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

<b>let </b><font color="Maroon" title="c1">O</font>, <font color="Maroon" title="c2">I</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">w</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c2">I</font><br/> for <font color="Olive" title="b2">i</font>, <font color="Olive" title="b3">f</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b4">o</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">f</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Maroon" title="c1">O</font><br/> for <font color="Olive" title="b5">j</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b5">j</font> &amp; <font color="Olive" title="b5">j</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="Olive" title="b1">w</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 holds <br/><span class="p1">(<span class="default"><a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Olive" title="b1">w</font>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="fsm_2.html#K1" title="FSM_2:func.1">-TwoStatesMooreSM</a> <font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">f</font>,<font color="Olive" title="b4">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">f</font></span><br/><b>let </b><font color="Maroon" title="c3">w</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c2">I</font>; <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">i</font>, <font color="Olive" title="b2">f</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b3">o</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">f</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Maroon" title="c1">O</font><br/> for <font color="Olive" title="b4">j</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b4">j</font> &amp; <font color="Olive" title="b4">j</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="c3">w</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 holds <br/><span class="p1">(<span class="default"><a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Maroon" title="c3">w</font>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="fsm_2.html#K1" title="FSM_2:func.1">-TwoStatesMooreSM</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">f</font>,<font color="Olive" title="b3">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f</font></span><br/><b>let </b><font color="Maroon" title="c4">i</font>, <font color="Maroon" title="c5">f</font> be    <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">o</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">f</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Maroon" title="c1">O</font><br/> for <font color="Olive" title="b2">j</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">j</font> &amp; <font color="Olive" title="b2">j</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="c3">w</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 holds <br/><span class="p1">(<span class="default"><a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Maroon" title="c3">w</font>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="fsm_2.html#K1" title="FSM_2:func.1">-TwoStatesMooreSM</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">f</font>,<font color="Olive" title="b1">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font></span><br/><b>let </b><font color="Maroon" title="c6">o</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">f</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Maroon" title="c1">O</font>; <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">j</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</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="c3">w</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 holds <br/><span class="p1">(<span class="default"><a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Maroon" title="c3">w</font>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="fsm_2.html#K1" title="FSM_2:func.1">-TwoStatesMooreSM</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font></span><br/><b>let </b><font color="Maroon" title="c7">j</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">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"> ( 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">j</font> &amp; <font color="Maroon" title="c7">j</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="c3">w</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies <span class="p1">(<span class="default"><a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Maroon" title="c3">w</font>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="fsm_2.html#K1" title="FSM_2:func.1">-TwoStatesMooreSM</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> )</span><br/>











<b>assume </b><a NAME="E1:32"/><i><font color="Green" title="E17">A1</font></i>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c7">j</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="c3">w</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 or <span class="p1">(<span class="default"><a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Maroon" title="c3">w</font>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="fsm_2.html#K1" title="FSM_2:func.1">-TwoStatesMooreSM</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> )</span><br/>

<b>set </b><font color="Maroon" title="c8">M</font> = <font color="Maroon" title="c2">I</font> <a href="fsm_2.html#K1" title="FSM_2:func.1">-TwoStatesMooreSM</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">o</font>;<br/>
<a NAME="E2:32"/><i><font color="Green" title="E18">A2</font></i>: 
( the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="fsm_2.html#K1" title="FSM_2:func.1">-TwoStatesMooreSM</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">o</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">i</font> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="fsm_2.html#K1" title="FSM_2:func.1">-TwoStatesMooreSM</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">o</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">f</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )
 
<b>by </b><i><a class="ref" href="fsm_2.html#D7" target="_self" title="FSM_2:def.7">Def7</a></i>;<br/>
<a NAME="E3:32"/><i><font color="Green" title="E19">A3</font></i>: 
the <a href="fsm_1.html#U1" title="FSM_1:sel.1">Tran</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="fsm_2.html#K1" title="FSM_2:func.1">-TwoStatesMooreSM</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">o</font></span>)</span> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">f</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Maroon" title="c2">I</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c5">f</font>
 
<b>by </b><i><a class="ref" href="fsm_2.html#D7" target="_self" title="FSM_2:def.7">Def7</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b>( $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="c3">w</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies <span class="p1">(<span class="default"><a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Maroon" title="c3">w</font>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="fsm_2.html#K1" title="FSM_2:func.1">-TwoStatesMooreSM</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> );<br/>
<a NAME="E4:32"/><i><font color="Green" title="E20">A4</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[2]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:32_1"/>
2 <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="c3">w</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"> <span class="p1">(<span class="default"><a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Maroon" title="c3">w</font>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="fsm_2.html#K1" title="FSM_2:func.1">-TwoStatesMooreSM</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font></span><br/>

