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<span class="kw">let </span><font color="Maroon" title="c1">OAlph</font>, <font color="Maroon" title="c2">IAlph</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">tfsm</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#L2" title="FSM_1:struct.2">Mealy-FSM</a> of <font color="Maroon" title="c2">IAlph</font>,<font color="Maroon" title="c1">OAlph</font>  ex <font color="Olive" title="b2">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="c2">IAlph</font>,<font color="Maroon" title="c1">OAlph</font> st <br/>( the <a href="fsm_1.html#U1" title="FSM_1:sel.1">Tran</a> of <font color="Olive" title="b2">Ctfsm</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="fsm_1.html#U1" title="FSM_1:sel.1">Tran</a> of <font color="Olive" title="b1">tfsm</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="fsm_1.html#K12" title="FSM_1:func.12">accessibleStates</a> <font color="Olive" title="b1">tfsm</font></span>)</span>,<font color="Maroon" title="c2">IAlph</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; the <a href="fsm_1.html#U3" title="FSM_1:sel.3">OFun</a> of <font color="Olive" title="b2">Ctfsm</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="fsm_1.html#U3" title="FSM_1:sel.3">OFun</a> of <font color="Olive" title="b1">tfsm</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="fsm_1.html#K12" title="FSM_1:func.12">accessibleStates</a> <font color="Olive" title="b1">tfsm</font></span>)</span>,<font color="Maroon" title="c2">IAlph</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Olive" title="b2">Ctfsm</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="Olive" title="b1">tfsm</font> &amp; <font color="Olive" title="b1">tfsm</font>,<font color="Olive" title="b2">Ctfsm</font> <a href="fsm_1.html#R4" title="FSM_1:pred.4">-are_equivalent</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">tfsm</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#L2" title="FSM_1:struct.2">Mealy-FSM</a> of <font color="Maroon" title="c2">IAlph</font>,<font color="Maroon" title="c1">OAlph</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <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="c2">IAlph</font>,<font color="Maroon" title="c1">OAlph</font> st <br/>( the <a href="fsm_1.html#U1" title="FSM_1:sel.1">Tran</a> of <font color="Olive" title="b1">Ctfsm</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="fsm_1.html#U1" title="FSM_1:sel.1">Tran</a> of <font color="Maroon" title="c3">tfsm</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="fsm_1.html#K12" title="FSM_1:func.12">accessibleStates</a> <font color="Maroon" title="c3">tfsm</font></span>)</span>,<font color="Maroon" title="c2">IAlph</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; the <a href="fsm_1.html#U3" title="FSM_1:sel.3">OFun</a> of <font color="Olive" title="b1">Ctfsm</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="fsm_1.html#U3" title="FSM_1:sel.3">OFun</a> of <font color="Maroon" title="c3">tfsm</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="fsm_1.html#K12" title="FSM_1:func.12">accessibleStates</a> <font color="Maroon" title="c3">tfsm</font></span>)</span>,<font color="Maroon" title="c2">IAlph</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Olive" title="b1">Ctfsm</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="c3">tfsm</font> &amp; <font color="Maroon" title="c3">tfsm</font>,<font color="Olive" title="b1">Ctfsm</font> <a href="fsm_1.html#R4" title="FSM_1:pred.4">-are_equivalent</a>  )</span><br/>





<span class="kw">set </span><font color="Maroon" title="c4">M</font> = <font color="Maroon" title="c3">tfsm</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">I</font> = the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c3">tfsm</font>;<br/>
<a NAME="E1:105"/>
 <a href="fsm_1.html#K12" title="FSM_1:func.12">accessibleStates</a> <font color="Maroon" title="c3">tfsm</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">tfsm</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fsm_1.html#T67" target="_self" title="FSM_1:th.67">Th67</a></span>;<br/>
<a NAME="E2:105"/><span class="kw">then </span>
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="fsm_1.html#K12" title="FSM_1:func.12">accessibleStates</a> <font color="Maroon" title="c3">tfsm</font></span>)</span>,<font color="Maroon" title="c2">IAlph</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">tfsm</font>,<font color="Maroon" title="c2">IAlph</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T119" title="ZFMISC_1:th.119">ZFMISC_1:119</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">cOFun</font> = the <a href="fsm_1.html#U3" title="FSM_1:sel.3">OFun</a> of <font color="Maroon" title="c3">tfsm</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="fsm_1.html#K12" title="FSM_1:func.12">accessibleStates</a> <font color="Maroon" title="c3">tfsm</font></span>)</span>,<font color="Maroon" title="c2">IAlph</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="fsm_1.html#K12" title="FSM_1:func.12">accessibleStates</a> <font color="Maroon" title="c3">tfsm</font></span>)</span>,<font color="Maroon" title="c2">IAlph</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c1">OAlph</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T38" title="FUNCT_2:th.38">FUNCT_2:38</a></span>;<br/>
<a NAME="E4:105"/>
the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c3">tfsm</font>, <a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> <font color="Maroon" title="c2">IAlph</font> <a href="fsm_1.html#R1" title="FSM_1:pred.1">-leads_to</a> the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c3">tfsm</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fsm_1.html#T17" target="_self" title="FSM_1:th.17">Th17</a></span>;<br/>
<a NAME="E5:105"/><span class="kw">then </span>
