<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">Ctfsm1</font>, <font color="Olive" title="b2">Ctfsm2</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>  st <font color="Olive" title="b1">Ctfsm1</font>,<font color="Olive" title="b2">Ctfsm2</font> <a href="fsm_1.html#R4" title="FSM_1:pred.4">-are_equivalent</a>  holds <br/> <a href="fsm_1.html#K11" title="FSM_1:func.11">the_reduction_of</a> <font color="Olive" title="b1">Ctfsm1</font>, <a href="fsm_1.html#K11" title="FSM_1:func.11">the_reduction_of</a> <font color="Olive" title="b2">Ctfsm2</font> <a href="fsm_1.html#R7" title="FSM_1:pred.7">-are_isomorphic</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c3">Ctfsm1</font>, <font color="Maroon" title="c4">Ctfsm2</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">Ctfsm1</font>,<font color="Maroon" title="c4">Ctfsm2</font> <a href="fsm_1.html#R4" title="FSM_1:pred.4">-are_equivalent</a>  implies  <a href="fsm_1.html#K11" title="FSM_1:func.11">the_reduction_of</a> <font color="Maroon" title="c3">Ctfsm1</font>, <a href="fsm_1.html#K11" title="FSM_1:func.11">the_reduction_of</a> <font color="Maroon" title="c4">Ctfsm2</font> <a href="fsm_1.html#R7" title="FSM_1:pred.7">-are_isomorphic</a>  )</span><br/>







<span class="kw">assume </span><a NAME="E1:121"/><span class="lab"><font color="Green" title="E85">A1</font></span>: 
<font color="Maroon" title="c3">Ctfsm1</font>,<font color="Maroon" title="c4">Ctfsm2</font> <a href="fsm_1.html#R4" title="FSM_1:pred.4">-are_equivalent</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="fsm_1.html#K11" title="FSM_1:func.11">the_reduction_of</a> <font color="Maroon" title="c3">Ctfsm1</font>, <a href="fsm_1.html#K11" title="FSM_1:func.11">the_reduction_of</a> <font color="Maroon" title="c4">Ctfsm2</font> <a href="fsm_1.html#R7" title="FSM_1:pred.7">-are_isomorphic</a> </span><br/>

<span class="kw">set </span><font color="Maroon" title="c5">rtfsm2</font> =  <a href="fsm_1.html#K11" title="FSM_1:func.11">the_reduction_of</a> <font color="Maroon" title="c4">Ctfsm2</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">rtfsm1</font> =  <a href="fsm_1.html#K11" title="FSM_1:func.11">the_reduction_of</a> <font color="Maroon" title="c3">Ctfsm1</font>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">fsm1</font>, <font color="Maroon" title="c8">fsm2</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="c1">IAlph</font>,<font color="Maroon" title="c2">OAlph</font><span class="kw"> such that </span><br/><a NAME="E3:121"/><span class="lab"><font color="Green" title="E86">A2</font></span>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">fsm1</font> <a href="subset_1.html#R1" title="SUBSET_1:pred.1">misses</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">fsm2</font>
 <span class="kw">and </span><br/><a NAME="E4:121"/><span class="lab"><font color="Green" title="E87">A3</font></span>: 
<font color="Maroon" title="c7">fsm1</font>, <a href="fsm_1.html#K11" title="FSM_1:func.11">the_reduction_of</a> <font color="Maroon" title="c3">Ctfsm1</font> <a href="fsm_1.html#R7" title="FSM_1:pred.7">-are_isomorphic</a> 
 <span class="kw">and </span><br/><a NAME="E5:121"/><span class="lab"><font color="Green" title="E88">A4</font></span>: 
<font color="Maroon" title="c8">fsm2</font>, <a href="fsm_1.html#K11" title="FSM_1:func.11">the_reduction_of</a> <font color="Maroon" title="c4">Ctfsm2</font> <a href="fsm_1.html#R7" title="FSM_1:pred.7">-are_isomorphic</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fsm_1.html#T81" target="_self" title="FSM_1:th.81">Th81</a></span>;<br/>
<a NAME="E6:121"/><span class="lab"><font color="Green" title="E89">A5</font></span>: 
 <a href="fsm_1.html#K11" title="FSM_1:func.11">the_reduction_of</a> <font color="Maroon" title="c3">Ctfsm1</font>,<font color="Maroon" title="c3">Ctfsm1</font> <a href="fsm_1.html#R4" title="FSM_1:pred.4">-are_equivalent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fsm_1.html#T58" target="_self" title="FSM_1:th.58">Th58</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">Srtfsm1</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="fsm_1.html#K11" title="FSM_1:func.11">the_reduction_of</a> <font color="Maroon" title="c3">Ctfsm1</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">Srtfsm2</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="fsm_1.html#K11" title="FSM_1:func.11">the_reduction_of</a> <font color="Maroon" title="c4">Ctfsm2</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c11">ISrtfsm1</font> = 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">Ctfsm1</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c12">ISrtfsm2</font> = 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="c4">Ctfsm2</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c13">Sfsm1</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">fsm1</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c14">Sfsm2</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">fsm2</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c15">ISfsm1</font> = the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c7">fsm1</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c16">ISfsm2</font> = the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c8">fsm2</font>;<br/>
