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

<b>let </b><font color="Maroon" title="c1">IAlph</font>, <font color="Maroon" title="c2">OAlph</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c3">rtfsm</font>, <font color="Maroon" title="c4">tfsm</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="group_1.html#V6" title="GROUP_1:attr.6">finite</a>  <a href="fsm_1.html#L2" title="FSM_1:struct.2">Mealy-FSM</a> of <font color="Maroon" title="c1">IAlph</font>,<font color="Maroon" title="c2">OAlph</font>;<br/><b>let </b><font color="Maroon" title="c5">qr1</font>, <font color="Maroon" title="c6">qr2</font> be   <a href="fsm_1.html#NM1" title="FSM_1:NM.1">State</a> of <font color="Maroon" title="c3">rtfsm</font>;<br/>











<b>assume </b><a NAME="E1:85"/><i><font color="Green" title="E63">A1</font></i>: 
( <font color="Maroon" title="c3">rtfsm</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fsm_1.html#K11" title="FSM_1:func.11">the_reduction_of</a> <font color="Maroon" title="c4">tfsm</font> &amp; <font color="Maroon" title="c5">qr1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">qr2</font> )
 ;<br/>

<a NAME="E2:85"/><b>then </b><i><font color="Green" title="E64">A2</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">rtfsm</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fsm_1.html#K8" title="FSM_1:func.8">final_states_partition</a> <font color="Maroon" title="c4">tfsm</font>
 
<b>by </b><i><a class="ref" href="fsm_1.html#D18" target="_self" title="FSM_1:def.18">Def18</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c7">q1'</font> = <font color="Maroon" title="c5">qr1</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c4">tfsm</font> <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">tfsm</font><b> such that </b><br/><a NAME="E5:85"/><i><font color="Green" title="E65">A3</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">q1'</font>
 <b>by </b><i><a class="txt" href="fsm_1.html#E2:85"><i><font color="Green" title="E64">A2</font></i></a>, <a class="ref" href="fsm_1.html#T11" target="_self" title="FSM_1:th.11">Th11</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">q2'</font> = <font color="Maroon" title="c6">qr2</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c4">tfsm</font> <b>by </b><i><a class="txt" href="fsm_1.html#E2:85"><i><font color="Green" title="E64">A2</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">tfsm</font><b> such that </b><br/><a NAME="E8:85"/><i><font color="Green" title="E66">A4</font></i>: 
<font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">q2'</font>
 <b>by </b><i><a class="txt" href="fsm_1.html#E2:85"><i><font color="Green" title="E64">A2</font></i></a>, <a class="ref" href="fsm_1.html#T11" target="_self" title="FSM_1:th.11">Th11</a></i>;<br/>
<a NAME="E9:85"/><i><font color="Green" title="E67">A5</font></i>: 
 <a href="fsm_1.html#K8" title="FSM_1:func.8">final_states_partition</a> <font color="Maroon" title="c4">tfsm</font> is <a href="fsm_1.html#V4" title="FSM_1:attr.4">final</a>
 
<b>by </b><i><a class="ref" href="fsm_1.html#D15" target="_self" title="FSM_1:def.15">Def15</a></i>;<br/>
<a NAME="E10:85"/>
not <font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c10">y</font> <a href="fsm_1.html#R5" title="FSM_1:pred.5">-are_equivalent</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="85_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:85_1"/>
<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c10">y</font> <a href="fsm_1.html#R5" title="FSM_1:pred.5">-are_equivalent</a> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c11">X</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">rtfsm</font><b> such that </b><br/><a NAME="E3:85_1"/><i><font color="Green" title="E68">A6</font></i>: 
( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">X</font> &amp; <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">X</font> )
 <b>by </b><i><a class="txt" href="fsm_1.html#E2:85"><i><font color="Green" title="E64">A2</font></i></a>, <a class="txt" href="fsm_1.html#E9:85"><i><font color="Green" title="E67">A5</font></i></a>, <a class="ref" href="fsm_1.html#D14" target="_self" title="FSM_1:def.14">Def14</a></i>;<br/>
<a NAME="E4:85_1"/>
<font color="Maroon" title="c11">X</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c4">tfsm</font>
 
<b>by </b><i><a class="txt" href="fsm_1.html#E2:85"><i><font color="Green" title="E64">A2</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E5:85_1"/><b>then </b><i><font color="Green" title="E69">A7</font></i>: 
( <font color="Maroon" title="c11">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">q1'</font> or <font color="Maroon" title="c11">X</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c7">q1'</font> )
 