the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c3">tfsm</font> is  <a href="fsm_1.html#V6" title="FSM_1:attr.6">accessible</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fsm_1.html#D21" target="_self" title="FSM_1:def.21">Def21</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">cInitS</font> = the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c3">tfsm</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="fsm_1.html#K12" title="FSM_1:func.12">accessibleStates</a> <font color="Maroon" title="c3">tfsm</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="fsm_1.html#T67" target="_self" title="FSM_1:th.67">Th67</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">cTran</font> = the <a href="fsm_1.html#U1" title="FSM_1:sel.1">Tran</a> of <font color="Maroon" title="c3">tfsm</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="fsm_1.html#K12" title="FSM_1:func.12">accessibleStates</a> <font color="Maroon" title="c3">tfsm</font></span>)</span>,<font color="Maroon" title="c2">IAlph</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="fsm_1.html#K12" title="FSM_1:func.12">accessibleStates</a> <font color="Maroon" title="c3">tfsm</font></span>)</span>,<font color="Maroon" title="c2">IAlph</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<span class="p1">(<span class="default"><a href="fsm_1.html#K12" title="FSM_1:func.12">accessibleStates</a> <font color="Maroon" title="c3">tfsm</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="fsm_1.html#T68" target="_self" title="FSM_1:th.68">Th68</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">cm</font> =  <a href="fsm_1.html#G2" title="FSM_1:aggr.2">Mealy-FSM</a>(# <span class="p1">(<span class="default"><a href="fsm_1.html#K12" title="FSM_1:func.12">accessibleStates</a> <font color="Maroon" title="c3">tfsm</font></span>)</span>,<font color="Maroon" title="c8">cTran</font>,<font color="Maroon" title="c6">cOFun</font>,<font color="Maroon" title="c7">cInitS</font> #);<br/>
<div><span class="lab"><font color="Green" title="E70">A1</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c10">w</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c2">IAlph</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c3">tfsm</font>,<font color="Maroon" title="c10">w</font> <a href="fsm_1.html#K2" title="FSM_1:func.2">-admissible</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 <a href="fsm_1.html#G2" title="FSM_1:aggr.2">Mealy-FSM</a>(# <span class="p1">(<span class="default"><a href="fsm_1.html#K12" title="FSM_1:func.12">accessibleStates</a> <font color="Maroon" title="c3">tfsm</font></span>)</span>,<font color="Maroon" title="c8">cTran</font>,<font color="Maroon" title="c6">cOFun</font>,<font color="Maroon" title="c7">cInitS</font> #),<font color="Maroon" title="c10">w</font> <a href="fsm_1.html#K2" title="FSM_1:func.2">-admissible</a> </span><br/><span class="kw">set </span><font color="Maroon" title="c11">ma</font> = the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c3">tfsm</font>,<font color="Maroon" title="c10">w</font> <a href="fsm_1.html#K2" title="FSM_1:func.2">-admissible</a> ;<br/><span class="kw">set </span><font color="Maroon" title="c12">cma</font> = the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <a href="fsm_1.html#G2" title="FSM_1:aggr.2">Mealy-FSM</a>(# <span class="p1">(<span class="default"><a href="fsm_1.html#K12" title="FSM_1:func.12">accessibleStates</a> <font color="Maroon" title="c3">tfsm</font></span>)</span>,<font color="Maroon" title="c8">cTran</font>,<font color="Maroon" title="c6">cOFun</font>,<font color="Maroon" title="c7">cInitS</font> #),<font color="Maroon" title="c10">w</font> <a href="fsm_1.html#K2" title="FSM_1:func.2">-admissible</a> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span>( $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 <font color="Maroon" title="c3">tfsm</font>,<font color="Maroon" title="c10">w</font> <a href="fsm_1.html#K2" title="FSM_1:func.2">-admissible</a> </span>)</span> implies <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">tfsm</font>,<font color="Maroon" title="c10">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> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <a href="fsm_1.html#G2" title="FSM_1:aggr.2">Mealy-FSM</a>(# <span class="p1">(<span class="default"><a href="fsm_1.html#K12" title="FSM_1:func.12">accessibleStates</a> <font color="Maroon" title="c3">tfsm</font></span>)</span>,<font color="Maroon" title="c8">cTran</font>,<font color="Maroon" title="c6">cOFun</font>,<font color="Maroon" title="c7">cInitS</font> #),<font color="Maroon" title="c10">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> $1 );<br/><span class="kw">thus </span><a NAME="E1:105_1_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <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">tfsm</font>,<font color="Maroon" title="c10">w</font> <a href="fsm_1.html#K2" title="FSM_1:func.2">-admissible</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">w</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default">the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <a href="fsm_1.html#G2" title="FSM_1:aggr.2">Mealy-FSM</a>(# <span class="p1">(<span class="default"><a href="fsm_1.html#K12" title="FSM_1:func.12">accessibleStates</a> <font color="Maroon" title="c3">tfsm</font></span>)</span>,<font color="Maroon" title="c8">cTran</font>,<font color="Maroon" title="c6">cOFun</font>,<font color="Maroon" title="c7">cInitS</font> #),<font color="Maroon" title="c10">w</font> <a href="fsm_1.html#K2" title="FSM_1:func.2">-admissible</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">w</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fsm_1.html#D2" target="_self" title="FSM_1:def.2">Def2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">j</font>]</span><br/><a NAME="E2:105_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A2</font></span>: 
