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<b>let </b><font color="Maroon" title="c1">k</font> be    <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> ;<br/><b>let </b><font color="Maroon" title="c2">IAlph</font>, <font color="Maroon" title="c3">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="c4">tfsm</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</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="c3">OAlph</font>;<br/>







<b>set </b><font color="Maroon" title="c5">S</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">tfsm</font>;<br/>
<b>set </b><font color="Maroon" title="c6">kpart</font> = <font color="Maroon" title="c1">k</font> <a href="fsm_1.html#K7" title="FSM_1:func.7">-eq_states_partition</a> <font color="Maroon" title="c4">tfsm</font>;<br/>
<b>set </b><font color="Maroon" title="c7">k1part</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="fsm_1.html#K7" title="FSM_1:func.7">-eq_states_partition</a> <font color="Maroon" title="c4">tfsm</font>;<br/>
<b>set </b><font color="Maroon" title="c8">keq</font> = <font color="Maroon" title="c1">k</font> <a href="fsm_1.html#K6" title="FSM_1:func.6">-eq_states_EqR</a> <font color="Maroon" title="c4">tfsm</font>;<br/>
<b>set </b><font color="Maroon" title="c9">k1eq</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="fsm_1.html#K6" title="FSM_1:func.6">-eq_states_EqR</a> <font color="Maroon" title="c4">tfsm</font>;<br/>
<b>assume </b><a NAME="E1:71"/><i><font color="Green" title="E51">A1</font></i>: 
<font color="Maroon" title="c1">k</font> <a href="fsm_1.html#K7" title="FSM_1:func.7">-eq_states_partition</a> <font color="Maroon" title="c4">tfsm</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="fsm_1.html#K7" title="FSM_1:func.7">-eq_states_partition</a> <font color="Maroon" title="c4">tfsm</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c10">qa</font>, <font color="Maroon" title="c11">qb</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">tfsm</font>;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:71_1_1"/>
<font color="Maroon" title="c10">qa</font>,<font color="Maroon" title="c11">qb</font> <a href="fsm_1.html#R5" title="FSM_1:pred.5">-are_equivalent</a> 
 ;<br/><a NAME="E2:71_1_1"/><b>then </b>
<font color="Maroon" title="c1">k</font> <a href="fsm_1.html#R6" title="FSM_1:pred.6">-equivalent</a> <font color="Maroon" title="c10">qa</font>,<font color="Maroon" title="c11">qb</font>
 <b>by </b><i><a class="ref" href="fsm_1.html#T41" target="_self" title="FSM_1:th.41">Th41</a></i>;<br/><a NAME="E3:71_1_1"/><b>then </b><i><font color="Green" title="E52">A2</font></i>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c10">qa</font>,<font color="Maroon" title="c11">qb</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">k</font> <a href="fsm_1.html#K6" title="FSM_1:func.6">-eq_states_EqR</a> <font color="Maroon" title="c4">tfsm</font>
 <b>by </b><i><a class="ref" href="fsm_1.html#D12" target="_self" title="FSM_1:def.12">Def12</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c12">X</font> =  <a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="fsm_1.html#K6" title="FSM_1:func.6">-eq_states_EqR</a> <font color="Maroon" title="c4">tfsm</font></span>)</span>,<font color="Maroon" title="c11">qb</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c1">k</font> <a href="fsm_1.html#K7" title="FSM_1:func.7">-eq_states_partition</a> <font color="Maroon" title="c4">tfsm</font> <b>by </b><i><a class="ref" href="eqrel_1.html#D5" title="EQREL_1:def.5">EQREL_1:def 5</a></i>;<br/><b>take </b><font color="Maroon" title="c13">X</font> = <font color="Maroon" title="c12">X</font>;<br/><b>thus </b><a NAME="E5:71_1_1"/>
( <font color="Maroon" title="c10">qa</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">X</font> &amp; <font color="Maroon" title="c11">qb</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">X</font> )
 <b>by </b><i><a class="txt" href="fsm_1.html#E3:71_1_1"><i><font color="Green" title="E52">A2</font></i></a>, <a class="ref" href="eqrel_1.html#T27" title="EQREL_1:th.27">EQREL_1:27</a>, <a class="ref" href="eqrel_1.html#T28" title="EQREL_1:th.28">EQREL_1:28</a></i>;<br/></div>
<b>end;</b></div><b>given </b><font color="Maroon" title="c12">X</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c1">k</font> <a href="fsm_1.html#K7" title="FSM_1:func.7">-eq_states_partition</a> <font color="Maroon" title="c4">tfsm</font><b> such that </b><a NAME="E2:71_1"/><i><font color="Green" title="E52">A3</font></i>: 