<b>by </b><i><a class="txt" href="fsm_1.html#E2:85"><i><font color="Green" title="E64">A2</font></i></a>, <a class="ref" href="eqrel_1.html#D6" title="EQREL_1:def.6">EQREL_1:def 6</a></i>;<br/>
<a NAME="E6:85_1"/>
<font color="Maroon" title="c9">q2'</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c7">q1'</font>
 
<b>by </b><i><a class="txt" href="fsm_1.html#E1:85"><i><font color="Green" title="E63">A1</font></i></a>, <a class="txt" href="fsm_1.html#E2:85"><i><font color="Green" title="E64">A2</font></i></a>, <a class="ref" href="eqrel_1.html#D6" title="EQREL_1:def.6">EQREL_1:def 6</a></i>;<br/>
<b>hence </b><a NAME="E7:85_1"/>
contradiction
 <b>by </b><i><a class="txt" href="fsm_1.html#E5:85"><i><font color="Green" title="E65">A3</font></i></a>, <a class="txt" href="fsm_1.html#E8:85"><i><font color="Green" title="E66">A4</font></i></a>, <a class="txt" href="fsm_1.html#E3:85_1"><i><font color="Green" title="E68">A6</font></i></a>, <a class="txt" href="fsm_1.html#E5:85_1"><i><font color="Green" title="E69">A7</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon" title="c11">w</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">IAlph</font><b> such that </b><br/><a NAME="E12:85"/><i><font color="Green" title="E68">A8</font></i>: 
<font color="Maroon" title="c8">x</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#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">y</font>,<font color="Maroon" title="c11">w</font> <a href="fsm_1.html#K4" title="FSM_1:func.4">-response</a> 
 <b>by </b><i><a class="ref" href="fsm_1.html#D10" target="_self" title="FSM_1:def.10">Def10</a></i>;<br/>
<b>set </b><font color="Maroon" title="c12">xresp</font> = <font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c11">w</font> <a href="fsm_1.html#K4" title="FSM_1:func.4">-response</a> ;<br/>
<b>set </b><font color="Maroon" title="c13">yresp</font> = <font color="Maroon" title="c10">y</font>,<font color="Maroon" title="c11">w</font> <a href="fsm_1.html#K4" title="FSM_1:func.4">-response</a> ;<br/>
<b>set </b><font color="Maroon" title="c14">xadm</font> = <font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c11">w</font> <a href="fsm_1.html#K2" title="FSM_1:func.2">-admissible</a> ;<br/>
<b>set </b><font color="Maroon" title="c15">yadm</font> = <font color="Maroon" title="c10">y</font>,<font color="Maroon" title="c11">w</font> <a href="fsm_1.html#K2" title="FSM_1:func.2">-admissible</a> ;<br/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">x</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="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">w</font>
<b>by </b><i><a class="ref" href="fsm_1.html#D6" target="_self" title="FSM_1:def.6">Def6</a></i>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">y</font>,<font color="Maroon" title="c11">w</font> <a href="fsm_1.html#K4" title="FSM_1:func.4">-response</a> </span>)</span>
<b>by </b><i><a class="ref" href="fsm_1.html#D6" target="_self" title="FSM_1:def.6">Def6</a></i>
;<br/>
<b>then consider </b><font color="Maroon" title="c16">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E15:85"/><i><font color="Green" title="E69">A9</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">k</font> &amp; <font color="Maroon" title="c16">k</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> <span class="p1">(<span class="default"><font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c11">w</font> <a href="fsm_1.html#K4" title="FSM_1:func.4">-response</a> </span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c8">x</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">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">y</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">k</font> )
 <b>by </b><i><a class="txt" href="fsm_1.html#E12:85"><i><font color="Green" title="E68">A8</font></i></a>, <a class="ref" href="finseq_1.html#T18" title="FINSEQ_1:th.18">FINSEQ_1:18</a></i>;<br/>
<a NAME="E16:85"/><i><font color="Green" title="E70">A10</font></i>: 
<font color="Maroon" title="c16">k</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"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c11">w</font> <a href="fsm_1.html#K4" title="FSM_1:func.4">-response</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="fsm_1.html#E15:85"><i><font color="Green" title="E69">A9</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<b>set </b><font color="Maroon" title="c17">q1resp</font> = <font color="Maroon" title="c5">qr1</font>,<font color="Maroon" title="c11">w</font> <a href="fsm_1.html#K4" title="FSM_1:func.4">-response</a> ;<br/>
<b>set </b><font color="Maroon" title="c18">q2resp</font> = <font color="Maroon" title="c6">qr2</font>,<font color="Maroon" title="c11">w</font> <a href="fsm_1.html#K4" title="FSM_1:func.4">-response</a> ;<br/>
<b>set </b><font color="Maroon" title="c19">q1adm</font> = <font color="Maroon" title="c5">qr1</font>,<font color="Maroon" title="c11">w</font> <a href="fsm_1.html#K2" title="FSM_1:func.2">-admissible</a> ;<br/>
<b>set </b><font color="Maroon" title="c20">q2adm</font> = <font color="Maroon" title="c6">qr2</font>,<font color="Maroon" title="c11">w</font> <a href="fsm_1.html#K2" title="FSM_1:func.2">-admissible</a> ;<br/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">qr1</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="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#K1" title="NAT_1:func.1">+</a> 1
<b>by </b><i><a class="ref" href="fsm_1.html#D2" target="_self" title="FSM_1:def.2">Def2</a></i>
<br/>.= 
<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"><font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c11">w</font> <a href="fsm_1.html#K4" title="FSM_1:func.4">-response</a> </span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
<b>by </b><i><a class="ref" href="fsm_1.html#D6" target="_self" title="FSM_1:def.6">Def6</a></i>
;<br/>
<a NAME="E18:85"/><b>then </b><i><font color="Green" title="E71">A11</font></i>: 
<font color="Maroon" title="c16">k</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"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">qr1</font>,<font color="Maroon" title="c11">w</font> <a href="fsm_1.html#K2" title="FSM_1:func.2">-admissible</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="fsm_1.html#E16:85"><i><font color="Green" title="E70">A10</font></i></a>, <a class="ref" href="finseq_2.html#T9" title="FINSEQ_2:th.9">FINSEQ_2:9</a></i>;<br/>
<a NAME="E19:85"/><b>then </b>
<font color="Maroon" title="c16">k</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"><font color="Maroon" title="c5">qr1</font>,<font color="Maroon" title="c11">w</font> <a href="fsm_1.html#K2" title="FSM_1:func.2">-admissible</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E20:85"/><b>then </b><i><font color="Green" title="E72">A12</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">qr1</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">k</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">rtfsm</font>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T13" title="FINSEQ_2:th.13">FINSEQ_2:13</a></i>;<br/>
<a NAME="E21:85"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">x</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>
 