<a NAME="E7:121"/><span class="lab"><font color="Green" title="E90">A6</font></span>: 
 <a href="fsm_1.html#K11" title="FSM_1:func.11">the_reduction_of</a> <font color="Maroon" title="c4">Ctfsm2</font>,<font color="Maroon" title="c4">Ctfsm2</font> <a href="fsm_1.html#R4" title="FSM_1:pred.4">-are_equivalent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fsm_1.html#T58" target="_self" title="FSM_1:th.58">Th58</a></span>;<br/>
<a NAME="E8:121"/>
<font color="Maroon" title="c8">fsm2</font>, <a href="fsm_1.html#K11" title="FSM_1:func.11">the_reduction_of</a> <font color="Maroon" title="c4">Ctfsm2</font> <a href="fsm_1.html#R4" title="FSM_1:pred.4">-are_equivalent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:121"><span class="lab"><font color="Green" title="E88">A4</font></span></a>, <a class="ref" href="fsm_1.html#T82" target="_self" title="FSM_1:th.82">Th82</a></span>;<br/>
<a NAME="E9:121"/><span class="kw">then </span><span class="lab"><font color="Green" title="E91">A7</font></span>: 
<font color="Maroon" title="c8">fsm2</font>,<font color="Maroon" title="c4">Ctfsm2</font> <a href="fsm_1.html#R4" title="FSM_1:pred.4">-are_equivalent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:121"><span class="lab"><font color="Green" title="E90">A6</font></span></a>, <a class="ref" href="fsm_1.html#T30" target="_self" title="FSM_1:th.30">Th30</a></span>;<br/>
<a NAME="E10:121"/><span class="lab"><font color="Green" title="E92">A8</font></span>: 
 <a href="fsm_1.html#K11" title="FSM_1:func.11">the_reduction_of</a> <font color="Maroon" title="c3">Ctfsm1</font> is  <a href="fsm_1.html#V7" title="FSM_1:attr.7">connected</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fsm_1.html#T66" target="_self" title="FSM_1:th.66">Th66</a></span>;<br/>
<a NAME="E11:121"/><span class="lab"><font color="Green" title="E93">A9</font></span>: 
<font color="Maroon" title="c7">fsm1</font> is  <a href="fsm_1.html#V7" title="FSM_1:attr.7">connected</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="121_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:121_1"/>
not <font color="Maroon" title="c7">fsm1</font> is  <a href="fsm_1.html#V7" title="FSM_1:attr.7">connected</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">q1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">fsm1</font><span class="kw"> such that </span><br/><a NAME="E3:121_1"/><span class="lab"><font color="Green" title="E94">A10</font></span>: 
not <font color="Maroon" title="c17">q1</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#D22" target="_self" title="FSM_1:def.22">Def22</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c18">Tf</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">fsm1</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="fsm_1.html#K11" title="FSM_1:func.11">the_reduction_of</a> <font color="Maroon" title="c3">Ctfsm1</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E5:121_1"/><span class="lab"><font color="Green" title="E95">A11</font></span>: 
<font color="Maroon" title="c18">Tf</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a> 
 <span class="kw">and </span><br/><a NAME="E6:121_1"/><span class="lab"><font color="Green" title="E96">A12</font></span>: 
( <font color="Maroon" title="c18">Tf</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c7">fsm1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <span class="p1">(<span class="default"><a href="fsm_1.html#K11" title="FSM_1:func.11">the_reduction_of</a> <font color="Maroon" title="c3">Ctfsm1</font></span>)</span> &amp; ( for <font color="Olive" title="b1">q</font> being  <a href="fsm_1.html#NM1" title="FSM_1:NM.1">State</a> of <font color="Maroon" title="c7">fsm1</font><br/> for <font color="Olive" title="b2">s</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">IAlph</font> holds <br/> ( <font color="Maroon" title="c18">Tf</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default">the <a href="fsm_1.html#U1" title="FSM_1:sel.1">Tran</a> of <font color="Maroon" title="c7">fsm1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">q</font>,<font color="Olive" title="b2">s</font></span>)</span> <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 <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">Ctfsm1</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">Tf</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">q</font></span>)</span>,<font color="Olive" title="b2">s</font> &amp; the <a href="fsm_1.html#U3" title="FSM_1:sel.3">OFun</a> of <font color="Maroon" title="c7">fsm1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">q</font>,<font color="Olive" title="b2">s</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 <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">Ctfsm1</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">Tf</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">q</font></span>)</span>,<font color="Olive" title="b2">s</font> ) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:121"><span class="lab"><font color="Green" title="E87">A3</font></span></a>, <a class="ref" href="fsm_1.html#D19" target="_self" title="FSM_1:def.19">Def19</a></span>;<br/>
