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<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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">tfsm1</font>, <font color="Olive" title="b2">tfsm2</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="c1">IAlph</font>,<font color="Maroon" title="c2">OAlph</font>  st <font color="Olive" title="b1">tfsm1</font>,<font color="Olive" title="b2">tfsm2</font> <a href="fsm_1.html#R7" title="FSM_1:pred.7">-are_isomorphic</a>  holds <br/><font color="Olive" title="b1">tfsm1</font>,<font color="Olive" title="b2">tfsm2</font> <a href="fsm_1.html#R4" title="FSM_1:pred.4">-are_equivalent</a> </span><br/><b>let </b><font color="Maroon" title="c3">tfsm1</font>, <font color="Maroon" title="c4">tfsm2</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="c1">IAlph</font>,<font color="Maroon" title="c2">OAlph</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">tfsm1</font>,<font color="Maroon" title="c4">tfsm2</font> <a href="fsm_1.html#R7" title="FSM_1:pred.7">-are_isomorphic</a>  implies <font color="Maroon" title="c3">tfsm1</font>,<font color="Maroon" title="c4">tfsm2</font> <a href="fsm_1.html#R4" title="FSM_1:pred.4">-are_equivalent</a>  )</span><br/>







<b>assume </b><a NAME="E1:112"/>
<font color="Maroon" title="c3">tfsm1</font>,<font color="Maroon" title="c4">tfsm2</font> <a href="fsm_1.html#R7" title="FSM_1:pred.7">-are_isomorphic</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">tfsm1</font>,<font color="Maroon" title="c4">tfsm2</font> <a href="fsm_1.html#R4" title="FSM_1:pred.4">-are_equivalent</a> </span><br/>

<b>then consider </b><font color="Maroon" title="c5">Tf</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">tfsm1</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">tfsm2</font><b> such that </b><br/><a NAME="E3:112"/><i><font color="Green" title="E84">A1</font></i>: 
( <font color="Maroon" title="c5">Tf</font> is <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a> &amp; <font color="Maroon" title="c5">Tf</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c3">tfsm1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c4">tfsm2</font> &amp; ( for <font color="Olive" title="b1">q</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">tfsm1</font><br/> for <font color="Olive" title="b2">s</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">IAlph</font> holds <br/> ( <font color="Maroon" title="c5">Tf</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default">the <a href="fsm_1.html#U1" title="FSM_1:sel.1">Tran</a> of <font color="Maroon" title="c3">tfsm1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">q</font>,<font color="Olive" title="b2">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="fsm_1.html#U1" title="FSM_1:sel.1">Tran</a> of <font color="Maroon" title="c4">tfsm2</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Tf</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">q</font></span>)</span>,<font color="Olive" title="b2">s</font> &amp; the <a href="fsm_1.html#U3" title="FSM_1:sel.3">OFun</a> of <font color="Maroon" title="c3">tfsm1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">q</font>,<font color="Olive" title="b2">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="fsm_1.html#U3" title="FSM_1:sel.3">OFun</a> of <font color="Maroon" title="c4">tfsm2</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Tf</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">q</font></span>)</span>,<font color="Olive" title="b2">s</font> ) ) )
 <b>by </b><i><a class="ref" href="fsm_1.html#D19" target="_self" title="FSM_1:def.19">Def19</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="112_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">w1</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">IAlph</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c3">tfsm1</font>,<font color="Maroon" title="c6">w1</font> <a href="fsm_1.html#K4" title="FSM_1:func.4">-response</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c4">tfsm2</font>,<font color="Maroon" title="c6">w1</font> <a href="fsm_1.html#K4" title="FSM_1:func.4">-response</a> </span><br/><b>set </b><font color="Maroon" title="c7">rw1</font> = the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c3">tfsm1</font>,<font color="Maroon" title="c6">w1</font> <a href="fsm_1.html#K4" title="FSM_1:func.4">-response</a> ;<br/><b>set </b><font color="Maroon" title="c8">rw2</font> = the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c4">tfsm2</font>,<font color="Maroon" title="c6">w1</font> <a href="fsm_1.html#K4" title="FSM_1:func.4">-response</a> ;<br/><b>set </b><font color="Maroon" title="c9">aw1</font> = the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c3">tfsm1</font>,<font color="Maroon" title="c6">w1</font> <a href="fsm_1.html#K2" title="FSM_1:func.2">-admissible</a> ;<br/><b>set </b><font color="Maroon" title="c10">aw2</font> = the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c4">tfsm2</font>,<font color="Maroon" title="c6">w1</font> <a href="fsm_1.html#K2" title="FSM_1:func.2">-admissible</a> ;<br/><a NAME="E1:112_1"/><i><font color="Green" title="E85">A2</font></i>: 