<b>by </b><i><a class="ref" href="fsm_1.html#D6" target="_self" title="FSM_1:def.6">Def6</a></i>;<br/>
<a NAME="E22:85"/><b>then </b><i><font color="Green" title="E73">A13</font></i>: 
<font color="Maroon" title="c16">k</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"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">w</font></span>)</span>
 
<b>by </b><i><a class="txt" href="fsm_1.html#E15:85"><i><font color="Green" title="E69">A9</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E23:85"/><b>then </b>
<font color="Maroon" title="c16">k</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>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E24:85"/><b>then </b><i><font color="Green" title="E74">A14</font></i>: 
<font color="Maroon" title="c11">w</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">k</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">IAlph</font>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T13" title="FINSEQ_2:th.13">FINSEQ_2:13</a></i>;<br/>
<a NAME="E25:85"/>
<font color="Maroon" title="c16">k</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="fsm_1.html#E22:85"><i><font color="Green" title="E73">A13</font></i></a>, <a class="ref" href="finseq_2.html#T9" title="FINSEQ_2:th.9">FINSEQ_2:9</a></i>;<br/>
<a NAME="E26:85"/><b>then </b><i><font color="Green" title="E75">A15</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">x</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">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">qr1</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">k</font>
 
<b>by </b><i><a class="txt" href="fsm_1.html#E1:85"><i><font color="Green" title="E63">A1</font></i></a>, <a class="txt" href="fsm_1.html#E5:85"><i><font color="Green" title="E65">A3</font></i></a>, <a class="ref" href="fsm_1.html#T57" target="_self" title="FSM_1:th.57">Th57</a></i>;<br/>
<i><font color="Green" title="E76">A16</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">qr1</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="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#K1" title="NAT_1:func.1">+</a> 1
<b>by </b><i><a class="ref" href="fsm_1.html#D2" target="_self" title="FSM_1:def.2">Def2</a></i>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c11">w</font> <a href="fsm_1.html#K2" title="FSM_1:func.2">-admissible</a> </span>)</span>
<b>by </b><i><a class="ref" href="fsm_1.html#D2" target="_self" title="FSM_1:def.2">Def2</a></i>
;<br/>
<a NAME="E28:85"/><b>then </b>
<font color="Maroon" title="c16">k</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"><font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c11">w</font> <a href="fsm_1.html#K2" title="FSM_1:func.2">-admissible</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="fsm_1.html#E18:85"><i><font color="Green" title="E71">A11</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E29:85"/><b>then </b><i><font color="Green" title="E77">A17</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">x</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">k</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">tfsm</font>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T13" title="FINSEQ_2:th.13">FINSEQ_2:13</a></i>;<br/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">qr2</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="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#K1" title="NAT_1:func.1">+</a> 1
<b>by </b><i><a class="ref" href="fsm_1.html#D2" target="_self" title="FSM_1:def.2">Def2</a></i>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">qr1</font>,<font color="Maroon" title="c11">w</font> <a href="fsm_1.html#K2" title="FSM_1:func.2">-admissible</a> </span>)</span>
<b>by </b><i><a class="ref" href="fsm_1.html#D2" target="_self" title="FSM_1:def.2">Def2</a></i>
;<br/>
<a NAME="E31:85"/><b>then </b>
<font color="Maroon" title="c16">k</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"><font color="Maroon" title="c6">qr2</font>,<font color="Maroon" title="c11">w</font> <a href="fsm_1.html#K2" title="FSM_1:func.2">-admissible</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="fsm_1.html#E18:85"><i><font color="Green" title="E71">A11</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E32:85"/><b>then </b><i><font color="Green" title="E78">A18</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">qr2</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">k</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">rtfsm</font>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T13" title="FINSEQ_2:th.13">FINSEQ_2:13</a></i>;<br/>
<a NAME="E33:85"/>
<font color="Maroon" title="c16">k</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="fsm_1.html#E22:85"><i><font color="Green" title="E73">A13</font></i></a>, <a class="ref" href="finseq_2.html#T9" title="FINSEQ_2:th.9">FINSEQ_2:9</a></i>;<br/>
<a NAME="E34:85"/><b>then </b><i><font color="Green" title="E79">A19</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">y</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">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">qr2</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">k</font>
 
