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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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">Ctfsm</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a> <a href="fsm_1.html#V7" title="FSM_1:attr.7">connected</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> holds   <a href="fsm_1.html#K11" title="FSM_1:func.11">the_reduction_of</a> <font color="Olive" title="b1">Ctfsm</font> is <a href="fsm_1.html#V7" title="FSM_1:attr.7">connected</a></span><br/><b>let </b><font color="Maroon" title="c3">Ctfsm</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a> <a href="fsm_1.html#V7" title="FSM_1:attr.7">connected</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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="fsm_1.html#K11" title="FSM_1:func.11">the_reduction_of</a> <font color="Maroon" title="c3">Ctfsm</font> is <a href="fsm_1.html#V7" title="FSM_1:attr.7">connected</a></span><br/>





<b>set </b><font color="Maroon" title="c4">c</font> = <font color="Maroon" title="c3">Ctfsm</font>;<br/>
<b>set </b><font color="Maroon" title="c5">rtfsm</font> =  <a href="fsm_1.html#K11" title="FSM_1:func.11">the_reduction_of</a> <font color="Maroon" title="c3">Ctfsm</font>;<br/>
<b>assume </b><a NAME="E1:91"/>
not  <a href="fsm_1.html#K11" title="FSM_1:func.11">the_reduction_of</a> <font color="Maroon" title="c3">Ctfsm</font> is <a href="fsm_1.html#V7" title="FSM_1:attr.7">connected</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<b>then consider </b><font color="Maroon" title="c6">Q</font> being   <a href="fsm_1.html#NM1" title="FSM_1:NM.1">State</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">Ctfsm</font></span>)</span><b> such that </b><br/><a NAME="E3:91"/><i><font color="Green" title="E69">A1</font></i>: 
not <font color="Maroon" title="c6">Q</font> is <a href="fsm_1.html#V6" title="FSM_1:attr.6">accessible</a>
 <b>by </b><i><a class="ref" href="fsm_1.html#D22" target="_self" title="FSM_1:def.22">Def22</a></i>;<br/>
<a NAME="E4:91"/><i><font color="Green" title="E70">A2</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">Ctfsm</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">Ctfsm</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="E5:91"/><i><font color="Green" title="E71">A3</font></i>: 
<font color="Maroon" title="c6">Q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 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">Ctfsm</font></span>)</span>
 
;<br/>
<b>reconsider </b><font color="Maroon" title="c7">Q'</font> = <font color="Maroon" title="c6">Q</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">Ctfsm</font> <b>by </b><i><a class="txt" href="fsm_1.html#E4:91"><i><font color="Green" title="E70">A2</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E7:91"/>
<font color="Maroon" title="c7">Q'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fsm_1.html#K8" title="FSM_1:func.8">final_states_partition</a> <font color="Maroon" title="c3">Ctfsm</font>
 
<b>by </b><i><a class="txt" href="fsm_1.html#E5:91"><i><font color="Green" title="E71">A3</font></i></a>, <a class="ref" href="fsm_1.html#D18" target="_self" title="FSM_1:def.18">Def18</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c8">q</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">Ctfsm</font><b> such that </b><br/><a NAME="E9:91"/><i><font color="Green" title="E72">A4</font></i>: 
<font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Q</font>
 <b>by </b><i><a class="ref" href="fsm_1.html#T11" target="_self" title="FSM_1:th.11">Th11</a></i>;<br/>
<a NAME="E10:91"/>
<font color="Maroon" title="c8">q</font> is <a href="fsm_1.html#V6" title="FSM_1:attr.6">accessible</a>
 
<b>by </b><i><a class="ref" href="fsm_1.html#D22" target="_self" title="FSM_1:def.22">Def22</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c9">w</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">IAlph</font><b> such that </b><br/><a NAME="E12:91"/><i><font color="Green" title="E73">A5</font></i>: 
the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c3">Ctfsm</font>,<font color="Maroon" title="c9">w</font> <a href="fsm_1.html#R1" title="FSM_1:pred.1">-leads_to</a> <font color="Maroon" title="c8">q</font>
 <b>by </b><i><a class="ref" href="fsm_1.html#D21" target="_self" title="FSM_1:def.21">Def21</a></i>;<br/>
<a NAME="E13:91"/><i><font color="Green" title="E74">A6</font></i>: 
<span class="p1">(<span class="default">the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c3">Ctfsm</font>,<font color="Maroon" title="c9">w</font> <a href="fsm_1.html#K2" title="FSM_1:func.2">-admissible</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">w</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">q</font>
 