for <font color="Olive" title="b1">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>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">w1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 holds <br/><font color="Maroon" title="c5">Tf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c3">tfsm1</font>,<font color="Maroon" title="c6">w1</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="Olive" title="b1">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c4">tfsm2</font>,<font color="Maroon" title="c6">w1</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="Olive" title="b1">k</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="112_1_1"><b>proof </b></a><div class="add">

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <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>means </b>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> $1 &amp; $1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">w1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies <font color="Maroon" title="c5">Tf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c3">tfsm1</font>,<font color="Maroon" title="c6">w1</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> $1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c4">tfsm2</font>,<font color="Maroon" title="c6">w1</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> $1 );<br/>
<a NAME="E1:112_1_1"/><i><font color="Green" title="E86">A3</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
;<br/>
<a NAME="E2:112_1_1"/><i><font color="Green" title="E87">A4</font></i>: 
for <font color="Olive" title="b1">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>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="112_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c11">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/>

<b>assume </b><a NAME="E1:112_1_1_1"/><i><font color="Green" title="E88">A5</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">k</font> &amp; <font color="Maroon" title="c11">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">w1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies <font color="Maroon" title="c5">Tf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c3">tfsm1</font>,<font color="Maroon" title="c6">w1</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="c11">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c4">tfsm2</font>,<font color="Maroon" title="c6">w1</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="c11">k</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/>

<b>assume </b><a NAME="E2:112_1_1_1"/><i><font color="Green" title="E89">A6</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">w1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">Tf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c3">tfsm1</font>,<font color="Maroon" title="c6">w1</font> <a href="fsm_1.html#K2" title="FSM_1:func.2">-admissible</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c4">tfsm2</font>,<font color="Maroon" title="c6">w1</font> <a href="fsm_1.html#K2" title="FSM_1:func.2">-admissible</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><br/>

<a NAME="E3:112_1_1_1"/><i><font color="Green" title="E90">A7</font></i>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">k</font> or (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">k</font> &amp; <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) )
 
;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="112_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:112_1_1_1_1"/>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">k</font> or 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">k</font> )
 </span><b>by </b><i><a class="txt" href="fsm_1.html#E3:112_1_1_1"><i><font color="Green" title="E90">A7</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="112_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:112_1_1_1_1_1"/><i><font color="Green" title="E91">A8</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">Tf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c3">tfsm1</font>,<font color="Maroon" title="c6">w1</font> <a href="fsm_1.html#K2" title="FSM_1:func.2">-admissible</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c4">tfsm2</font>,<font color="Maroon" title="c6">w1</font> <a href="fsm_1.html#K2" title="FSM_1:func.2">-admissible</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><br/><div class="add"><b>hence </b><font color="Maroon" title="c5">Tf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c3">tfsm1</font>,<font color="Maroon" title="c6">w1</font> <a href="fsm_1.html#K2" title="FSM_1:func.2">-admissible</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> = 
the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c4">tfsm2</font>