 <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 <font color="Maroon" title="c3">tfsm</font>,<font color="Maroon" title="c10">w</font> <a href="fsm_1.html#K2" title="FSM_1:func.2">-admissible</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c10">w</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E3:105_1_1"/><span class="lab"><font color="Green" title="E72">A3</font></span>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c13">j</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c13">j</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c13">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:105_1_1_1"/><span class="lab"><font color="Green" title="E73">A4</font></span>: 
( <font color="Maroon" title="c13">j</font> <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 <font color="Maroon" title="c3">tfsm</font>,<font color="Maroon" title="c10">w</font> <a href="fsm_1.html#K2" title="FSM_1:func.2">-admissible</a> </span>)</span> implies <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">tfsm</font>,<font color="Maroon" title="c10">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> <font color="Maroon" title="c13">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <a href="fsm_1.html#G2" title="FSM_1:aggr.2">Mealy-FSM</a>(# <span class="p1">(<span class="default"><a href="fsm_1.html#K12" title="FSM_1:func.12">accessibleStates</a> <font color="Maroon" title="c3">tfsm</font></span>)</span>,<font color="Maroon" title="c8">cTran</font>,<font color="Maroon" title="c6">cOFun</font>,<font color="Maroon" title="c7">cInitS</font> #),<font color="Maroon" title="c10">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> <font color="Maroon" title="c13">j</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c13">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<a NAME="E2:105_1_1_1"/><span class="lab"><font color="Green" title="E74">A5</font></span>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">j</font> or (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">j</font> &amp; <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) )
 
;<br/>
<span class="kw">assume </span><a NAME="E3:105_1_1_1"/>
<font color="Maroon" title="c13">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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 <font color="Maroon" title="c3">tfsm</font>,<font color="Maroon" title="c10">w</font> <a href="fsm_1.html#K2" title="FSM_1:func.2">-admissible</a> </span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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">tfsm</font>,<font color="Maroon" title="c10">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"><font color="Maroon" title="c13">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <a href="fsm_1.html#G2" title="FSM_1:aggr.2">Mealy-FSM</a>(# <span class="p1">(<span class="default"><a href="fsm_1.html#K12" title="FSM_1:func.12">accessibleStates</a> <font color="Maroon" title="c3">tfsm</font></span>)</span>,<font color="Maroon" title="c8">cTran</font>,<font color="Maroon" title="c6">cOFun</font>,<font color="Maroon" title="c7">cInitS</font> #),<font color="Maroon" title="c10">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"><font color="Maroon" title="c13">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><br/>

<a NAME="E4:105_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A6</font></span>: 
<font color="Maroon" title="c13">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c10">w</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:105_1_1"><span class="lab"><font color="Green" title="E71">A2</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E5:105_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A7</font></span>: 
<font color="Maroon" title="c13">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">w</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/>
<a NAME="E6:105_1_1_1"/><span class="lab"><font color="Green" title="E77">A8</font></span>: 
<font color="Maroon" title="c13">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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="c10">w</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:105_1_1_1"><span class="lab"><font color="Green" title="E75">A6</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_1_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:105_1_1_1_1"/>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">j</font> or 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">j</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:105_1_1_1"><span class="lab"><font color="Green" title="E74">A5</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:105_1_1_1_1_1"/><span class="lab"><font color="Green" title="E78">A9</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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">tfsm</font>,<font color="Maroon" title="c10">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"><font color="Maroon" title="c13">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <a href="fsm_1.html#G2" title="FSM_1:aggr.2">Mealy-FSM</a>(# <span class="p1">(<span class="default"><a href="fsm_1.html#K12" title="FSM_1:func.12">accessibleStates</a> <font color="Maroon" title="c3">tfsm</font></span>)</span>,<font color="Maroon" title="c8">cTran</font>,<font color="Maroon" title="c6">cOFun</font>,<font color="Maroon" title="c7">cInitS</font> #),<font color="Maroon" title="c10">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"><font color="Maroon" title="c13">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><br/><div class="add"><span class="kw">hence </span><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">tfsm</font>,<font color="Maroon" title="c10">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"><font color="Maroon" title="c13">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c3">tfsm</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="fsm_1.html#D2" target="_self" title="FSM_1:def.2">Def2</a></span>