<b>by </b><i><a class="txt" href="fsm_1.html#E1:85"><i><font color="Green" title="E63">A1</font></i></a>, <a class="txt" href="fsm_1.html#E8:85"><i><font color="Green" title="E66">A4</font></i></a>, <a class="ref" href="fsm_1.html#T57" target="_self" title="FSM_1:th.57">Th57</a></i>;<br/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">y</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="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#K1" title="NAT_1:func.1">+</a> 1
<b>by </b><i><a class="ref" href="fsm_1.html#D2" target="_self" title="FSM_1:def.2">Def2</a></i>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c11">w</font> <a href="fsm_1.html#K2" title="FSM_1:func.2">-admissible</a> </span>)</span>
<b>by </b><i><a class="ref" href="fsm_1.html#D2" target="_self" title="FSM_1:def.2">Def2</a></i>
;<br/>
<a NAME="E36:85"/><b>then </b>
<font color="Maroon" title="c16">k</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"><font color="Maroon" title="c10">y</font>,<font color="Maroon" title="c11">w</font> <a href="fsm_1.html#K2" title="FSM_1:func.2">-admissible</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="fsm_1.html#E18:85"><i><font color="Green" title="E71">A11</font></i></a>, <a class="txt" href="fsm_1.html#E27:85"><i><font color="Green" title="E76">A16</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E37:85"/><b>then </b><i><font color="Green" title="E80">A20</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">y</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">k</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">tfsm</font>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T13" title="FINSEQ_2:th.13">FINSEQ_2:13</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="85_7"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:85_7"/><i><font color="Green" title="E81">A21</font></i>: 
<font color="Maroon" title="c5">qr1</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> <font color="Maroon" title="c6">qr2</font>,<font color="Maroon" title="c11">w</font> <a href="fsm_1.html#K4" title="FSM_1:func.4">-response</a> 
 ;<br/><a NAME="E2:85_7"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c11">w</font> <a href="fsm_1.html#K4" title="FSM_1:func.4">-response</a> </span>)</span>
 <b>by </b><i><a class="ref" href="fsm_1.html#D6" target="_self" title="FSM_1:def.6">Def6</a></i>;<br/><a NAME="E3:85_7"/><b>then </b><i><font color="Green" title="E82">A22</font></i>: 
<font color="Maroon" title="c16">k</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>
 <b>by </b><i><a class="txt" href="fsm_1.html#E15:85"><i><font color="Green" title="E69">A9</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>then </b><i><font color="Green" title="E83">A23</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c5">qr1</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">k</font> = 
the <a href="fsm_1.html#U3" title="FSM_1:sel.3">OFun</a> of <font color="Maroon" title="c3">rtfsm</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">qr1</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">k</font></span>)</span>,<span class="p1">(<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">k</font></span>)</span>
<b>by </b><i><a class="ref" href="fsm_1.html#D6" target="_self" title="FSM_1:def.6">Def6</a></i>
<br/>.= 
the <a href="fsm_1.html#U3" title="FSM_1:sel.3">OFun</a> of <font color="Maroon" title="c4">tfsm</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">x</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">k</font></span>)</span>,<span class="p1">(<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">k</font></span>)</span>