<b>by </b><i><a class="txt" href="fsm_1.html#E3:112"><i><font color="Green" title="E84">A1</font></i></a>, <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">the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c4">tfsm2</font>,<font color="Maroon" title="c6">w1</font> <a href="fsm_1.html#K2" title="FSM_1:func.2">-admissible</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="fsm_1.html#E1:112_1_1_1_1_1"><i><font color="Green" title="E91">A8</font></i></a>, <a class="ref" href="fsm_1.html#D2" target="_self" title="FSM_1:def.2">Def2</a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="112_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:112_1_1_1_1_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">Tf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c3">tfsm1</font>,<font color="Maroon" title="c6">w1</font> <a href="fsm_1.html#K2" title="FSM_1:func.2">-admissible</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c4">tfsm2</font>,<font color="Maroon" title="c6">w1</font> <a href="fsm_1.html#K2" title="FSM_1:func.2">-admissible</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><br/><div class="add"><a NAME="E2:112_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E91">A9</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">k</font> &amp; <font color="Maroon" title="c11">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> <font color="Maroon" title="c6">w1</font> )
 <b>by </b><i><a class="txt" href="fsm_1.html#E2:112_1_1_1"><i><font color="Green" title="E89">A6</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><b>then consider </b><font color="Maroon" title="c12">w1k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">IAlph</font>, <font color="Maroon" title="c13">qk</font>, <font color="Maroon" title="c14">qk1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">tfsm1</font><b> such that </b><br/><a NAME="E4:112_1_1_1_1_2"/><i><font color="Green" title="E92">A10</font></i>: 
( <font color="Maroon" title="c12">w1k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">w1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">k</font> &amp; <font color="Maroon" title="c13">qk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c3">tfsm1</font>,<font color="Maroon" title="c6">w1</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="c11">k</font> &amp; <font color="Maroon" title="c14">qk1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c3">tfsm1</font>,<font color="Maroon" title="c6">w1</font> <a href="fsm_1.html#K2" title="FSM_1:func.2">-admissible</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c12">w1k</font> <a href="fsm_1.html#K1" title="FSM_1:func.1">-succ_of</a> <font color="Maroon" title="c13">qk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">qk1</font> )
 <b>by </b><i><a class="ref" href="fsm_1.html#D2" target="_self" title="FSM_1:def.2">Def2</a></i>;<br/><b>consider </b><font color="Maroon" title="c15">w2k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">IAlph</font>, <font color="Maroon" title="c16">q2k</font>, <font color="Maroon" title="c17">q2k1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">tfsm2</font><b> such that </b><br/><a NAME="E6:112_1_1_1_1_2"/><i><font color="Green" title="E93">A11</font></i>: 
( <font color="Maroon" title="c15">w2k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">w1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">k</font> &amp; <font color="Maroon" title="c16">q2k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c4">tfsm2</font>,<font color="Maroon" title="c6">w1</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="c11">k</font> &amp; <font color="Maroon" title="c17">q2k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c4">tfsm2</font>,<font color="Maroon" title="c6">w1</font> <a href="fsm_1.html#K2" title="FSM_1:func.2">-admissible</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c15">w2k</font> <a href="fsm_1.html#K1" title="FSM_1:func.1">-succ_of</a> <font color="Maroon" title="c16">q2k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">q2k1</font> )
 <b>by </b><i><a class="txt" href="fsm_1.html#E2:112_1_1_1_1_2"><i><font color="Green" title="E91">A9</font></i></a>, <a class="ref" href="fsm_1.html#D2" target="_self" title="FSM_1:def.2">Def2</a></i>;<br/><a NAME="E7:112_1_1_1_1_2"/><i><font color="Green" title="E94">A12</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">w1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">w1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><b>thus </b><font color="Maroon" title="c5">Tf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c3">tfsm1</font>,<font color="Maroon" title="c6">w1</font> <a href="fsm_1.html#K2" title="FSM_1:func.2">-admissible</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> = 
<font color="Maroon" title="c5">Tf</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default">the <a href="fsm_1.html#U1" title="FSM_1:sel.1">Tran</a> of <font color="Maroon" title="c3">tfsm1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c13">qk</font>,<font color="Maroon" title="c12">w1k</font></span>)</span>
<b>by </b><i><a class="txt" href="fsm_1.html#E4:112_1_1_1_1_2"><i><font color="Green" title="E92">A10</font></i></a></i>
<br/>.= 
the <a href="fsm_1.html#U1" title="FSM_1:sel.1">Tran</a> of <font color="Maroon" title="c4">tfsm2</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c16">q2k</font>,<font color="Maroon" title="c15">w2k</font>
