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<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="group_1.html#V6">finite</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>let </b><font color="Maroon">Rtfsm1</font> be  non <a href="struct_0.html#V3">empty</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>let </b><font color="Maroon">Rtfsm2</font> be  non <a href="struct_0.html#V3">empty</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">rtfsm1</font> = <font color="Maroon">Rtfsm1</font>;<br/>
<b>set </b><font color="Maroon">rtfsm2</font> = <font color="Maroon">Rtfsm2</font>;<br/>
<b>assume </b><a NAME="E1:108"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">tfsm</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Rtfsm1</font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">Rtfsm2</font> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Rtfsm1</font> <a href="subset_1.html#R1">misses</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Rtfsm2</font> &amp; <font color="Maroon">Rtfsm1</font>,<font color="Maroon">Rtfsm2</font> <a href="fsm_1.html#R4" target="_self">-are_equivalent</a>  )
 ;<br/>

<b>set </b><font color="Maroon">RUN</font> =  <a href="fsm_1.html#K11" target="_self">the_reduction_of</a> <font color="Maroon">tfsm</font>;<br/>
<b>set </b><font color="Maroon">SRUN</font> = 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">tfsm</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">Stfsm</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">tfsm</font>;<br/>
<b>set </b><font color="Maroon">Srtfsm1</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Rtfsm1</font>;<br/>
<b>set </b><font color="Maroon">Srtfsm2</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Rtfsm2</font>;<br/>
<a NAME="E2:108"/><i><font color="Green">E23</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">tfsm</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Rtfsm1</font> <a href="xboole_0.html#K2">\/</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Rtfsm2</font>
 
<b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">IS1</font> = the <a href="fsm_1.html#U2" target="_self">InitS</a> of <font color="Maroon">Rtfsm1</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>, </i>;<br/>
<b>reconsider </b><font color="Maroon">IS2</font> = the <a href="fsm_1.html#U2" target="_self">InitS</a> of <font color="Maroon">Rtfsm2</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>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E5:108"/><i><font color="Green">E38</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">tfsm</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="fsm_1.html#K8" target="_self">final_states_partition</a> <font color="Maroon">tfsm</font>
 
<b>by </b><i/>;<br/>
<a NAME="E6:108"/><i><font color="Green">E39</font></i>: 
 <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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="108_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">w</font> be    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">IAlph</font>;<br/><a NAME="E1:108_1"/>
the <a href="fsm_1.html#U2" target="_self">InitS</a> of <font color="Maroon">Rtfsm1</font>,<font color="Maroon">w</font> <a href="fsm_1.html#K4" target="_self">-response</a>  <a href="hidden.html#R1">=</a> the <a href="fsm_1.html#U2" target="_self">InitS</a> of <font color="Maroon">Rtfsm2</font>,<font color="Maroon">w</font> <a href="fsm_1.html#K4" target="_self">-response</a> 
 <b>by </b><i>, </i>;<br/><b>hence </b><font color="Maroon">IS1</font>,<font color="Maroon">w</font> <a href="fsm_1.html#K4" target="_self">-response</a>  = 
the <a href="fsm_1.html#U2" target="_self">InitS</a> of <font color="Maroon">Rtfsm2</font>,<font color="Maroon">w</font> <a href="fsm_1.html#K4" target="_self">-response</a> 
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">IS2</font>,<font color="Maroon">w</font> <a href="fsm_1.html#K4" target="_self">-response</a> 
<b>by </b><i>, </i>
;<br/><br/></div><b>end;</b></div>
<a NAME="E8:108"/><b>then </b>
<font color="Maroon">IS1</font>,<font color="Maroon">IS2</font> <a href="fsm_1.html#R5" target="_self">-are_equivalent</a> 
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">X</font> being    <a href="subset_1.html#M1">Element</a> of 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">tfsm</font></span>)</span><b> such that </b><br/><a NAME="E10:108"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">IS1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">IS2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 <b>by </b><i>, , </i>;<br/>
<b>take </b>
<font color="Maroon">X</font>
;<br/>

<b>thus </b><a NAME="E11:108"/>
( the <a href="fsm_1.html#U2" target="_self">InitS</a> of <font color="Maroon">Rtfsm1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; the <a href="fsm_1.html#U2" target="_self">InitS</a> of <font color="Maroon">Rtfsm2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 <b>by </b><i/>;<br/>

<a NAME="E12:108"/>
the <a href="fsm_1.html#U2" target="_self">InitS</a> of <font color="Maroon">tfsm</font> <a href="hidden.html#R1">=</a> the <a href="fsm_1.html#U2" target="_self">InitS</a> of <font color="Maroon">Rtfsm1</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E13:108"/><b>then </b><i><font color="Green">E42</font></i>: 
the <a href="fsm_1.html#U2" target="_self">InitS</a> of <font color="Maroon">Rtfsm1</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">tfsm</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E14:108"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">X</font> is   <a href="subset_1.html#NM2">Subset</a> of the <a href="struct_0.html#U1">carrier</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="E15:108"/>
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">tfsm</font></span>)</span> is   <a href="subset_1.html#NM2">Subset</a> of the <a href="struct_0.html#U1">carrier</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="E16:108"/><b>then </b>
( <font color="Maroon">X</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">tfsm</font></span>)</span> or <font color="Maroon">X</font> <a href="xboole_0.html#R1">misses</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">tfsm</font></span>)</span> )
 
<b>by </b><i>, , <a class="ref" href="eqrel_1.html#D6">EQREL_1:def 6</a></i>;<br/>
<b>hence </b><a NAME="E17:108"/>
<font color="Maroon">X</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">tfsm</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>


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