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<span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">IAlph</font>, <font color="Olive" title="b2">OAlph</font> being   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/>  for <font color="Olive" title="b3">tfsm</font> being   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="fsm_1.html#L2" title="FSM_1:struct.2">Mealy-FSM</a> of <font color="Olive" title="b1">IAlph</font>,<font color="Olive" title="b2">OAlph</font>  st <font color="Maroon" title="c1">k</font> <a href="fsm_1.html#K7" title="FSM_1:func.7">-eq_states_partition</a> <font color="Olive" title="b3">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#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="fsm_1.html#K7" title="FSM_1:func.7">-eq_states_partition</a> <font color="Olive" title="b3">tfsm</font> holds <br/><font color="Maroon" title="c1">k</font> <a href="fsm_1.html#K7" title="FSM_1:func.7">-eq_states_partition</a> <font color="Olive" title="b3">tfsm</font> is  <a href="fsm_1.html#V4" title="FSM_1:attr.4">final</a> </span><br/><span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">tfsm</font> being   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="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>  st <font color="Maroon" title="c1">k</font> <a href="fsm_1.html#K7" title="FSM_1:func.7">-eq_states_partition</a> <font color="Olive" title="b1">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#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="fsm_1.html#K7" title="FSM_1:func.7">-eq_states_partition</a> <font color="Olive" title="b1">tfsm</font> holds <br/><font color="Maroon" title="c1">k</font> <a href="fsm_1.html#K7" title="FSM_1:func.7">-eq_states_partition</a> <font color="Olive" title="b1">tfsm</font> is  <a href="fsm_1.html#V4" title="FSM_1:attr.4">final</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c4">tfsm</font> be   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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#K2" title="NAT_1:func.2">+</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> implies <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>  )</span><br/>







<span class="kw">set </span><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/>
<span class="kw">set </span><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/>
<span class="kw">set </span><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#K2" title="NAT_1:func.2">+</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/>
<span class="kw">set </span><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/>
<span class="kw">assume </span><a NAME="E1:58"/><span class="lab"><font color="Green" title="E41">A1</font></span>: 
<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#K2" title="NAT_1:func.2">+</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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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> </span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c9">qa</font>, <font color="Maroon" title="c10">qb</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">tfsm</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c9">qa</font>,<font color="Maroon" title="c10">qb</font> <a href="fsm_1.html#R5" title="FSM_1:pred.5">-are_equivalent</a>  implies  ex <font color="Olive" title="b1">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> st <br/>( <font color="Maroon" title="c9">qa</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">X</font> &amp; <font color="Maroon" title="c10">qb</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">X</font> ) ) &amp; (  ex <font color="Olive" title="b1">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> st <br/>( <font color="Maroon" title="c9">qa</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">X</font> &amp; <font color="Maroon" title="c10">qb</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">X</font> ) implies <font color="Maroon" title="c9">qa</font>,<font color="Maroon" title="c10">qb</font> <a href="fsm_1.html#R5" title="FSM_1:pred.5">-are_equivalent</a>  ) )</span><br/><div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  ex <font color="Olive" title="b1">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> st <br/>( <font color="Maroon" title="c9">qa</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">X</font> &amp; <font color="Maroon" title="c10">qb</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">X</font> ) implies <font color="Maroon" title="c9">qa</font>,<font color="Maroon" title="c10">qb</font> <a href="fsm_1.html#R5" title="FSM_1:pred.5">-are_equivalent</a>  )</span>
<div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c11">X</font> =  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">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="c10">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> <span class="kw">by</span> <span class="lab"><a class="ref" href="eqrel_1.html#D3" title="EQREL_1:def.3">EQREL_1:def 3</a></span>;<br/><span class="kw">assume </span><a NAME="E2:58_1_1"/><span class="lab"><font color="Green" title="E42">A2</font></span>: 
