconsider </b><font color="Maroon">T1'</font> = <font color="Maroon">T1</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>, <font color="Maroon">T2'</font> = <font color="Maroon">T2</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> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">tfsm</font> <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E6:77_1_2_1_1"/>
( <font color="Maroon">T1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">T2'</font> or <font color="Maroon">T1'</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">T2'</font> )
 <b>by </b><i><a class="ref" href="eqrel_1.html#D6">EQREL_1:def 6</a></i>;<br/><b>hence </b><a NAME="E7:77_1_2_1_1"/>
<font color="Maroon">T1</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">Q</font>,<font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">T2</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="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:77_1_2_1"/>
the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">it1</font> <a href="hidden.html#R1">=</a> the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">it2</font>
 <b>by </b><i><a class="ref" href="binop_1.html#T2">BINOP_1:2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:77_1_2"/><i><font color="Green">E86</font></i>: 
the <a href="fsm_1.html#U3" target="_self">OFun</a> of <font color="Maroon">it1</font> <a href="funct_2.html#R1">=</a> the <a href="fsm_1.html#U3" target="_self">OFun</a> of <font color="Maroon">it2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1_2_2"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">OF1</font> = the <a href="fsm_1.html#U3" target="_self">OFun</a> of <font color="Maroon">it1</font> as   <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>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">OF2</font> = the <a href="fsm_1.html#U3" target="_self">OFun</a> of <font color="Maroon">it2</font> as   <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>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1_2_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">Q</font> be    <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>let </b><font color="Maroon">s</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">IAlph</font>;<br/><b>consider </b><font color="Maroon">q</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">tfsm</font><b> such that </b><br/><a NAME="E2:77_1_2_2_1"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font>
 <b>by </b><i/>;<br/><b>thus </b><font color="Maroon">OF1</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">Q</font>,<font color="Maroon">s</font> = 
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="Maroon">q</font>,<font color="Maroon">s</font>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">OF2</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">Q</font>,<font color="Maroon">s</font>
<b>by </b><i>, </i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:77_1_2_2"/>
the <a href="fsm_1.html#U3" target="_self">OFun</a> of <font color="Maroon">it1</font> <a href="funct_2.html#R1">=</a> the <a href="fsm_1.html#U3" target="_self">OFun</a> of <font color="Maroon">it2</font>
 <b>by </b><i><a class="ref" href="binop_1.html#T2">BINOP_1:2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:77_1_2"/>
the <a href="fsm_1.html#U2" target="_self">InitS</a> of <font color="Maroon">it1</font> <a href="hidden.html#R1">=</a> the <a href="fsm_1.html#U2" target="_self">InitS</a> of <font color="Maroon">it2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1_2_3"><b>proof </b></a><div class="add">

<a NAME="E1:77_1_2_3"/>
the <a href="fsm_1.html#U2" target="_self">InitS</a> of <font color="Maroon">it1</font> <a href="hidden.html#R2">in</a>  <a href="fsm_1.html#K8" target="_self">final_states_partition</a> <font color="Maroon">tfsm</font>
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">IS1</font> = the <a href="fsm_1.html#U2" target="_self">InitS</a> of <font color="Maroon">it1</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">tfsm</font> ;<br/>
<a NAME="E3:77_1_2_3"/>
the <a href="fsm_1.html#U2" target="_self">InitS</a> of <font color="Maroon">it2</font> <a href="hidden.html#R2">in</a>  <a href="fsm_1.html#K8" target="_self">final_states_partition</a> <font color="Maroon">tfsm</font>
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">IS2</font> = the <a href="fsm_1.html#U2" target="_self">InitS</a> of <font color="Maroon">it2</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">tfsm</font> ;<br/>
<a NAME="E5:77_1_2_3"/>
( <font color="Maroon">IS1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">IS2</font> or <font color="Maroon">IS1</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">IS2</font> )
 
<b>by </b><i>, , <a class="ref" href="eqrel_1.html#D6">EQREL_1:def 6</a></i>;<br/>
<b>hence </b><a NAME="E6:77_1_2_3"/>
the <a href="fsm_1.html#U2" target="_self">InitS</a> of <font color="Maroon">it1</font> <a href="hidden.html#R1">=</a> the <a href="fsm_1.html#U2" target="_self">InitS</a> of <font color="Maroon">it2</font>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:77_1_2"/>
<font color="Maroon">it1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">it2</font>
 <b>by </b><i>, , , <a class="ref" href="fsm_1.html#T11" target="_self">Th11</a></i>;<br/>


</div>