<b>by </b><i><a class="txt" href="fsm_1.html#E3:112"><i><font color="Green" title="E84">A1</font></i></a>, <a class="txt" href="fsm_1.html#E1:112_1_1_1"><i><font color="Green" title="E88">A5</font></i></a>, <a class="txt" href="fsm_1.html#E2:112_1_1_1_1_2"><i><font color="Green" title="E91">A9</font></i></a>, <a class="txt" href="fsm_1.html#E4:112_1_1_1_1_2"><i><font color="Green" title="E92">A10</font></i></a>, <a class="txt" href="fsm_1.html#E6:112_1_1_1_1_2"><i><font color="Green" title="E93">A11</font></i></a>, <a class="txt" href="fsm_1.html#E7:112_1_1_1_1_2"><i><font color="Green" title="E94">A12</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c4">tfsm2</font>,<font color="Maroon" title="c6">w1</font> <a href="fsm_1.html#K2" title="FSM_1:func.2">-admissible</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="fsm_1.html#E6:112_1_1_1_1_2"><i><font color="Green" title="E93">A11</font></i></a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>thus </b><a NAME="E3:112_1_1"/>
for <font color="Olive" title="b1">i</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>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="fsm_1.html#E1:112_1_1"><i><font color="Green" title="E86">A3</font></i></a>, <a class="txt" href="fsm_1.html#E2:112_1_1"><i><font color="Green" title="E87">A4</font></i></a>);</i> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><i><font color="Green" title="E86">A13</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default">the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c3">tfsm1</font>,<font color="Maroon" title="c6">w1</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="c6">w1</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">the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c4">tfsm2</font>,<font color="Maroon" title="c6">w1</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/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="112_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c11">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">k</font> &amp; <font color="Maroon" title="c11">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">the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c3">tfsm1</font>,<font color="Maroon" title="c6">w1</font> <a href="fsm_1.html#K4" title="FSM_1:func.4">-response</a> </span>)</span> implies <span class="p1">(<span class="default">the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c4">tfsm2</font>,<font color="Maroon" title="c6">w1</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="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c3">tfsm1</font>,<font color="Maroon" title="c6">w1</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="c11">k</font> )</span><br/><b>assume </b><a NAME="E1:112_1_3"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">k</font> &amp; <font color="Maroon" title="c11">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">the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c3">tfsm1</font>,<font color="Maroon" title="c6">w1</font> <a href="fsm_1.html#K4" title="FSM_1:func.4">-response</a> </span>)</span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default">the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c4">tfsm2</font>,<font color="Maroon" title="c6">w1</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="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c3">tfsm1</font>,<font color="Maroon" title="c6">w1</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="c11">k</font></span><br/><a NAME="E2:112_1_3"/><b>then </b>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">k</font> &amp; <font color="Maroon" title="c11">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> <font color="Maroon" title="c6">w1</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="E3:112_1_3"/><b>then </b><i><font color="Green" title="E87">A14</font></i>: 
<font color="Maroon" title="c11">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="c6">w1</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E4:112_1_3"/><b>then </b>
<font color="Maroon" title="c11">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="c6">w1</font></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="E5:112_1_3"/><b>then </b>
<font color="Maroon" title="c11">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="c6">w1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="finseq_2.html#T9" title="FINSEQ_2:th.9">FINSEQ_2:9</a></i>;<br/><a NAME="E6:112_1_3"/><b>then </b>
<font color="Maroon" title="c11">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">the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c3">tfsm1</font>,<font color="Maroon" title="c6">w1</font> <a href="fsm_1.html#K2" title="FSM_1:func.2">-admissible</a> </span>)</span></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="E7:112_1_3"/><b>then </b><i><font color="Green" title="E88">A15</font></i>: 