<a NAME="E7:121_1"/><span class="lab"><font color="Green" title="E97">A13</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c18">Tf</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">fsm1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c19">q</font> = <font color="Maroon" title="c18">Tf</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">q1</font>;<br/>
<a NAME="E8:121_1"/>
<font color="Maroon" title="c18">Tf</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">q1</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="txt" href="#E10:121"><span class="lab"><font color="Green" title="E92">A8</font></span></a>, <a class="ref" href="fsm_1.html#D22" target="_self" title="FSM_1:def.22">Def22</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c20">w</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">IAlph</font><span class="kw"> such that </span><br/><a NAME="E10:121_1"/><span class="lab"><font color="Green" title="E98">A14</font></span>: 
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">Ctfsm1</font></span>)</span>,<font color="Maroon" title="c20">w</font> <a href="fsm_1.html#R1" title="FSM_1:pred.1">-leads_to</a> <font color="Maroon" title="c18">Tf</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">q1</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="E11:121_1"/><span class="lab"><font color="Green" title="E99">A15</font></span>: 
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="c20">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="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/>
<a NAME="E12:121_1"/><span class="kw">then </span>
<font color="Maroon" title="c18">Tf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c7">fsm1</font>,<font color="Maroon" title="c20">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="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">w</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></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 <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">Ctfsm1</font></span>)</span>,<font color="Maroon" title="c20">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="c20">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="#E6:121_1"><span class="lab"><font color="Green" title="E96">A12</font></span></a>, <a class="ref" href="fsm_1.html#T60" target="_self" title="FSM_1:th.60">Th60</a></span>;<br/>
<a NAME="E13:121_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E100">A16</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c18">Tf</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</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="c18">Tf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <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="c7">fsm1</font>,<font color="Maroon" title="c20">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="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">w</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">Tf</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</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="c18">Tf</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">q1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:121_1"><span class="lab"><font color="Green" title="E98">A14</font></span></a>, <a class="ref" href="fsm_1.html#D3" target="_self" title="FSM_1:def.3">Def3</a></span>;<br/>
<a NAME="E14:121_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="c7">fsm1</font>,<font color="Maroon" title="c20">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="c20">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/>
<a NAME="E15:121_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">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 <font color="Maroon" title="c7">fsm1</font>,<font color="Maroon" title="c20">w</font> <a href="fsm_1.html#K2" title="FSM_1:func.2">-admissible</a> </span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:121_1"><span class="lab"><font color="Green" title="E99">A15</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E16:121_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">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 <font color="Maroon" title="c7">fsm1</font>,<font color="Maroon" title="c20">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="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E17:121_1"/><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 <font color="Maroon" title="c7">fsm1</font>,<font color="Maroon" title="c20">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="c20">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>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c18">Tf</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:121_1"><span class="lab"><font color="Green" title="E97">A13</font></span></a>, <a class="ref" href="finseq_2.html#T13" title="FINSEQ_2:th.13">FINSEQ_2:13</a></span>;<br/>
<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 <font color="Maroon" title="c7">fsm1</font>,<font color="Maroon" title="c20">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="c20">w</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c18">Tf</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</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="c18">Tf</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">q1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:121_1"><span class="lab"><font color="Green" title="E95">A11</font></span></a>, <a class="txt" href="#E13:121_1"><span class="lab"><font color="Green" title="E100">A16</font></span></a>, <a class="ref" href="funct_1.html#T56" title="FUNCT_1:th.56">FUNCT_1:56</a></span>