<b>by </b><i><a class="txt" href="fsm_1.html#E1:85"><i><font color="Green" title="E63">A1</font></i></a>, <a class="txt" href="fsm_1.html#E20:85"><i><font color="Green" title="E72">A12</font></i></a>, <a class="txt" href="fsm_1.html#E24:85"><i><font color="Green" title="E74">A14</font></i></a>, <a class="txt" href="fsm_1.html#E26:85"><i><font color="Green" title="E75">A15</font></i></a>, <a class="txt" href="fsm_1.html#E29:85"><i><font color="Green" title="E77">A17</font></i></a>, <a class="ref" href="fsm_1.html#D18" target="_self" title="FSM_1:def.18">Def18</a></i>
;<br/><i><font color="Green" title="E84">A24</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c6">qr2</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">k</font> = 
the <a href="fsm_1.html#U3" title="FSM_1:sel.3">OFun</a> of <font color="Maroon" title="c3">rtfsm</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">qr2</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">k</font></span>)</span>,<span class="p1">(<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">k</font></span>)</span>
<b>by </b><i><a class="txt" href="fsm_1.html#E3:85_7"><i><font color="Green" title="E82">A22</font></i></a>, <a class="ref" href="fsm_1.html#D6" target="_self" title="FSM_1:def.6">Def6</a></i>
<br/>.= 
the <a href="fsm_1.html#U3" title="FSM_1:sel.3">OFun</a> of <font color="Maroon" title="c4">tfsm</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">y</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">k</font></span>)</span>,<span class="p1">(<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">k</font></span>)</span>
<b>by </b><i><a class="txt" href="fsm_1.html#E1:85"><i><font color="Green" title="E63">A1</font></i></a>, <a class="txt" href="fsm_1.html#E24:85"><i><font color="Green" title="E74">A14</font></i></a>, <a class="txt" href="fsm_1.html#E32:85"><i><font color="Green" title="E78">A18</font></i></a>, <a class="txt" href="fsm_1.html#E34:85"><i><font color="Green" title="E79">A19</font></i></a>, <a class="txt" href="fsm_1.html#E37:85"><i><font color="Green" title="E80">A20</font></i></a>, <a class="ref" href="fsm_1.html#D18" target="_self" title="FSM_1:def.18">Def18</a></i>
;<br/><a NAME="E6:85_7"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">x</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">k</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="c4">tfsm</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">x</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">k</font></span>)</span>,<span class="p1">(<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">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="fsm_1.html#E3:85_7"><i><font color="Green" title="E82">A22</font></i></a>, <a class="ref" href="fsm_1.html#D6" target="_self" title="FSM_1:def.6">Def6</a></i>;<br/><b>hence </b><a NAME="E7:85_7"/>
contradiction
 <b>by </b><i><a class="txt" href="fsm_1.html#E15:85"><i><font color="Green" title="E69">A9</font></i></a>, <a class="txt" href="fsm_1.html#E1:85_7"><i><font color="Green" title="E81">A21</font></i></a>, <a class="txt" href="fsm_1.html#E3:85_7"><i><font color="Green" title="E82">A22</font></i></a>, <a class="txt" href="fsm_1.html#E4:85_7"><i><font color="Green" title="E83">A23</font></i></a>, <a class="txt" href="fsm_1.html#E5:85_7"><i><font color="Green" title="E84">A24</font></i></a>, <a class="ref" href="fsm_1.html#D6" target="_self" title="FSM_1:def.6">Def6</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E39:85"/>
not <font color="Maroon" title="c5">qr1</font>,<font color="Maroon" title="c6">qr2</font> <a href="fsm_1.html#R5" title="FSM_1:pred.5">-are_equivalent</a> 
 <b>by </b><i><a class="ref" href="fsm_1.html#D10" target="_self" title="FSM_1:def.10">Def10</a></i>;<br/>


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