<a NAME="E2:32_1"/><b>then </b>
1 <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="c3">w</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
;<br/>
<a NAME="E3:32_1"/><b>then </b>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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="c3">w</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#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E4:32_1"/><b>then </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>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">w</font> )
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c9">WI</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">I</font>, <font color="Maroon" title="c10">QI</font>, <font color="Maroon" title="c11">QI1</font> being   <a href="fsm_1.html#NM1" title="FSM_1:NM.1">State</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="fsm_2.html#K1" title="FSM_2:func.1">-TwoStatesMooreSM</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">o</font></span>)</span><b> such that </b><br/><a NAME="E6:32_1"/><i><font color="Green" title="E21">A5</font></i>: 
( <font color="Maroon" title="c9">WI</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">w</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 &amp; <font color="Maroon" title="c10">QI</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Maroon" title="c3">w</font>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="fsm_2.html#K1" title="FSM_2:func.1">-TwoStatesMooreSM</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 &amp; <font color="Maroon" title="c11">QI1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Maroon" title="c3">w</font>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="fsm_2.html#K1" title="FSM_2:func.1">-TwoStatesMooreSM</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c9">WI</font> <a href="fsm_1.html#K1" title="FSM_1:func.1">-succ_of</a> <font color="Maroon" title="c10">QI</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">QI1</font> )
 <b>by </b><i><a class="ref" href="fsm_1.html#D2" title="FSM_1:def.2">FSM_1:def 2</a></i>;<br/>
<b>thus </b><a NAME="E7:32_1"/>
<span class="p1">(<span class="default"><a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Maroon" title="c3">w</font>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="fsm_2.html#K1" title="FSM_2:func.1">-TwoStatesMooreSM</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font>
 <b>by </b><i><a class="txt" href="#E2:32"><i><font color="Green" title="E18">A2</font></i></a>, <a class="txt" href="#E3:32"><i><font color="Green" title="E19">A3</font></i></a>, <a class="txt" href="#E6:32_1"><i><font color="Green" title="E21">A5</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</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>
<a NAME="E5:32"/><i><font color="Green" title="E21">A6</font></i>: 
for <font color="Olive" title="b1">k</font> being non <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a> <a href="nat_1.html#NM1" title="NAT_1:NM.1">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#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">k</font> be  non <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a> <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"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:32_2"/>
( <font color="Maroon" title="c9">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="c3">w</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies <span class="p1">(<span class="default"><a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Maroon" title="c3">w</font>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="fsm_2.html#K1" title="FSM_2:func.1">-TwoStatesMooreSM</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> )
 <b>and </b><br/><a NAME="E2:32_2"/><i><font color="Green" title="E22">A7</font></i>: 
<font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c3">w</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"> <span class="p1">(<span class="default"><a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Maroon" title="c3">w</font>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="fsm_2.html#K1" title="FSM_2:func.1">-TwoStatesMooreSM</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font></span><br/>

<a NAME="E3:32_2"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">k</font> &amp; <font color="Maroon" title="c9">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="c3">w</font> )
 
<b>by </b><i><a class="txt" href="#E2:32_2"><i><font color="Green" title="E22">A7</font></i></a>, <a class="ref" href="nat_2.html#T21" title="NAT_2:th.21">NAT_2:21</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="c10">WI</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">I</font>, <font color="Maroon" title="c11">QI</font>, <font color="Maroon" title="c12">QI1</font> being   <a href="fsm_1.html#NM1" title="FSM_1:NM.1">State</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="fsm_2.html#K1" title="FSM_2:func.1">-TwoStatesMooreSM</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">o</font></span>)</span><b> such that </b><br/><a NAME="E5:32_2"/><i><font color="Green" title="E23">A8</font></i>: 
( <font color="Maroon" title="c10">WI</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">w</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> &amp; <font color="Maroon" title="c11">QI</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Maroon" title="c3">w</font>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="fsm_2.html#K1" title="FSM_2:func.1">-TwoStatesMooreSM</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> &amp; <font color="Maroon" title="c12">QI1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Maroon" title="c3">w</font>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="fsm_2.html#K1" title="FSM_2:func.1">-TwoStatesMooreSM</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c10">WI</font> <a href="fsm_1.html#K1" title="FSM_1:func.1">-succ_of</a> <font color="Maroon" title="c11">QI</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">QI1</font> )
 <b>by </b><i><a class="ref" href="fsm_1.html#D2" title="FSM_1:def.2">FSM_1:def 2</a></i>;<br/>
<b>thus </b><a NAME="E6:32_2"/>
<span class="p1">(<span class="default"><a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Maroon" title="c3">w</font>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="fsm_2.html#K1" title="FSM_2:func.1">-TwoStatesMooreSM</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font>
 <b>by </b><i><a class="txt" href="#E2:32"><i><font color="Green" title="E18">A2</font></i></a>, <a class="txt" href="#E3:32"><i><font color="Green" title="E19">A3</font></i></a>, <a class="txt" href="#E5:32_2"><i><font color="Green" title="E23">A8</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</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>
<a NAME="E6:32"/><i><font color="Green" title="E22">A9</font></i>: 
<font color="Maroon" title="c7">j</font> is  non <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a> <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>
 
<b>by </b><i><a class="txt" href="#E1:32"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="nat_2.html#D1" title="NAT_2:def.1">NAT_2:def 1</a></i>;<br/>
<a NAME="E7:32"/>
for <font color="Olive" title="b1">k</font> being non <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a> <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<b>from </b><i><a class="ref" href="nat_2.html#S2" title="NAT_2:sch.2">NAT_2:sch 2</a>(<a class="txt" href="#E4:32"><i><font color="Green" title="E20">A4</font></i></a>, <a class="txt" href="#E5:32"><i><font color="Green" title="E21">A6</font></i></a>);<br/></i>
<b>hence </b><a NAME="E8:32"/>
(  not <font color="Maroon" title="c7">j</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="c3">w</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 or <span class="p1">(<span class="default"><a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Maroon" title="c3">w</font>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="fsm_2.html#K1" title="FSM_2:func.1">-TwoStatesMooreSM</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> )
 <b>by </b><i><a class="txt" href="#E6:32"><i><font color="Green" title="E22">A9</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>