<br/>.= 
<font color="Maroon" title="c17">q1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:121_1"><span class="lab"><font color="Green" title="E95">A11</font></span></a>, <a class="txt" href="#E7:121_1"><span class="lab"><font color="Green" title="E97">A13</font></span></a>, <a class="ref" href="funct_1.html#T56" title="FUNCT_1:th.56">FUNCT_1:56</a></span>
;<br/>
<a NAME="E19:121_1"/><span class="kw">then </span>
the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c7">fsm1</font>,<font color="Maroon" title="c20">w</font> <a href="fsm_1.html#R1" title="FSM_1:pred.1">-leads_to</a> <font color="Maroon" title="c17">q1</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="E20:121_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:121_1"><span class="lab"><font color="Green" title="E94">A10</font></span></a>, <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>
<a NAME="E12:121"/><span class="lab"><font color="Green" title="E94">A17</font></span>: 
 <a href="fsm_1.html#K11" title="FSM_1:func.11">the_reduction_of</a> <font color="Maroon" title="c4">Ctfsm2</font> is  <a href="fsm_1.html#V7" title="FSM_1:attr.7">connected</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fsm_1.html#T66" target="_self" title="FSM_1:th.66">Th66</a></span>;<br/>
<a NAME="E13:121"/><span class="lab"><font color="Green" title="E95">A18</font></span>: 
<font color="Maroon" title="c8">fsm2</font> is  <a href="fsm_1.html#V7" title="FSM_1:attr.7">connected</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="121_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:121_2"/>
not <font color="Maroon" title="c8">fsm2</font> is  <a href="fsm_1.html#V7" title="FSM_1:attr.7">connected</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">q1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">fsm2</font><span class="kw"> such that </span><br/><a NAME="E3:121_2"/><span class="lab"><font color="Green" title="E96">A19</font></span>: 
not <font color="Maroon" title="c17">q1</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#D22" target="_self" title="FSM_1:def.22">Def22</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c18">Tf</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">fsm2</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="fsm_1.html#K11" title="FSM_1:func.11">the_reduction_of</a> <font color="Maroon" title="c4">Ctfsm2</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E5:121_2"/><span class="lab"><font color="Green" title="E97">A20</font></span>: 
<font color="Maroon" title="c18">Tf</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a> 
 <span class="kw">and </span><br/><a NAME="E6:121_2"/><span class="lab"><font color="Green" title="E98">A21</font></span>: 
( <font color="Maroon" title="c18">Tf</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c8">fsm2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <span class="p1">(<span class="default"><a href="fsm_1.html#K11" title="FSM_1:func.11">the_reduction_of</a> <font color="Maroon" title="c4">Ctfsm2</font></span>)</span> &amp; ( for <font color="Olive" title="b1">q</font> being  <a href="fsm_1.html#NM1" title="FSM_1:NM.1">State</a> of <font color="Maroon" title="c8">fsm2</font><br/> for <font color="Olive" title="b2">s</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">IAlph</font> holds <br/> ( <font color="Maroon" title="c18">Tf</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default">the <a href="fsm_1.html#U1" title="FSM_1:sel.1">Tran</a> of <font color="Maroon" title="c8">fsm2</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">q</font>,<font color="Olive" title="b2">s</font></span>)</span> <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 <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="c4">Ctfsm2</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">Tf</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">q</font></span>)</span>,<font color="Olive" title="b2">s</font> &amp; the <a href="fsm_1.html#U3" title="FSM_1:sel.3">OFun</a> of <font color="Maroon" title="c8">fsm2</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">q</font>,<font color="Olive" title="b2">s</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 <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="c4">Ctfsm2</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">Tf</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">q</font></span>)</span>,<font color="Olive" title="b2">s</font> ) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:121"><span class="lab"><font color="Green" title="E88">A4</font></span></a>, <a class="ref" href="fsm_1.html#D19" target="_self" title="FSM_1:def.19">Def19</a></span>;<br/>
<a NAME="E7:121_2"/><span class="lab"><font color="Green" title="E99">A22</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c18">Tf</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">fsm2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c19">q</font> = <font color="Maroon" title="c18">Tf</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">q1</font>;<br/>