<font color="Maroon" title="c11">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">the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c3">tfsm1</font>,<font color="Maroon" title="c6">w1</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="E8:112_1_3"/><b>then </b>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">k</font> &amp; <font color="Maroon" title="c11">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">the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c3">tfsm1</font>,<font color="Maroon" title="c6">w1</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_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E9:112_1_3"/><b>then </b><i><font color="Green" title="E89">A16</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">k</font> &amp; <font color="Maroon" title="c11">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">w1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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 NAME="E10:112_1_3"/><i><font color="Green" title="E90">A17</font></i>: 
<font color="Maroon" title="c11">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E11:112_1_3"/><i><font color="Green" title="E91">A18</font></i>: 
( <span class="p1">(<span class="default">the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c3">tfsm1</font>,<font color="Maroon" title="c6">w1</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="c11">k</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">tfsm1</font> &amp; <font color="Maroon" title="c6">w1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">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="txt" href="fsm_1.html#E3:112_1_3"><i><font color="Green" title="E87">A14</font></i></a>, <a class="txt" href="fsm_1.html#E7:112_1_3"><i><font color="Green" title="E88">A15</font></i></a>, <a class="ref" href="finseq_2.html#T13" title="FINSEQ_2:th.13">FINSEQ_2:13</a></i>;<br/><b>thus </b><span class="p1">(<span class="default">the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c4">tfsm2</font>,<font color="Maroon" title="c6">w1</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="c11">k</font> = 
the <a href="fsm_1.html#U3" title="FSM_1:sel.3">OFun</a> of <font color="Maroon" title="c4">tfsm2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c4">tfsm2</font>,<font color="Maroon" title="c6">w1</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="c11">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">w1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">k</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="fsm_1.html#E3:112_1_3"><i><font color="Green" title="E87">A14</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">tfsm2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Tf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c3">tfsm1</font>,<font color="Maroon" title="c6">w1</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="c11">k</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">w1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">k</font></span>)</span>
<b>by </b><i><a class="txt" href="fsm_1.html#E1:112_1"><i><font color="Green" title="E85">A2</font></i></a>, <a class="txt" href="fsm_1.html#E9:112_1_3"><i><font color="Green" title="E89">A16</font></i></a>, <a class="txt" href="fsm_1.html#E10:112_1_3"><i><font color="Green" title="E90">A17</font></i></a></i>
<br/>.= 
the <a href="fsm_1.html#U3" title="FSM_1:sel.3">OFun</a> of <font color="Maroon" title="c3">tfsm1</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">the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c3">tfsm1</font>,<font color="Maroon" title="c6">w1</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="c11">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">w1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">k</font></span>)</span>
<b>by </b><i><a class="txt" href="fsm_1.html#E3:112"><i><font color="Green" title="E84">A1</font></i></a>, <a class="txt" href="fsm_1.html#E11:112_1_3"><i><font color="Green" title="E91">A18</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c3">tfsm1</font>,<font color="Maroon" title="c6">w1</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="c11">k</font>
<b>by </b><i><a class="txt" href="fsm_1.html#E3:112_1_3"><i><font color="Green" title="E87">A14</font></i></a>, <a class="ref" href="fsm_1.html#D6" target="_self" title="FSM_1:def.6">Def6</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>hence </b><a NAME="E4:112_1"/>
the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c3">tfsm1</font>,<font color="Maroon" title="c6">w1</font> <a href="fsm_1.html#K4" title="FSM_1:func.4">-response</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c4">tfsm2</font>,<font color="Maroon" title="c6">w1</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#E2:112_1"><i><font color="Green" title="E86">A13</font></i></a>, <a class="ref" href="finseq_1.html#T18" title="FINSEQ_1:th.18">FINSEQ_1:18</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:112"/>
<font color="Maroon" title="c3">tfsm1</font>,<font color="Maroon" title="c4">tfsm2</font> <a href="fsm_1.html#R4" title="FSM_1:pred.4">-are_equivalent</a> 
 <b>by </b><i><a class="ref" href="fsm_1.html#D9" target="_self" title="FSM_1:def.9">Def9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
