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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">Ctfsm</font> be  non <a href="struct_0.html#V3">empty</a> <a href="group_1.html#V6">finite</a> <a href="fsm_1.html#V7" target="_self">connected</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">c</font> = <font color="Maroon">Ctfsm</font>;<br/>
<b>set </b><font color="Maroon">rtfsm</font> =  <a href="fsm_1.html#K11" target="_self">the_reduction_of</a> <font color="Maroon">Ctfsm</font>;<br/>
<b>assume </b><a NAME="E1:94"/>
not  <a href="fsm_1.html#K11" target="_self">the_reduction_of</a> <font color="Maroon">Ctfsm</font> is <a href="fsm_1.html#V7" target="_self">connected</a>
 ;<br/>

<b>then consider </b><font color="Maroon">Q</font> being   <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">Ctfsm</font></span>)</span><b> such that </b><br/><a NAME="E3:94"/><i><font color="Green">E22</font></i>: 
not <font color="Maroon">Q</font> is <a href="fsm_1.html#V6" target="_self">accessible</a>
 <b>by </b><i/>;<br/>
<a NAME="E4:94"/><i><font color="Green">E23</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">Ctfsm</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="fsm_1.html#K8" target="_self">final_states_partition</a> <font color="Maroon">Ctfsm</font>
 
<b>by </b><i/>;<br/>
<a NAME="E5:94"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">Q</font> <a href="hidden.html#R2">in</a> 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">Ctfsm</font></span>)</span>
 
;<br/>
<b>reconsider </b><font color="Maroon">Q'</font> = <font color="Maroon">Q</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Ctfsm</font> <b>by </b><i>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E7:94"/>
<font color="Maroon">Q'</font> <a href="hidden.html#R2">in</a>  <a href="fsm_1.html#K8" target="_self">final_states_partition</a> <font color="Maroon">Ctfsm</font>
 
<b>by </b><i>, </i>;<br/>
<b>then consider </b><font color="Maroon">q</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Ctfsm</font><b> such that </b><br/><a NAME="E9:94"/><i><font color="Green">E39</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="E10:94"/>
<font color="Maroon">q</font> is <a href="fsm_1.html#V6" target="_self">accessible</a>
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">w</font> being    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">IAlph</font><b> such that </b><br/><a NAME="E12:94"/><i><font color="Green">E40</font></i>: 
the <a href="fsm_1.html#U2" target="_self">InitS</a> of <font color="Maroon">Ctfsm</font>,<font color="Maroon">w</font> <a href="fsm_1.html#R1" target="_self">-leads_to</a> <font color="Maroon">q</font>
 <b>by </b><i/>;<br/>
<a NAME="E13:94"/><i><font color="Green">E42</font></i>: 
<span class="p1">(<span class="default">the <a href="fsm_1.html#U2" target="_self">InitS</a> of <font color="Maroon">Ctfsm</font>,<font color="Maroon">w</font> <a href="fsm_1.html#K2" target="_self">-admissible</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">w</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E14:94"/><i><font color="Green">E43</font></i>: 
the <a href="fsm_1.html#U2" target="_self">InitS</a> of <font color="Maroon">Ctfsm</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">Ctfsm</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E15:94"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">w</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">w</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">w</font></span>)</span> <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="E16:94"/><b>then </b><i><font color="Green">E44</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">w</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">w</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E17:94"/><b>then </b><i><font color="Green">E45</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default">the <a href="fsm_1.html#U2" target="_self">InitS</a> of <span class="p2">(<span class="default"><a href="fsm_1.html#K11" target="_self">the_reduction_of</a> <font color="Maroon">Ctfsm</font></span>)</span>,<font color="Maroon">w</font> <a href="fsm_1.html#K2" target="_self">-admissible</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">w</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E18:94"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">w</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default">the <a href="fsm_1.html#U2" target="_self">InitS</a> of <span class="p3">(<span class="default"><a href="fsm_1.html#K11" target="_self">the_reduction_of</a> <font color="Maroon">Ctfsm</font></span>)</span>,<font color="Maroon">w</font> <a href="fsm_1.html#K2" target="_self">-admissible</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="fsm_1.html#T15" target="_self">Th15</a>, </i>;<br/>
<a NAME="E19:94"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">w</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default">the <a href="fsm_1.html#U2" target="_self">InitS</a> of <span class="p2">(<span class="default"><a href="fsm_1.html#K11" target="_self">the_reduction_of</a> <font color="Maroon">Ctfsm</font></span>)</span>,<font color="Maroon">w</font> <a href="fsm_1.html#K2" target="_self">-admissible</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E20:94"/><b>then </b><i><font color="Green">E46</font></i>: 
<span class="p1">(<span class="default">the <a href="fsm_1.html#U2" target="_self">InitS</a> of <span class="p2">(<span class="default"><a href="fsm_1.html#K11" target="_self">the_reduction_of</a> <font color="Maroon">Ctfsm</font></span>)</span>,<font color="Maroon">w</font> <a href="fsm_1.html#K2" target="_self">-admissible</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">w</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> 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">Ctfsm</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T13">FINSEQ_2:13</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">QQ</font> = <span class="p1">(<span class="default">the <a href="fsm_1.html#U2" target="_self">InitS</a> of <span class="p2">(<span class="default"><a href="fsm_1.html#K11" target="_self">the_reduction_of</a> <font color="Maroon">Ctfsm</font></span>)</span>,<font color="Maroon">w</font> <a href="fsm_1.html#K2" target="_self">-admissible</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">w</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Ctfsm</font> <b>by </b><i/>;<br/>
<a NAME="E22:94"/>
( <font color="Maroon">Q'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">QQ</font> or <font color="Maroon">Q'</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">QQ</font> )
 
<b>by </b><i>, , <a class="ref" href="eqrel_1.html#D6">EQREL_1:def 6</a></i>;<br/>
<a NAME="E23:94"/><b>then </b>
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">Ctfsm</font></span>)</span>,<font color="Maroon">w</font> <a href="fsm_1.html#R1" target="_self">-leads_to</a> <font color="Maroon">Q</font>
 
<b>by </b><i>, , , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<b>hence </b><a NAME="E24:94"/>
contradiction
 <b>by </b><i><a class="ref" href="fsm_1.html#T14" target="_self">Th14</a>, </i>;<br/>


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