<a NAME="E8:121_2"/>
<font color="Maroon" title="c18">Tf</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">q1</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="txt" href="#E12:121"><span class="lab"><font color="Green" title="E94">A17</font></span></a>, <a class="ref" href="fsm_1.html#D22" target="_self" title="FSM_1:def.22">Def22</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c20">w</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">IAlph</font><span class="kw"> such that </span><br/><a NAME="E10:121_2"/><span class="lab"><font color="Green" title="E100">A23</font></span>: 
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="c4">Ctfsm2</font></span>)</span>,<font color="Maroon" title="c20">w</font> <a href="fsm_1.html#R1" title="FSM_1:pred.1">-leads_to</a> <font color="Maroon" title="c18">Tf</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">q1</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="E11:121_2"/><span class="lab"><font color="Green" title="E101">A24</font></span>: 
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="c20">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="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/>
<a NAME="E12:121_2"/><span class="kw">then </span>
<font color="Maroon" title="c18">Tf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c8">fsm2</font>,<font color="Maroon" title="c20">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="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">w</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></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 <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="c4">Ctfsm2</font></span>)</span>,<font color="Maroon" title="c20">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="c20">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="#E6:121_2"><span class="lab"><font color="Green" title="E98">A21</font></span></a>, <a class="ref" href="fsm_1.html#T60" target="_self" title="FSM_1:th.60">Th60</a></span>;<br/>
<a NAME="E13:121_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E102">A25</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c18">Tf</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</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="c18">Tf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <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="c8">fsm2</font>,<font color="Maroon" title="c20">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="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">w</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">Tf</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</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="c18">Tf</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">q1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:121_2"><span class="lab"><font color="Green" title="E100">A23</font></span></a>, <a class="ref" href="fsm_1.html#D3" target="_self" title="FSM_1:def.3">Def3</a></span>;<br/>
<a NAME="E14:121_2"/>
 <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="c8">fsm2</font>,<font color="Maroon" title="c20">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="c20">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/>
<a NAME="E15:121_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">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 <font color="Maroon" title="c8">fsm2</font>,<font color="Maroon" title="c20">w</font> <a href="fsm_1.html#K2" title="FSM_1:func.2">-admissible</a> </span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:121_2"><span class="lab"><font color="Green" title="E101">A24</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E16:121_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">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 <font color="Maroon" title="c8">fsm2</font>,<font color="Maroon" title="c20">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="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E17:121_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 <font color="Maroon" title="c8">fsm2</font>,<font color="Maroon" title="c20">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="c20">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>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c18">Tf</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:121_2"><span class="lab"><font color="Green" title="E99">A22</font></span></a>, <a class="ref" href="finseq_2.html#T13" title="FINSEQ_2:th.13">FINSEQ_2:13</a></span>;<br/>
<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 <font color="Maroon" title="c8">fsm2</font>,<font color="Maroon" title="c20">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="c20">w</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c18">Tf</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</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="c18">Tf</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">q1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:121_2"><span class="lab"><font color="Green" title="E97">A20</font></span></a>, <a class="txt" href="#E13:121_2"><span class="lab"><font color="Green" title="E102">A25</font></span></a>, <a class="ref" href="funct_1.html#T56" title="FUNCT_1:th.56">FUNCT_1:56</a></span>