<b>by </b><i><a class="txt" href="fsm_1.html#E12:91"><i><font color="Green" title="E73">A5</font></i></a>, <a class="ref" href="fsm_1.html#D3" target="_self" title="FSM_1:def.3">Def3</a></i>;<br/>
<a NAME="E14:91"/><i><font color="Green" title="E75">A7</font></i>: 
the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c3">Ctfsm</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">Ctfsm</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="E15:91"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">w</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">w</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">w</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 
<b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<a NAME="E16:91"/><b>then </b><i><font color="Green" title="E76">A8</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">w</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">w</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E17:91"/><b>then </b><i><font color="Green" title="E77">A9</font></i>: 
<font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default">the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <span class="p2">(<span class="default"><a href="fsm_1.html#K11" title="FSM_1:func.11">the_reduction_of</a> <font color="Maroon" title="c3">Ctfsm</font></span>)</span>,<font color="Maroon" title="c9">w</font> <a href="fsm_1.html#K2" title="FSM_1:func.2">-admissible</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">w</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="fsm_1.html#E13:91"><i><font color="Green" title="E74">A6</font></i></a>, <a class="txt" href="fsm_1.html#E14:91"><i><font color="Green" title="E75">A7</font></i></a>, <a class="ref" href="fsm_1.html#T57" target="_self" title="FSM_1:th.57">Th57</a></i>;<br/>
<a NAME="E18:91"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">w</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default">the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <span class="p3">(<span class="default"><a href="fsm_1.html#K11" title="FSM_1:func.11">the_reduction_of</a> <font color="Maroon" title="c3">Ctfsm</font></span>)</span>,<font color="Maroon" title="c9">w</font> <a href="fsm_1.html#K2" title="FSM_1:func.2">-admissible</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="fsm_1.html#E16:91"><i><font color="Green" title="E76">A8</font></i></a>, <a class="ref" href="fsm_1.html#D2" target="_self" title="FSM_1:def.2">Def2</a></i>;<br/>
<a NAME="E19:91"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">w</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default">the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <span class="p2">(<span class="default"><a href="fsm_1.html#K11" title="FSM_1:func.11">the_reduction_of</a> <font color="Maroon" title="c3">Ctfsm</font></span>)</span>,<font color="Maroon" title="c9">w</font> <a href="fsm_1.html#K2" title="FSM_1:func.2">-admissible</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E20:91"/><b>then </b><i><font color="Green" title="E78">A10</font></i>: 
<span class="p1">(<span class="default">the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <span class="p2">(<span class="default"><a href="fsm_1.html#K11" title="FSM_1:func.11">the_reduction_of</a> <font color="Maroon" title="c3">Ctfsm</font></span>)</span>,<font color="Maroon" title="c9">w</font> <a href="fsm_1.html#K2" title="FSM_1:func.2">-admissible</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">w</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 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">Ctfsm</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T13" title="FINSEQ_2:th.13">FINSEQ_2:13</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c10">QQ</font> = <span class="p1">(<span class="default">the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <span class="p2">(<span class="default"><a href="fsm_1.html#K11" title="FSM_1:func.11">the_reduction_of</a> <font color="Maroon" title="c3">Ctfsm</font></span>)</span>,<font color="Maroon" title="c9">w</font> <a href="fsm_1.html#K2" title="FSM_1:func.2">-admissible</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">w</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">Ctfsm</font> <b>by </b><i><a class="txt" href="fsm_1.html#E4:91"><i><font color="Green" title="E70">A2</font></i></a></i>;<br/>
<a NAME="E22:91"/>
( <font color="Maroon" title="c7">Q'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">QQ</font> or <font color="Maroon" title="c7">Q'</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c10">QQ</font> )
 
<b>by </b><i><a class="txt" href="fsm_1.html#E4:91"><i><font color="Green" title="E70">A2</font></i></a>, <a class="txt" href="fsm_1.html#E20:91"><i><font color="Green" title="E78">A10</font></i></a>, <a class="ref" href="eqrel_1.html#D6" title="EQREL_1:def.6">EQREL_1:def 6</a></i>;<br/>
<a NAME="E23:91"/><b>then </b>
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">Ctfsm</font></span>)</span>,<font color="Maroon" title="c9">w</font> <a href="fsm_1.html#R1" title="FSM_1:pred.1">-leads_to</a> <font color="Maroon" title="c6">Q</font>
 
<b>by </b><i><a class="txt" href="fsm_1.html#E9:91"><i><font color="Green" title="E72">A4</font></i></a>, <a class="txt" href="fsm_1.html#E17:91"><i><font color="Green" title="E77">A9</font></i></a>, <a class="ref" href="fsm_1.html#D3" target="_self" title="FSM_1:def.3">Def3</a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>
<b>hence </b><a NAME="E24:91"/>
contradiction
 <b>by </b><i><a class="txt" href="fsm_1.html#E3:91"><i><font color="Green" title="E69">A1</font></i></a>, <a class="ref" href="fsm_1.html#D21" target="_self" title="FSM_1:def.21">Def21</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