<br/>.= 
<span class="p1">(<span class="default">the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <a href="fsm_1.html#G2" title="FSM_1:aggr.2">Mealy-FSM</a>(# <span class="p1">(<span class="default"><a href="fsm_1.html#K12" title="FSM_1:func.12">accessibleStates</a> <font color="Maroon" title="c3">tfsm</font></span>)</span>,<font color="Maroon" title="c8">cTran</font>,<font color="Maroon" title="c6">cOFun</font>,<font color="Maroon" title="c7">cInitS</font> #),<font color="Maroon" title="c10">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"><font color="Maroon" title="c13">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:105_1_1_1_1_1"><span class="lab"><font color="Green" title="E78">A9</font></span></a>, <a class="ref" href="fsm_1.html#D2" target="_self" title="FSM_1:def.2">Def2</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:105_1_1_1_1_2"/><span class="lab"><font color="Green" title="E78">A10</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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">tfsm</font>,<font color="Maroon" title="c10">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"><font color="Maroon" title="c13">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <a href="fsm_1.html#G2" title="FSM_1:aggr.2">Mealy-FSM</a>(# <span class="p1">(<span class="default"><a href="fsm_1.html#K12" title="FSM_1:func.12">accessibleStates</a> <font color="Maroon" title="c3">tfsm</font></span>)</span>,<font color="Maroon" title="c8">cTran</font>,<font color="Maroon" title="c6">cOFun</font>,<font color="Maroon" title="c7">cInitS</font> #),<font color="Maroon" title="c10">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"><font color="Maroon" title="c13">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><br/><div class="add"><a NAME="E2:105_1_1_1_1_2"/><span class="kw">then </span>
(  ex <font color="Olive" title="b1">mwj</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">IAlph</font> ex <font color="Olive" title="b2">mqj</font>, <font color="Olive" title="b3">mqj1</font> being  <a href="fsm_1.html#NM1" title="FSM_1:NM.1">State</a> of <font color="Maroon" title="c3">tfsm</font> st <br/>( <font color="Olive" title="b1">mwj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">w</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</font> &amp; <font color="Olive" title="b2">mqj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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">tfsm</font>,<font color="Maroon" title="c10">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> <font color="Maroon" title="c13">j</font> &amp; <font color="Olive" title="b3">mqj1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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">tfsm</font>,<font color="Maroon" title="c10">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"><font color="Maroon" title="c13">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Olive" title="b1">mwj</font> <a href="fsm_1.html#K1" title="FSM_1:func.1">-succ_of</a> <font color="Olive" title="b2">mqj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">mqj1</font> ) &amp;  ex <font color="Olive" title="b1">cmwj</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">IAlph</font> ex <font color="Olive" title="b2">cmqj</font>, <font color="Olive" title="b3">cmqj1</font> being  <a href="fsm_1.html#NM1" title="FSM_1:NM.1">State</a> of <a href="fsm_1.html#G2" title="FSM_1:aggr.2">Mealy-FSM</a>(# <span class="p1">(<span class="default"><a href="fsm_1.html#K12" title="FSM_1:func.12">accessibleStates</a> <font color="Maroon" title="c3">tfsm</font></span>)</span>,<font color="Maroon" title="c8">cTran</font>,<font color="Maroon" title="c6">cOFun</font>,<font color="Maroon" title="c7">cInitS</font> #) st <br/>( <font color="Olive" title="b1">cmwj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">w</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</font> &amp; <font color="Olive" title="b2">cmqj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <a href="fsm_1.html#G2" title="FSM_1:aggr.2">Mealy-FSM</a>(# <span class="p1">(<span class="default"><a href="fsm_1.html#K12" title="FSM_1:func.12">accessibleStates</a> <font color="Maroon" title="c3">tfsm</font></span>)</span>,<font color="Maroon" title="c8">cTran</font>,<font color="Maroon" title="c6">cOFun</font>,<font color="Maroon" title="c7">cInitS</font> #),<font color="Maroon" title="c10">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> <font color="Maroon" title="c13">j</font> &amp; <font color="Olive" title="b3">cmqj1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <a href="fsm_1.html#G2" title="FSM_1:aggr.2">Mealy-FSM</a>(# <span class="p1">(<span class="default"><a href="fsm_1.html#K12" title="FSM_1:func.12">accessibleStates</a> <font color="Maroon" title="c3">tfsm</font></span>)</span>,<font color="Maroon" title="c8">cTran</font>,<font color="Maroon" title="c6">cOFun</font>,<font color="Maroon" title="c7">cInitS</font> #),<font color="Maroon" title="c10">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"><font color="Maroon" title="c13">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Olive" title="b1">cmwj</font> <a href="fsm_1.html#K1" title="FSM_1:func.1">-succ_of</a> <font color="Olive" title="b2">cmqj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">cmqj1</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:105_1_1_1"><span class="lab"><font color="Green" title="E76">A7</font></span></a>, <a class="ref" href="fsm_1.html#D2" target="_self" title="FSM_1:def.2">Def2</a></span>;<br/><span class="kw">hence </span><a NAME="E3:105_1_1_1_1_2"/>
