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<b>let </b><font color="Maroon" title="c1">IAlph</font>, <font color="Maroon" title="c2">OAlph</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c3">tfsm</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="group_1.html#V6" title="GROUP_1:attr.6">finite</a>  <a href="fsm_1.html#L2" title="FSM_1:struct.2">Mealy-FSM</a> of <font color="Maroon" title="c1">IAlph</font>,<font color="Maroon" title="c2">OAlph</font>;<br/><b>let </b><font color="Maroon" title="c4">Rtfsm1</font>, <font color="Maroon" title="c5">Rtfsm2</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="fsm_1.html#V5" title="FSM_1:attr.5">reduced</a>  <a href="fsm_1.html#L2" title="FSM_1:struct.2">Mealy-FSM</a> of <font color="Maroon" title="c1">IAlph</font>,<font color="Maroon" title="c2">OAlph</font>;<br/>









<b>set </b><font color="Maroon" title="c6">rtfsm1</font> = <font color="Maroon" title="c4">Rtfsm1</font>;<br/>
<b>set </b><font color="Maroon" title="c7">rtfsm2</font> = <font color="Maroon" title="c5">Rtfsm2</font>;<br/>
<b>assume </b><a NAME="E1:108"/><i><font color="Green" title="E78">A1</font></i>: 
( <font color="Maroon" title="c3">tfsm</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">Rtfsm1</font> <a href="fsm_1.html#K13" title="FSM_1:func.13">-Mealy_union</a> <font color="Maroon" title="c5">Rtfsm2</font> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">Rtfsm1</font> <a href="subset_1.html#R1" title="SUBSET_1:pred.1">misses</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">Rtfsm2</font> &amp; <font color="Maroon" title="c4">Rtfsm1</font>,<font color="Maroon" title="c5">Rtfsm2</font> <a href="fsm_1.html#R4" title="FSM_1:pred.4">-are_equivalent</a>  )
 ;<br/>

<b>set </b><font color="Maroon" title="c8">RUN</font> =  <a href="fsm_1.html#K11" title="FSM_1:func.11">the_reduction_of</a> <font color="Maroon" title="c3">tfsm</font>;<br/>
<b>set </b><font color="Maroon" title="c9">SRUN</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="fsm_1.html#K11" title="FSM_1:func.11">the_reduction_of</a> <font color="Maroon" title="c3">tfsm</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c10">Stfsm</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">tfsm</font>;<br/>
<b>set </b><font color="Maroon" title="c11">Srtfsm1</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">Rtfsm1</font>;<br/>
<b>set </b><font color="Maroon" title="c12">Srtfsm2</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">Rtfsm2</font>;<br/>
<a NAME="E2:108"/><i><font color="Green" title="E79">A2</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">tfsm</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">Rtfsm1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">Rtfsm2</font>
 
<b>by </b><i><a class="txt" href="fsm_1.html#E1:108"><i><font color="Green" title="E78">A1</font></i></a>, <a class="ref" href="fsm_1.html#D24" target="_self" title="FSM_1:def.24">Def24</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">IS1</font> = the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c4">Rtfsm1</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">tfsm</font> <b>by </b><i><a class="txt" href="fsm_1.html#E1:108"><i><font color="Green" title="E78">A1</font></i></a>, <a class="ref" href="fsm_1.html#D24" target="_self" title="FSM_1:def.24">Def24</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">IS2</font> = the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c5">Rtfsm2</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">tfsm</font> <b>by </b><i><a class="txt" href="fsm_1.html#E2:108"><i><font color="Green" title="E79">A2</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E5:108"/><i><font color="Green" title="E80">A3</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="fsm_1.html#K11" title="FSM_1:func.11">the_reduction_of</a> <font color="Maroon" title="c3">tfsm</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fsm_1.html#K8" title="FSM_1:func.8">final_states_partition</a> <font color="Maroon" title="c3">tfsm</font>
 
<b>by </b><i><a class="ref" href="fsm_1.html#D18" target="_self" title="FSM_1:def.18">Def18</a></i>;<br/>
<a NAME="E6:108"/><i><font color="Green" title="E81">A4</font></i>: 
 <a href="fsm_1.html#K8" title="FSM_1:func.8">final_states_partition</a> <font color="Maroon" title="c3">tfsm</font> is <a href="fsm_1.html#V4" title="FSM_1:attr.4">final</a>
 
