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<b>let </b><font color="Maroon" title="c6">it1</font>, <font color="Maroon" title="c7">it2</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="fsm_1.html#K8" title="FSM_1:func.8">final_states_partition</a> <font color="Maroon" title="c3">tfsm</font>;<br/>

<b>given </b><font color="Maroon" title="c8">q1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">tfsm</font>, <font color="Maroon" title="c9">n1</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><a NAME="E2:73_1_2"/><i><font color="Green" title="E52">A4</font></i>: 
( <font color="Maroon" title="c8">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">qf</font> &amp; <font color="Maroon" title="c9">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">tfsm</font> &amp; <font color="Maroon" title="c6">it1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n1</font> <a href="fsm_1.html#K6" title="FSM_1:func.6">-eq_states_EqR</a> <font color="Maroon" title="c3">tfsm</font></span>)</span>,<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="c3">tfsm</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c8">q1</font>,<font color="Maroon" title="c5">s</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> )
 ;<br/>

<b>given </b><font color="Maroon" title="c10">q2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">tfsm</font>, <font color="Maroon" title="c11">n2</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><a NAME="E4:73_1_2"/><i><font color="Green" title="E53">A5</font></i>: 
( <font color="Maroon" title="c10">q2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">qf</font> &amp; <font color="Maroon" title="c11">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">tfsm</font> &amp; <font color="Maroon" title="c7">it2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">n2</font> <a href="fsm_1.html#K6" title="FSM_1:func.6">-eq_states_EqR</a> <font color="Maroon" title="c3">tfsm</font></span>)</span>,<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="c3">tfsm</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c10">q2</font>,<font color="Maroon" title="c5">s</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> )
 ;<br/>

<b>set </b><font color="Maroon" title="c12">m</font> = <font color="Maroon" title="c9">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1;<br/>
<a NAME="E5:73_1_2"/><i><font color="Green" title="E54">A6</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="int_1.html#T19" title="INT_1:th.19">INT_1:19</a></i>;<br/>
<a NAME="E6:73_1_2"/><i><font color="Green" title="E55">A7</font></i>: 
<font color="Maroon" title="c9">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 
;<br/>
<b>set </b><font color="Maroon" title="c13">q1'</font> = the <a href="fsm_1.html#U1" title="FSM_1:sel.1">Tran</a> of <font color="Maroon" title="c3">tfsm</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c8">q1</font>,<font color="Maroon" title="c5">s</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>;<br/>
<b>set </b><font color="Maroon" title="c14">q2'</font> = the <a href="fsm_1.html#U1" title="FSM_1:sel.1">Tran</a> of <font color="Maroon" title="c3">tfsm</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c10">q2</font>,<font color="Maroon" title="c5">s</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>;<br/>
<a NAME="E7:73_1_2"/>
 <a href="fsm_1.html#K8" title="FSM_1:func.8">final_states_partition</a> <font color="Maroon" title="c3">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="E8:73_1_2"/><b>then </b>
<font color="Maroon" title="c8">q1</font>,<font color="Maroon" title="c10">q2</font> <a href="fsm_1.html#R5" title="FSM_1:pred.5">-are_equivalent</a> 
 
<b>by </b><i><a class="txt" href="fsm_1.html#E2:73_1_2"><i><font color="Green" title="E52">A4</font></i></a>, <a class="txt" href="fsm_1.html#E4:73_1_2"><i><font color="Green" title="E53">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="E9:73_1_2"/><b>then </b>
<font color="Maroon" title="c9">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="fsm_1.html#R6" title="FSM_1:pred.6">-equivalent</a> <font color="Maroon" title="c8">q1</font>,<font color="Maroon" title="c10">q2</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="E10:73_1_2"/><b>then </b>
<font color="Maroon" title="c9">n1</font> <a href="fsm_1.html#R6" title="FSM_1:pred.6">-equivalent</a> the <a href="fsm_1.html#U1" title="FSM_1:sel.1">Tran</a> of <font color="Maroon" title="c3">tfsm</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c8">q1</font>,<font color="Maroon" title="c5">s</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,the <a href="fsm_1.html#U1" title="FSM_1:sel.1">Tran</a> of <font color="Maroon" title="c3">tfsm</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c10">q2</font>,<font color="Maroon" title="c5">s</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 
<b>by </b><i><a class="txt" href="fsm_1.html#E5:73_1_2"><i><font color="Green" title="E54">A6</font></i></a>, <a class="txt" href="fsm_1.html#E6:73_1_2"><i><font color="Green" title="E55">A7</font></i></a>, <a class="ref" href="fsm_1.html#T44" target="_self" title="FSM_1:th.44">Th44</a></i>;<br/>
<a NAME="E11:73_1_2"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default">the <a href="fsm_1.html#U1" title="FSM_1:sel.1">Tran</a> of <font color="Maroon" title="c3">tfsm</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c8">q1</font>,<font color="Maroon" title="c5">s</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span>,<span class="p2">(<span class="default">the <a href="fsm_1.html#U1" title="FSM_1:sel.1">Tran</a> of <font color="Maroon" title="c3">tfsm</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c10">q2</font>,<font color="Maroon" title="c5">s</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span></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="c9">n1</font> <a href="fsm_1.html#K6" title="FSM_1:func.6">-eq_states_EqR</a> <font color="Maroon" title="c3">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>hence </b><a NAME="E12:73_1_2"/>
<font color="Maroon" title="c6">it1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">it2</font>
 <b>by </b><i><a class="txt" href="fsm_1.html#E2:73_1_2"><i><font color="Green" title="E52">A4</font></i></a>, <a class="txt" href="fsm_1.html#E4:73_1_2"><i><font color="Green" title="E53">A5</font></i></a>, <a class="ref" href="eqrel_1.html#T44" title="EQREL_1:th.44">EQREL_1:44</a></i>;<br/>


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