<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">tfsm</font>,<font color="Maroon" title="c10">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"><font color="Maroon" title="c13">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <a href="fsm_1.html#G2" title="FSM_1:aggr.2">Mealy-FSM</a>(# <span class="p1">(<span class="default"><a href="fsm_1.html#K12" title="FSM_1:func.12">accessibleStates</a> <font color="Maroon" title="c3">tfsm</font></span>)</span>,<font color="Maroon" title="c8">cTran</font>,<font color="Maroon" title="c6">cOFun</font>,<font color="Maroon" title="c7">cInitS</font> #),<font color="Maroon" title="c10">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"><font color="Maroon" title="c13">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:105_1_1"><span class="lab"><font color="Green" title="E71">A2</font></span></a>, <a class="txt" href="#E1:105_1_1_1"><span class="lab"><font color="Green" title="E73">A4</font></span></a>, <a class="txt" href="#E6:105_1_1_1"><span class="lab"><font color="Green" title="E77">A8</font></span></a>, <a class="txt" href="#E1:105_1_1_1_1_2"><span class="lab"><font color="Green" title="E78">A10</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><a NAME="E4:105_1_1"/><span class="lab"><font color="Green" title="E73">A11</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:105_1_1"><span class="lab"><font color="Green" title="E71">A2</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><span class="kw">thus </span><a NAME="E5:105_1_1"/>
for <font color="Olive" title="b1">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">j</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="#E4:105_1_1"><span class="lab"><font color="Green" title="E73">A11</font></span></a>, <a class="txt" href="#E3:105_1_1"><span class="lab"><font color="Green" title="E72">A3</font></span></a>);</span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E2:105_1"/>
the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c3">tfsm</font>,<font color="Maroon" title="c10">w</font> <a href="fsm_1.html#K2" title="FSM_1:func.2">-admissible</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 <a href="fsm_1.html#G2" title="FSM_1:aggr.2">Mealy-FSM</a>(# <span class="p1">(<span class="default"><a href="fsm_1.html#K12" title="FSM_1:func.12">accessibleStates</a> <font color="Maroon" title="c3">tfsm</font></span>)</span>,<font color="Maroon" title="c8">cTran</font>,<font color="Maroon" title="c6">cOFun</font>,<font color="Maroon" title="c7">cInitS</font> #),<font color="Maroon" title="c10">w</font> <a href="fsm_1.html#K2" title="FSM_1:func.2">-admissible</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T10" title="FINSEQ_2:th.10">FINSEQ_2:10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c10">q</font> be   <a href="fsm_1.html#NM1" title="FSM_1:NM.1">State</a> of <a href="fsm_1.html#G2" title="FSM_1:aggr.2">Mealy-FSM</a>(# <span class="p1">(<span class="default"><a href="fsm_1.html#K12" title="FSM_1:func.12">accessibleStates</a> <font color="Maroon" title="c3">tfsm</font></span>)</span>,<font color="Maroon" title="c8">cTran</font>,<font color="Maroon" title="c6">cOFun</font>,<font color="Maroon" title="c7">cInitS</font> #); <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">q</font> is  <a href="fsm_1.html#V6" title="FSM_1:attr.6">accessible</a> </span><br/><a NAME="E1:105_2"/>
( <font color="Maroon" title="c10">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fsm_1.html#K12" title="FSM_1:func.12">accessibleStates</a> <font color="Maroon" title="c3">tfsm</font> &amp;  <a href="fsm_1.html#K12" title="FSM_1:func.12">accessibleStates</a> <font color="Maroon" title="c3">tfsm</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">tfsm</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fsm_1.html#T67" target="_self" title="FSM_1:th.67">Th67</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">q9</font> = <font color="Maroon" title="c10">q</font> as   <a href="fsm_1.html#NM1" title="FSM_1:NM.1">State</a> of <font color="Maroon" title="c3">tfsm</font> ;<br/><a NAME="E3:105_2"/>
<font color="Maroon" title="c11">q9</font> is  <a href="fsm_1.html#V6" title="FSM_1:attr.6">accessible</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fsm_1.html#T67" target="_self" title="FSM_1:th.67">Th67</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">w</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c2">IAlph</font><span class="kw"> such that </span><br/><a NAME="E5:105_2"/><span class="lab"><font color="Green" title="E71">A12</font></span>: 
the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c3">tfsm</font>,<font color="Maroon" title="c12">w</font> <a href="fsm_1.html#R1" title="FSM_1:pred.1">-leads_to</a> <font color="Maroon" title="c11">q9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fsm_1.html#D21" target="_self" title="FSM_1:def.21">Def21</a></span>;<br/><a NAME="E6:105_2"/>
<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">tfsm</font>,<font color="Maroon" title="c12">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="c12">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="c11">q9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:105_2"><span class="lab"><font color="Green" title="E71">A12</font></span></a>, <a class="ref" href="fsm_1.html#D3" target="_self" title="FSM_1:def.3">Def3</a></span>;<br/><a NAME="E7:105_2"/><span class="kw">then </span>
<span class="p1">(<span class="default">the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <a href="fsm_1.html#G2" title="FSM_1:aggr.2">Mealy-FSM</a>(# <span class="p1">(<span class="default"><a href="fsm_1.html#K12" title="FSM_1:func.12">accessibleStates</a> <font color="Maroon" title="c3">tfsm</font></span>)</span>,<font color="Maroon" title="c8">cTran</font>,<font color="Maroon" title="c6">cOFun</font>,<font color="Maroon" title="c7">cInitS</font> #),<font color="Maroon" title="c12">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="c12">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="c10">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:105"><span class="lab"><font color="Green" title="E70">A1</font></span></a></span>;<br/><a NAME="E8:105_2"/><span class="kw">then </span>