( <font color="Maroon" title="c10">qa</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">X</font> &amp; <font color="Maroon" title="c11">qb</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">X</font> )
 ;<br/><b>consider </b><font color="Maroon" title="c13">qc</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:71_1"/><i><font color="Green" title="E53">A4</font></i>: 
( <font color="Maroon" title="c13">qc</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">tfsm</font> &amp; <font color="Maroon" title="c12">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="fsm_1.html#K6" title="FSM_1:func.6">-eq_states_EqR</a> <font color="Maroon" title="c4">tfsm</font></span>)</span>,<font color="Maroon" title="c13">qc</font> )
 <b>by </b><i><a class="ref" href="eqrel_1.html#D5" title="EQREL_1:def.5">EQREL_1:def 5</a></i>;<br/><a NAME="E5:71_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">qb</font>,<font color="Maroon" title="c13">qc</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">k</font> <a href="fsm_1.html#K6" title="FSM_1:func.6">-eq_states_EqR</a> <font color="Maroon" title="c4">tfsm</font>
 <b>by </b><i><a class="txt" href="fsm_1.html#E2:71_1"><i><font color="Green" title="E52">A3</font></i></a>, <a class="txt" href="fsm_1.html#E4:71_1"><i><font color="Green" title="E53">A4</font></i></a>, <a class="ref" href="eqrel_1.html#T27" title="EQREL_1:th.27">EQREL_1:27</a></i>;<br/><a NAME="E6:71_1"/><b>then </b><i><font color="Green" title="E54">A5</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">qc</font>,<font color="Maroon" title="c11">qb</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">k</font> <a href="fsm_1.html#K6" title="FSM_1:func.6">-eq_states_EqR</a> <font color="Maroon" title="c4">tfsm</font>
 <b>by </b><i><a class="ref" href="eqrel_1.html#T12" title="EQREL_1:th.12">EQREL_1:12</a></i>;<br/><a NAME="E7:71_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">qa</font>,<font color="Maroon" title="c13">qc</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">k</font> <a href="fsm_1.html#K6" title="FSM_1:func.6">-eq_states_EqR</a> <font color="Maroon" title="c4">tfsm</font>
 <b>by </b><i><a class="txt" href="fsm_1.html#E2:71_1"><i><font color="Green" title="E52">A3</font></i></a>, <a class="txt" href="fsm_1.html#E4:71_1"><i><font color="Green" title="E53">A4</font></i></a>, <a class="ref" href="eqrel_1.html#T27" title="EQREL_1:th.27">EQREL_1:27</a></i>;<br/><a NAME="E8:71_1"/><b>then </b><i><font color="Green" title="E55">A6</font></i>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c10">qa</font>,<font color="Maroon" title="c11">qb</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">k</font> <a href="fsm_1.html#K6" title="FSM_1:func.6">-eq_states_EqR</a> <font color="Maroon" title="c4">tfsm</font>
 <b>by </b><i><a class="txt" href="fsm_1.html#E6:71_1"><i><font color="Green" title="E54">A5</font></i></a>, <a class="ref" href="eqrel_1.html#T13" title="EQREL_1:th.13">EQREL_1:13</a></i>;<br/><a NAME="E9:71_1"/><b>then </b><i><font color="Green" title="E56">A7</font></i>: 
<font color="Maroon" title="c1">k</font> <a href="fsm_1.html#R6" title="FSM_1:pred.6">-equivalent</a> <font color="Maroon" title="c10">qa</font>,<font color="Maroon" title="c11">qb</font>
 <b>by </b><i><a class="ref" href="fsm_1.html#D12" target="_self" title="FSM_1:def.12">Def12</a></i>;<br/><b>thus </b><a NAME="E10:71_1"/>
<font color="Maroon" title="c10">qa</font>,<font color="Maroon" title="c11">qb</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="71_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">w</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c2">IAlph</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="fsm_1.html#D10" target="_self" title="FSM_1:def.10">FSM_1:def 10</a><br/>

<b>consider </b><font color="Maroon" title="c15">n</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="E2:71_1_2"/><i><font color="Green" title="E57">A8</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c14">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c15">n</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D2" title="FINSEQ_1:def.2">FINSEQ_1:def 2</a></i>;<br/>
<a NAME="E3:71_1_2"/><i><font color="Green" title="E58">A9</font></i>: 