<b>by </b><i><a class="ref" href="fsm_1.html#D15" target="_self" title="FSM_1:def.15">Def15</a></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" title="c15">w</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">IAlph</font>;<br/><a NAME="E1:108_1"/>
the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c4">Rtfsm1</font>,<font color="Maroon" title="c15">w</font> <a href="fsm_1.html#K4" title="FSM_1:func.4">-response</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c5">Rtfsm2</font>,<font color="Maroon" title="c15">w</font> <a href="fsm_1.html#K4" title="FSM_1:func.4">-response</a> 
 <b>by </b><i><a class="txt" href="fsm_1.html#E1:108"><i><font color="Green" title="E78">A1</font></i></a>, <a class="ref" href="fsm_1.html#D9" target="_self" title="FSM_1:def.9">Def9</a></i>;<br/><b>hence </b><font color="Maroon" title="c13">IS1</font>,<font color="Maroon" title="c15">w</font> <a href="fsm_1.html#K4" title="FSM_1:func.4">-response</a>  = 
the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c5">Rtfsm2</font>,<font color="Maroon" title="c15">w</font> <a href="fsm_1.html#K4" title="FSM_1:func.4">-response</a> 
<b>by </b><i><a class="txt" href="fsm_1.html#E1:108"><i><font color="Green" title="E78">A1</font></i></a>, <a class="ref" href="fsm_1.html#T72" target="_self" title="FSM_1:th.72">Th72</a></i>
<br/>.= 
<font color="Maroon" title="c14">IS2</font>,<font color="Maroon" title="c15">w</font> <a href="fsm_1.html#K4" title="FSM_1:func.4">-response</a> 
<b>by </b><i><a class="txt" href="fsm_1.html#E1:108"><i><font color="Green" title="E78">A1</font></i></a>, <a class="ref" href="fsm_1.html#T74" target="_self" title="FSM_1:th.74">Th74</a></i>
;<br/><br/></div><b>end;</b></div>
<a NAME="E8:108"/><b>then </b>
<font color="Maroon" title="c13">IS1</font>,<font color="Maroon" title="c14">IS2</font> <a href="fsm_1.html#R5" title="FSM_1:pred.5">-are_equivalent</a> 
 
<b>by </b><i><a class="ref" href="fsm_1.html#D10" target="_self" title="FSM_1:def.10">Def10</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c15">X</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="fsm_1.html#K11" title="FSM_1:func.11">the_reduction_of</a> <font color="Maroon" title="c3">tfsm</font></span>)</span><b> such that </b><br/><a NAME="E10:108"/><i><font color="Green" title="E82">A5</font></i>: 
( <font color="Maroon" title="c13">IS1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">X</font> &amp; <font color="Maroon" title="c14">IS2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">X</font> )
 <b>by </b><i><a class="txt" href="fsm_1.html#E5:108"><i><font color="Green" title="E80">A3</font></i></a>, <a class="txt" href="fsm_1.html#E6:108"><i><font color="Green" title="E81">A4</font></i></a>, <a class="ref" href="fsm_1.html#D14" target="_self" title="FSM_1:def.14">Def14</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c15">X</font>
;<br/>

<b>thus </b><a NAME="E11:108"/>
( the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c4">Rtfsm1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">X</font> &amp; the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c5">Rtfsm2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">X</font> )
 <b>by </b><i><a class="txt" href="fsm_1.html#E10:108"><i><font color="Green" title="E82">A5</font></i></a></i>;<br/>

<a NAME="E12:108"/>
the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c3">tfsm</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c4">Rtfsm1</font>
 
<b>by </b><i><a class="txt" href="fsm_1.html#E1:108"><i><font color="Green" title="E78">A1</font></i></a>, <a class="ref" href="fsm_1.html#D24" target="_self" title="FSM_1:def.24">Def24</a></i>;<br/>
<a NAME="E13:108"/><b>then </b><i><font color="Green" title="E83">A6</font></i>: 
the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c4">Rtfsm1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <span class="p1">(<span class="default"><a href="fsm_1.html#K11" title="FSM_1:func.11">the_reduction_of</a> <font color="Maroon" title="c3">tfsm</font></span>)</span>
 
<b>by </b><i><a class="ref" href="fsm_1.html#D18" target="_self" title="FSM_1:def.18">Def18</a></i>;<br/>
<a NAME="E14:108"/><i><font color="Green" title="E84">A7</font></i>: 
<font color="Maroon" title="c15">X</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">tfsm</font>
 
<b>by </b><i><a class="txt" href="fsm_1.html#E5:108"><i><font color="Green" title="E80">A3</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E15:108"/>
the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <span class="p1">(<span class="default"><a href="fsm_1.html#K11" title="FSM_1:func.11">the_reduction_of</a> <font color="Maroon" title="c3">tfsm</font></span>)</span> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">tfsm</font>
 
<b>by </b><i><a class="txt" href="fsm_1.html#E5:108"><i><font color="Green" title="E80">A3</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E16:108"/><b>then </b>
( <font color="Maroon" title="c15">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <span class="p1">(<span class="default"><a href="fsm_1.html#K11" title="FSM_1:func.11">the_reduction_of</a> <font color="Maroon" title="c3">tfsm</font></span>)</span> or <font color="Maroon" title="c15">X</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <span class="p1">(<span class="default"><a href="fsm_1.html#K11" title="FSM_1:func.11">the_reduction_of</a> <font color="Maroon" title="c3">tfsm</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="fsm_1.html#E5:108"><i><font color="Green" title="E80">A3</font></i></a>, <a class="txt" href="fsm_1.html#E14:108"><i><font color="Green" title="E84">A7</font></i></a>, <a class="ref" href="eqrel_1.html#D6" title="EQREL_1:def.6">EQREL_1:def 6</a></i>;<br/>
<b>hence </b><a NAME="E17:108"/>
<font color="Maroon" title="c15">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <span class="p1">(<span class="default"><a href="fsm_1.html#K11" title="FSM_1:func.11">the_reduction_of</a> <font color="Maroon" title="c3">tfsm</font></span>)</span>
 <b>by </b><i><a class="txt" href="fsm_1.html#E10:108"><i><font color="Green" title="E82">A5</font></i></a>, <a class="txt" href="fsm_1.html#E13:108"><i><font color="Green" title="E83">A6</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>


</div>