the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <a href="fsm_1.html#G2" title="FSM_1:aggr.2">Mealy-FSM</a>(# <span class="p1">(<span class="default"><a href="fsm_1.html#K12" title="FSM_1:func.12">accessibleStates</a> <font color="Maroon" title="c3">tfsm</font></span>)</span>,<font color="Maroon" title="c8">cTran</font>,<font color="Maroon" title="c6">cOFun</font>,<font color="Maroon" title="c7">cInitS</font> #),<font color="Maroon" title="c12">w</font> <a href="fsm_1.html#R1" title="FSM_1:pred.1">-leads_to</a> <font color="Maroon" title="c10">q</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fsm_1.html#D3" target="_self" title="FSM_1:def.3">Def3</a></span>;<br/><span class="kw">hence </span><a NAME="E9:105_2"/>
<font color="Maroon" title="c10">q</font> is  <a href="fsm_1.html#V6" title="FSM_1:attr.6">accessible</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fsm_1.html#D21" target="_self" title="FSM_1:def.21">Def21</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">cm</font> =  <a href="fsm_1.html#G2" title="FSM_1:aggr.2">Mealy-FSM</a>(# <span class="p1">(<span class="default"><a href="fsm_1.html#K12" title="FSM_1:func.12">accessibleStates</a> <font color="Maroon" title="c3">tfsm</font></span>)</span>,<font color="Maroon" title="c8">cTran</font>,<font color="Maroon" title="c6">cOFun</font>,<font color="Maroon" title="c7">cInitS</font> #) as  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="c2">IAlph</font>,<font color="Maroon" title="c1">OAlph</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="fsm_1.html#D22" target="_self" title="FSM_1:def.22">Def22</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c10">cm</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( the <a href="fsm_1.html#U1" title="FSM_1:sel.1">Tran</a> of <font color="Maroon" title="c10">cm</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="fsm_1.html#U1" title="FSM_1:sel.1">Tran</a> of <font color="Maroon" title="c3">tfsm</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="fsm_1.html#K12" title="FSM_1:func.12">accessibleStates</a> <font color="Maroon" title="c3">tfsm</font></span>)</span>,<font color="Maroon" title="c2">IAlph</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; the <a href="fsm_1.html#U3" title="FSM_1:sel.3">OFun</a> of <font color="Maroon" title="c10">cm</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="fsm_1.html#U3" title="FSM_1:sel.3">OFun</a> of <font color="Maroon" title="c3">tfsm</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="fsm_1.html#K12" title="FSM_1:func.12">accessibleStates</a> <font color="Maroon" title="c3">tfsm</font></span>)</span>,<font color="Maroon" title="c2">IAlph</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c10">cm</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="c3">tfsm</font> &amp; <font color="Maroon" title="c3">tfsm</font>,<font color="Maroon" title="c10">cm</font> <a href="fsm_1.html#R4" title="FSM_1:pred.4">-are_equivalent</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E11:105"/>
the <a href="fsm_1.html#U1" title="FSM_1:sel.1">Tran</a> of <font color="Maroon" title="c10">cm</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="fsm_1.html#U1" title="FSM_1:sel.1">Tran</a> of <font color="Maroon" title="c3">tfsm</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="fsm_1.html#K12" title="FSM_1:func.12">accessibleStates</a> <font color="Maroon" title="c3">tfsm</font></span>)</span>,<font color="Maroon" title="c2">IAlph</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( the <a href="fsm_1.html#U3" title="FSM_1:sel.3">OFun</a> of <font color="Maroon" title="c10">cm</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="fsm_1.html#U3" title="FSM_1:sel.3">OFun</a> of <font color="Maroon" title="c3">tfsm</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="fsm_1.html#K12" title="FSM_1:func.12">accessibleStates</a> <font color="Maroon" title="c3">tfsm</font></span>)</span>,<font color="Maroon" title="c2">IAlph</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c10">cm</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="c3">tfsm</font> &amp; <font color="Maroon" title="c3">tfsm</font>,<font color="Maroon" title="c10">cm</font> <a href="fsm_1.html#R4" title="FSM_1:pred.4">-are_equivalent</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E12:105"/>
the <a href="fsm_1.html#U3" title="FSM_1:sel.3">OFun</a> of <font color="Maroon" title="c10">cm</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="fsm_1.html#U3" title="FSM_1:sel.3">OFun</a> of <font color="Maroon" title="c3">tfsm</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="fsm_1.html#K12" title="FSM_1:func.12">accessibleStates</a> <font color="Maroon" title="c3">tfsm</font></span>)</span>,<font color="Maroon" title="c2">IAlph</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c10">cm</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="c3">tfsm</font> &amp; <font color="Maroon" title="c3">tfsm</font>,<font color="Maroon" title="c10">cm</font> <a href="fsm_1.html#R4" title="FSM_1:pred.4">-are_equivalent</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E13:105"/>
the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c10">cm</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="c3">tfsm</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">tfsm</font>,<font color="Maroon" title="c10">cm</font> <a href="fsm_1.html#R4" title="FSM_1:pred.4">-are_equivalent</a> </span><br/>