<font color="Maroon" title="c9">qa</font>,<font color="Maroon" title="c10">qb</font> <a href="fsm_1.html#R5" title="FSM_1:pred.5">-are_equivalent</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">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> st <br/>( <font color="Maroon" title="c9">qa</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">X</font> &amp; <font color="Maroon" title="c10">qb</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">X</font> )</span><br/><span class="kw">take </span><font color="Maroon" title="c12">X</font> = <font color="Maroon" title="c11">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">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="c10">qb</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">X</font> )</span><br/><a NAME="E3:58_1_1"/>
<font color="Maroon" title="c1">k</font> <a href="fsm_1.html#R6" title="FSM_1:pred.6">-equivalent</a> <font color="Maroon" title="c9">qa</font>,<font color="Maroon" title="c10">qb</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:58_1_1"><span class="lab"><font color="Green" title="E42">A2</font></span></a>, <a class="ref" href="fsm_1.html#T24" target="_self" title="FSM_1:th.24">Th41</a></span>;<br/><a NAME="E4:58_1_1"/><span class="kw">then </span>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c9">qa</font>,<font color="Maroon" title="c10">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fsm_1.html#D12" target="_self" title="FSM_1:def.12">Def12</a></span>;<br/><span class="kw">hence </span><a NAME="E5:58_1_1"/>
( <font color="Maroon" title="c9">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="c10">qb</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">X</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="eqrel_1.html#T19" title="EQREL_1:th.19">EQREL_1:19</a>, <a class="ref" href="eqrel_1.html#T20" title="EQREL_1:th.20">EQREL_1:20</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div><span class="kw">given </span><font color="Maroon" title="c11">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><span class="kw"> such that </span><a NAME="E3:58_1"/><span class="lab"><font color="Green" title="E42">A3</font></span>: 
<font color="Maroon" title="c9">qa</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">X</font>
 <span class="kw">and </span><br/><a NAME="E4:58_1"/><span class="lab"><font color="Green" title="E43">A4</font></span>: 
<font color="Maroon" title="c10">qb</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">qa</font>,<font color="Maroon" title="c10">qb</font> <a href="fsm_1.html#R5" title="FSM_1:pred.5">-are_equivalent</a> </span><br/><span class="kw">consider </span><font color="Maroon" title="c12">qc</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E6:58_1"/>
<font color="Maroon" title="c12">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>
 <span class="kw">and </span><br/><a NAME="E7:58_1"/><span class="lab"><font color="Green" title="E44">A5</font></span>: 
<font color="Maroon" title="c11">X</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="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="c12">qc</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="eqrel_1.html#D3" title="EQREL_1:def.3">EQREL_1:def 3</a></span>;<br/><a NAME="E8:58_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">qb</font>,<font color="Maroon" title="c12">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:58_1"><span class="lab"><font color="Green" title="E43">A4</font></span></a>, <a class="txt" href="#E7:58_1"><span class="lab"><font color="Green" title="E44">A5</font></span></a>, <a class="ref" href="eqrel_1.html#T19" title="EQREL_1:th.19">EQREL_1:19</a></span>;<br/><a NAME="E9:58_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A6</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">qc</font>,<font color="Maroon" title="c10">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="eqrel_1.html#T6" title="EQREL_1:th.6">EQREL_1:6</a></span>;<br/><a NAME="E10:58_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">qa</font>,<font color="Maroon" title="c12">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:58_1"><span class="lab"><font color="Green" title="E42">A3</font></span></a>, <a class="txt" href="#E7:58_1"><span class="lab"><font color="Green" title="E44">A5</font></span></a>, <a class="ref" href="eqrel_1.html#T19" title="EQREL_1:th.19">EQREL_1:19</a></span>;<br/><a NAME="E11:58_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A7</font></span>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c9">qa</font>,<font color="Maroon" title="c10">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:58_1"><span class="lab"><font color="Green" title="E45">A6</font></span></a>, <a class="ref" href="eqrel_1.html#T7" title="EQREL_1:th.7">EQREL_1:7</a></span>;<br/><a NAME="E12:58_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A8</font></span>: 
<font color="Maroon" title="c1">k</font> <a href="fsm_1.html#R6" title="FSM_1:pred.6">-equivalent</a> <font color="Maroon" title="c9">qa</font>,<font color="Maroon" title="c10">qb</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fsm_1.html#D12" target="_self" title="FSM_1:def.12">Def12</a></span>;<br/><span class="kw">thus </span><a NAME="E13:58_1"/>