<br/>.= 
<font color="Maroon" title="c17">q1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:121_2"><span class="lab"><font color="Green" title="E97">A20</font></span></a>, <a class="txt" href="#E7:121_2"><span class="lab"><font color="Green" title="E99">A22</font></span></a>, <a class="ref" href="funct_1.html#T56" title="FUNCT_1:th.56">FUNCT_1:56</a></span>
;<br/>
<a NAME="E19:121_2"/><span class="kw">then </span>
the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c8">fsm2</font>,<font color="Maroon" title="c20">w</font> <a href="fsm_1.html#R1" title="FSM_1:pred.1">-leads_to</a> <font color="Maroon" title="c17">q1</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="E20:121_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:121_2"><span class="lab"><font color="Green" title="E96">A19</font></span></a>, <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>
<a NAME="E14:121"/>
<font color="Maroon" title="c7">fsm1</font>, <a href="fsm_1.html#K11" title="FSM_1:func.11">the_reduction_of</a> <font color="Maroon" title="c3">Ctfsm1</font> <a href="fsm_1.html#R4" title="FSM_1:pred.4">-are_equivalent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:121"><span class="lab"><font color="Green" title="E87">A3</font></span></a>, <a class="ref" href="fsm_1.html#T82" target="_self" title="FSM_1:th.82">Th82</a></span>;<br/>
<a NAME="E15:121"/><span class="kw">then </span>
<font color="Maroon" title="c7">fsm1</font>,<font color="Maroon" title="c3">Ctfsm1</font> <a href="fsm_1.html#R4" title="FSM_1:pred.4">-are_equivalent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:121"><span class="lab"><font color="Green" title="E89">A5</font></span></a>, <a class="ref" href="fsm_1.html#T30" target="_self" title="FSM_1:th.30">Th30</a></span>;<br/>
<a NAME="E16:121"/><span class="kw">then </span><span class="lab"><font color="Green" title="E96">A26</font></span>: 
<font color="Maroon" title="c7">fsm1</font>,<font color="Maroon" title="c4">Ctfsm2</font> <a href="fsm_1.html#R4" title="FSM_1:pred.4">-are_equivalent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:121"><span class="lab"><font color="Green" title="E85">A1</font></span></a>, <a class="ref" href="fsm_1.html#T30" target="_self" title="FSM_1:th.30">Th30</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c17">fsm1</font> = <font color="Maroon" title="c7">fsm1</font>, <font color="Maroon" title="c18">fsm2</font> = <font color="Maroon" title="c8">fsm2</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#V5" title="FSM_1:attr.5">reduced</a>   <a href="fsm_1.html#L2" title="FSM_1:struct.2">Mealy-FSM</a> of <font color="Maroon" title="c1">IAlph</font>,<font color="Maroon" title="c2">OAlph</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:121"><span class="lab"><font color="Green" title="E87">A3</font></span></a>, <a class="txt" href="#E5:121"><span class="lab"><font color="Green" title="E88">A4</font></span></a>, <a class="ref" href="fsm_1.html#T65" target="_self" title="FSM_1:th.65">Th65</a></span>;<br/>
<a NAME="E18:121"/>
<font color="Maroon" title="c17">fsm1</font>,<font color="Maroon" title="c18">fsm2</font> <a href="fsm_1.html#R7" title="FSM_1:pred.7">-are_isomorphic</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:121"><span class="lab"><font color="Green" title="E86">A2</font></span></a>, <a class="txt" href="#E11:121"><span class="lab"><font color="Green" title="E93">A9</font></span></a>, <a class="txt" href="#E13:121"><span class="lab"><font color="Green" title="E95">A18</font></span></a>, <a class="txt" href="#E9:121"><span class="lab"><font color="Green" title="E91">A7</font></span></a>, <a class="txt" href="#E16:121"><span class="lab"><font color="Green" title="E96">A26</font></span></a>, <a class="ref" href="fsm_1.html#T30" target="_self" title="FSM_1:th.30">Th30</a>, <a class="ref" href="fsm_1.html#T83" target="_self" title="FSM_1:th.83">Th83</a></span>;<br/>
<a NAME="E19:121"/><span class="kw">then </span>
<font color="Maroon" title="c17">fsm1</font>, <a href="fsm_1.html#K11" title="FSM_1:func.11">the_reduction_of</a> <font color="Maroon" title="c4">Ctfsm2</font> <a href="fsm_1.html#R7" title="FSM_1:pred.7">-are_isomorphic</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:121"><span class="lab"><font color="Green" title="E88">A4</font></span></a>, <a class="ref" href="fsm_1.html#T59" target="_self" title="FSM_1:th.59">Th59</a></span>;<br/>
<span class="kw">hence </span><a NAME="E20:121"/>
 <a href="fsm_1.html#K11" title="FSM_1:func.11">the_reduction_of</a> <font color="Maroon" title="c3">Ctfsm1</font>, <a href="fsm_1.html#K11" title="FSM_1:func.11">the_reduction_of</a> <font color="Maroon" title="c4">Ctfsm2</font> <a href="fsm_1.html#R7" title="FSM_1:pred.7">-are_isomorphic</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:121"><span class="lab"><font color="Green" title="E87">A3</font></span></a>, <a class="ref" href="fsm_1.html#T59" target="_self" title="FSM_1:th.59">Th59</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>