<span class="kw">let </span><font color="Maroon" title="c11">w</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c2">IAlph</font>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="fsm_1.html#D9" target="_self" title="FSM_1:def.9">FSM_1:def 9</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c3">tfsm</font>,<font color="Maroon" title="c11">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="c10">cm</font>,<font color="Maroon" title="c11">w</font> <a href="fsm_1.html#K4" title="FSM_1:func.4">-response</a> </span><br/>

<span class="kw">set </span><font color="Maroon" title="c12">ma</font> = the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c3">tfsm</font>,<font color="Maroon" title="c11">w</font> <a href="fsm_1.html#K2" title="FSM_1:func.2">-admissible</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c13">cma</font> = the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c10">cm</font>,<font color="Maroon" title="c11">w</font> <a href="fsm_1.html#K2" title="FSM_1:func.2">-admissible</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c14">mr</font> = the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c3">tfsm</font>,<font color="Maroon" title="c11">w</font> <a href="fsm_1.html#K4" title="FSM_1:func.4">-response</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c15">cmr</font> = the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c10">cm</font>,<font color="Maroon" title="c11">w</font> <a href="fsm_1.html#K4" title="FSM_1:func.4">-response</a> ;<br/>
<a NAME="E14:105"/><span class="lab"><font color="Green" title="E71">A13</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <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="c10">cm</font>,<font color="Maroon" title="c11">w</font> <a href="fsm_1.html#K2" title="FSM_1:func.2">-admissible</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">w</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fsm_1.html#D2" target="_self" title="FSM_1:def.2">Def2</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_3"><span class="kw">now </span></a><div class="add"><span class="kw">thus </span><a NAME="E1:105_3"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <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">tfsm</font>,<font color="Maroon" title="c11">w</font> <a href="fsm_1.html#K4" title="FSM_1:func.4">-response</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">w</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <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="c10">cm</font>,<font color="Maroon" title="c11">w</font> <a href="fsm_1.html#K4" title="FSM_1:func.4">-response</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">w</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fsm_1.html#D6" target="_self" title="FSM_1:def.6">Def6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">j</font> <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 <font color="Maroon" title="c3">tfsm</font>,<font color="Maroon" title="c11">w</font> <a href="fsm_1.html#K4" title="FSM_1:func.4">-response</a> </span>)</span> holds <br/><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">tfsm</font>,<font color="Maroon" title="c11">w</font> <a href="fsm_1.html#K4" title="FSM_1:func.4">-response</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="c10">cm</font>,<font color="Maroon" title="c11">w</font> <a href="fsm_1.html#K4" title="FSM_1:func.4">-response</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font></span><br/><a NAME="E2:105_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A14</font></span>: 
 <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 <font color="Maroon" title="c3">tfsm</font>,<font color="Maroon" title="c11">w</font> <a href="fsm_1.html#K4" title="FSM_1:func.4">-response</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> <font color="Maroon" title="c11">w</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><span class="kw">let </span><font color="Maroon" title="c16">j</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c16">j</font> <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 <font color="Maroon" title="c3">tfsm</font>,<font color="Maroon" title="c11">w</font> <a href="fsm_1.html#K4" title="FSM_1:func.4">-response</a> </span>)</span> implies <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">tfsm</font>,<font color="Maroon" title="c11">w</font> <a href="fsm_1.html#K4" title="FSM_1:func.4">-response</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="c10">cm</font>,<font color="Maroon" title="c11">w</font> <a href="fsm_1.html#K4" title="FSM_1:func.4">-response</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">j</font> )</span><br/><span class="kw">assume </span><a NAME="E3:105_3"/><span class="lab"><font color="Green" title="E73">A15</font></span>: 
<font color="Maroon" title="c16">j</font> <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 <font color="Maroon" title="c3">tfsm</font>,<font color="Maroon" title="c11">w</font> <a href="fsm_1.html#K4" title="FSM_1:func.4">-response</a> </span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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">tfsm</font>,<font color="Maroon" title="c11">w</font> <a href="fsm_1.html#K4" title="FSM_1:func.4">-response</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="c10">cm</font>,<font color="Maroon" title="c11">w</font> <a href="fsm_1.html#K4" title="FSM_1:func.4">-response</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">j</font></span><br/><a NAME="E4:105_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A16</font></span>: 
<font color="Maroon" title="c16">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">w</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:105_3"><span class="lab"><font color="Green" title="E72">A14</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E5:105_3"/>