<font color="Maroon" title="c15">n</font> <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="c14">w</font>
 
<b>by </b><i><a class="txt" href="fsm_1.html#E2:71_1_2"><i><font color="Green" title="E57">A8</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="71_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:71_1_2_1"/>
( <font color="Maroon" title="c15">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">k</font> or <font color="Maroon" title="c15">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">k</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="71_1_2_1_1"><b>suppose </b></a><a NAME="E1:71_1_2_1_1"/>
<font color="Maroon" title="c15">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">k</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:71_1_2_1_1"/>
<font color="Maroon" title="c10">qa</font>,<font color="Maroon" title="c14">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="c11">qb</font>,<font color="Maroon" title="c14">w</font> <a href="fsm_1.html#K4" title="FSM_1:func.4">-response</a> 
 <b>by </b><i><a class="txt" href="fsm_1.html#E9:71_1"><i><font color="Green" title="E56">A7</font></i></a>, <a class="txt" href="fsm_1.html#E3:71_1_2"><i><font color="Green" title="E58">A9</font></i></a>, <a class="ref" href="fsm_1.html#D11" target="_self" title="FSM_1:def.11">Def11</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="71_1_2_1_2"><b>suppose </b></a><a NAME="E1:71_1_2_1_2"/>
<font color="Maroon" title="c15">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">k</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c16">m</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="E3:71_1_2_1_2"/><i><font color="Green" title="E59">A10</font></i>: 
( <font color="Maroon" title="c15">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c16">m</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">m</font> )
 <b>by </b><i><a class="ref" href="fsm_1.html#T1" target="_self" title="FSM_1:th.1">Th1</a></i>;<br/><a NAME="E4:71_1_2_1_2"/><i><font color="Green" title="E60">A11</font></i>: 
<font color="Maroon" title="c15">n</font> <a href="fsm_1.html#K7" title="FSM_1:func.7">-eq_states_partition</a> <font color="Maroon" title="c4">tfsm</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="fsm_1.html#K7" title="FSM_1:func.7">-eq_states_partition</a> <font color="Maroon" title="c4">tfsm</font>
 <b>by </b><i><a class="txt" href="fsm_1.html#E1:71"><i><font color="Green" title="E51">A1</font></i></a>, <a class="txt" href="fsm_1.html#E3:71_1_2_1_2"><i><font color="Green" title="E59">A10</font></i></a>, <a class="ref" href="fsm_1.html#T46" target="_self" title="FSM_1:th.46">Th46</a></i>;<br/><a NAME="E5:71_1_2_1_2"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c10">qa</font>,<font color="Maroon" title="c11">qb</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">n</font> <a href="fsm_1.html#K6" title="FSM_1:func.6">-eq_states_EqR</a> <font color="Maroon" title="c4">tfsm</font>
 <b>by </b><i><a class="txt" href="fsm_1.html#E8:71_1"><i><font color="Green" title="E55">A6</font></i></a>, <a class="ref" href="fsm_1.html#T8" target="_self" title="FSM_1:th.8">Th8</a>, <a class="txt" href="fsm_1.html#E4:71_1_2_1_2"><i><font color="Green" title="E60">A11</font></i></a></i>;<br/><a NAME="E6:71_1_2_1_2"/><b>then </b>
<font color="Maroon" title="c15">n</font> <a href="fsm_1.html#R6" title="FSM_1:pred.6">-equivalent</a> <font color="Maroon" title="c10">qa</font>,<font color="Maroon" title="c11">qb</font>
 <b>by </b><i><a class="ref" href="fsm_1.html#D12" target="_self" title="FSM_1:def.12">Def12</a></i>;<br/><b>hence </b><a NAME="E7:71_1_2_1_2"/>
<font color="Maroon" title="c10">qa</font>,<font color="Maroon" title="c14">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="c11">qb</font>,<font color="Maroon" title="c14">w</font> <a href="fsm_1.html#K4" title="FSM_1:func.4">-response</a> 
 <b>by </b><i><a class="txt" href="fsm_1.html#E3:71_1_2"><i><font color="Green" title="E58">A9</font></i></a>, <a class="ref" href="fsm_1.html#D11" target="_self" title="FSM_1:def.11">Def11</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:71"/>
<font color="Maroon" title="c1">k</font> <a href="fsm_1.html#K7" title="FSM_1:func.7">-eq_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#D14" target="_self" title="FSM_1:def.14">Def14</a></i>;<br/>


</div>