<font color="Maroon" title="c9">qa</font>,<font color="Maroon" title="c10">qb</font> <a href="fsm_1.html#R5" title="FSM_1:pred.5">-are_equivalent</a> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c13">w</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c2">IAlph</font>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="fsm_1.html#D10" target="_self" title="FSM_1:def.10">FSM_1:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (<font color="Maroon" title="c9">qa</font>,<font color="Maroon" title="c13">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="c10">qb</font>,<font color="Maroon" title="c13">w</font>) <a href="fsm_1.html#K4" title="FSM_1:func.4">-response</a> </span><br/>

<span class="kw">consider </span><font color="Maroon" title="c14">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E2:58_1_2"/><span class="lab"><font color="Green" title="E48">A9</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c13">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="c14">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D2" title="FINSEQ_1:def.2">FINSEQ_1:def 2</a></span>;<br/>
<a NAME="E3:58_1_2"/>
<font color="Maroon" title="c14">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/>
<a NAME="E4:58_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A10</font></span>: 
<font color="Maroon" title="c14">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="c13">w</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:58_1_2"><span class="lab"><font color="Green" title="E48">A9</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:58_1_2_1"/>
( <font color="Maroon" title="c14">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="c14">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="58_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:58_1_2_1_1"/>
<font color="Maroon" title="c14">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (<font color="Maroon" title="c9">qa</font>,<font color="Maroon" title="c13">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="c10">qb</font>,<font color="Maroon" title="c13">w</font>) <a href="fsm_1.html#K4" title="FSM_1:func.4">-response</a> </span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:58_1_2_1_1"/>
(<font color="Maroon" title="c9">qa</font>,<font color="Maroon" title="c13">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="c10">qb</font>,<font color="Maroon" title="c13">w</font>) <a href="fsm_1.html#K4" title="FSM_1:func.4">-response</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:58_1"><span class="lab"><font color="Green" title="E47">A8</font></span></a>, <a class="txt" href="#E4:58_1_2"><span class="lab"><font color="Green" title="E49">A10</font></span></a>, <a class="ref" href="fsm_1.html#D11" target="_self" title="FSM_1:def.11">Def11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:58_1_2_1_2"/>
<font color="Maroon" title="c14">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (<font color="Maroon" title="c9">qa</font>,<font color="Maroon" title="c13">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="c10">qb</font>,<font color="Maroon" title="c13">w</font>) <a href="fsm_1.html#K4" title="FSM_1:func.4">-response</a> </span><br/><div class="add"><a NAME="E2:58_1_2_1_2"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">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>  st <br/>( <font color="Maroon" title="c14">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">m</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_4.html#T84" title="FINSEQ_4:th.84">FINSEQ_4:84</a></span>;<br/><a NAME="E3:58_1_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c14">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:58"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="ref" href="fsm_1.html#T29" target="_self" title="FSM_1:th.29">Th46</a></span>;<br/><a NAME="E4:58_1_2_1_2"/><span class="kw">then </span>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c9">qa</font>,<font color="Maroon" title="c10">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="c14">n</font> <a href="fsm_1.html#K6" title="FSM_1:func.6">-eq_states_EqR</a> <font color="Maroon" title="c4">tfsm</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:58_1"><span class="lab"><font color="Green" title="E46">A7</font></span></a>, <a class="ref" href="finseq_4.html#T86" title="FINSEQ_4:th.86">FINSEQ_4:86</a></span>;<br/><a NAME="E5:58_1_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c14">n</font> <a href="fsm_1.html#R6" title="FSM_1:pred.6">-equivalent</a> <font color="Maroon" title="c9">qa</font>,<font color="Maroon" title="c10">qb</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fsm_1.html#D12" target="_self" title="FSM_1:def.12">Def12</a></span>;<br/><span class="kw">hence </span><a NAME="E6:58_1_2_1_2"/>
(<font color="Maroon" title="c9">qa</font>,<font color="Maroon" title="c13">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="c10">qb</font>,<font color="Maroon" title="c13">w</font>) <a href="fsm_1.html#K4" title="FSM_1:func.4">-response</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:58_1_2"><span class="lab"><font color="Green" title="E49">A10</font></span></a>, <a class="ref" href="fsm_1.html#D11" target="_self" title="FSM_1:def.11">Def11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:58"/>
<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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fsm_1.html#D14" target="_self" title="FSM_1:def.14">Def14</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