<font color="Maroon" title="c16">j</font> <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="c11">w</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:105_3"><span class="lab"><font color="Green" title="E72">A14</font></span></a>, <a class="txt" href="#E3:105_3"><span class="lab"><font color="Green" title="E73">A15</font></span></a>, <a class="ref" href="finseq_2.html#T9" title="FINSEQ_2:th.9">FINSEQ_2:9</a></span>;<br/><a NAME="E6:105_3"/><span class="kw">then </span>
<font color="Maroon" title="c16">j</font> <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 <font color="Maroon" title="c10">cm</font>,<font color="Maroon" title="c11">w</font> <a href="fsm_1.html#K2" title="FSM_1:func.2">-admissible</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:105"><span class="lab"><font color="Green" title="E71">A13</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E7:105_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A17</font></span>: 
<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="c10">cm</font>,<font color="Maroon" title="c11">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> <font color="Maroon" title="c16">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fsm_1.html#K12" title="FSM_1:func.12">accessibleStates</a> <font color="Maroon" title="c3">tfsm</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T13" title="FINSEQ_2:th.13">FINSEQ_2:13</a></span>;<br/><a NAME="E8:105_3"/>
<font color="Maroon" title="c11">w</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">IAlph</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:105_3"><span class="lab"><font color="Green" title="E74">A16</font></span></a>, <a class="ref" href="finseq_2.html#T13" title="FINSEQ_2:th.13">FINSEQ_2:13</a></span>;<br/><a NAME="E9:105_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A18</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c10">cm</font>,<font color="Maroon" title="c11">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> <font color="Maroon" title="c16">j</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">w</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">j</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="fsm_1.html#K12" title="FSM_1:func.12">accessibleStates</a> <font color="Maroon" title="c3">tfsm</font></span>)</span>,<font color="Maroon" title="c2">IAlph</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:105_3"><span class="lab"><font color="Green" title="E75">A17</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/><a NAME="E10:105_3"/><span class="lab"><font color="Green" title="E77">A19</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c3">tfsm</font>,<font color="Maroon" title="c11">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> <font color="Maroon" title="c16">j</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">w</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">j</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c10">cm</font>,<font color="Maroon" title="c11">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> <font color="Maroon" title="c16">j</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">w</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">j</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:105"><span class="lab"><font color="Green" title="E70">A1</font></span></a></span>;<br/><span class="kw">thus </span><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">tfsm</font>,<font color="Maroon" title="c11">w</font> <a href="fsm_1.html#K4" title="FSM_1:func.4">-response</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">j</font> = 
the <a href="fsm_1.html#U3" title="FSM_1:sel.3">OFun</a> of <font color="Maroon" title="c3">tfsm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c3">tfsm</font>,<font color="Maroon" title="c11">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> <font color="Maroon" title="c16">j</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">w</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">j</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:105_3"><span class="lab"><font color="Green" title="E74">A16</font></span></a>, <a class="ref" href="fsm_1.html#D6" target="_self" title="FSM_1:def.6">Def6</a></span>
<br/>.= 
<font color="Maroon" title="c6">cOFun</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c10">cm</font>,<font color="Maroon" title="c11">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> <font color="Maroon" title="c16">j</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">w</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">j</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:105_3"><span class="lab"><font color="Green" title="E77">A19</font></span></a>, <a class="txt" href="#E9:105_3"><span class="lab"><font color="Green" title="E76">A18</font></span></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>
<br/>.= 
<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="c10">cm</font>,<font color="Maroon" title="c11">w</font> <a href="fsm_1.html#K4" title="FSM_1:func.4">-response</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">j</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:105_3"><span class="lab"><font color="Green" title="E74">A16</font></span></a>, <a class="ref" href="fsm_1.html#D6" target="_self" title="FSM_1:def.6">Def6</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E16:105"/>
the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c3">tfsm</font>,<font color="Maroon" title="c11">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="c10">cm</font>,<font color="Maroon" title="c11">w</font> <a href="fsm_1.html#K4" title="FSM_1:func.4">-response</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T10" title="FINSEQ_2:th.10">FINSEQ_2:10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
