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

<b>let </b><font color="Maroon">O</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">I</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">w</font> be    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">I</font>;<br/><b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">f</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">o</font> be   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">f</font></span><a href="tarski.html#K2">}</a></span>,<font color="Maroon">O</font>;<br/><b>let </b><font color="Maroon">j</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>











<b>assume </b><a NAME="E1:32"/><i><font color="Green">E38</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">j</font>
 ;<br/>

<b>set </b><font color="Maroon">M</font> = <font color="Maroon">I</font> <a href="fsm_2.html#K1" target="_self">-TwoStatesMooreSM</a> <font color="Maroon">i</font>,<font color="Maroon">f</font>,<font color="Maroon">o</font>;<br/>
<a NAME="E2:32"/><i><font color="Green">E39</font></i>: 
( the <a href="fsm_1.html#U2">InitS</a> of <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="fsm_2.html#K1" target="_self">-TwoStatesMooreSM</a> <font color="Maroon">i</font>,<font color="Maroon">f</font>,<font color="Maroon">o</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> &amp; the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="fsm_2.html#K1" target="_self">-TwoStatesMooreSM</a> <font color="Maroon">i</font>,<font color="Maroon">f</font>,<font color="Maroon">o</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">f</font></span><a href="tarski.html#K2">}</a></span> )
 
<b>by </b><i/>;<br/>
<a NAME="E3:32"/><i><font color="Green">E40</font></i>: 
the <a href="fsm_1.html#U1">Tran</a> of <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="fsm_2.html#K1" target="_self">-TwoStatesMooreSM</a> <font color="Maroon">i</font>,<font color="Maroon">f</font>,<font color="Maroon">o</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">f</font></span><a href="tarski.html#K2">}</a></span>,<font color="Maroon">I</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">f</font>
 
<b>by </b><i/>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">w</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 implies <span class="p1">(<span class="default"><a href="fsm_2.html#NK1" target="_self">GEN</a> <font color="Maroon">w</font>,the <a href="fsm_1.html#U2">InitS</a> of <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="fsm_2.html#K1" target="_self">-TwoStatesMooreSM</a> <font color="Maroon">i</font>,<font color="Maroon">f</font>,<font color="Maroon">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> );<br/>
<a NAME="E4:32"/><i><font color="Green">E44</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">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">w</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 ;<br/>

<a NAME="E2:32_1"/><b>then </b>
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">w</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
;<br/>
<a NAME="E3:32_1"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">w</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E4:32_1"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">w</font> )
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<b>then consider </b><font color="Maroon">WI</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font>, <font color="Maroon">QI</font> being   <a href="fsm_1.html#NM1">State</a> of <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="fsm_2.html#K1" target="_self">-TwoStatesMooreSM</a> <font color="Maroon">i</font>,<font color="Maroon">f</font>,<font color="Maroon">o</font></span>)</span>, <font color="Maroon">QI1</font> being   <a href="fsm_1.html#NM1">State</a> of <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="fsm_2.html#K1" target="_self">-TwoStatesMooreSM</a> <font color="Maroon">i</font>,<font color="Maroon">f</font>,<font color="Maroon">o</font></span>)</span><b> such that </b><br/><a NAME="E6:32_1"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">WI</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w</font> <a href="funct_1.html#K1">.</a> 1 &amp; <font color="Maroon">QI</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="fsm_2.html#NK1" target="_self">GEN</a> <font color="Maroon">w</font>,the <a href="fsm_1.html#U2">InitS</a> of <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="fsm_2.html#K1" target="_self">-TwoStatesMooreSM</a> <font color="Maroon">i</font>,<font color="Maroon">f</font>,<font color="Maroon">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> 1 &amp; <font color="Maroon">QI1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="fsm_2.html#NK1" target="_self">GEN</a> <font color="Maroon">w</font>,the <a href="fsm_1.html#U2">InitS</a> of <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="fsm_2.html#K1" target="_self">-TwoStatesMooreSM</a> <font color="Maroon">i</font>,<font color="Maroon">f</font>,<font color="Maroon">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">WI</font> <a href="fsm_1.html#K1">-succ_of</a> <font color="Maroon">QI</font> <a href="hidden.html#R1">=</a> <font color="Maroon">QI1</font> )
 <b>by </b><i><a class="ref" href="fsm_1.html#D2">FSM_1:def 2</a></i>;<br/>
<a NAME="E7:32_1"/>
<span class="p1">(<span class="default"><a href="fsm_2.html#NK1" target="_self">GEN</a> <font color="Maroon">w</font>,the <a href="fsm_1.html#U2">InitS</a> of <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="fsm_2.html#K1" target="_self">-TwoStatesMooreSM</a> <font color="Maroon">i</font>,<font color="Maroon">f</font>,<font color="Maroon">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">f</font>
 
<b>by </b><i>, , , <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>
<b>hence </b><a NAME="E8:32_1"/>
<span class="p1">(<span class="default"><a href="fsm_2.html#NK1" target="_self">GEN</a> <font color="Maroon">w</font>,the <a href="fsm_1.html#U2">InitS</a> of <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="fsm_2.html#K1" target="_self">-TwoStatesMooreSM</a> <font color="Maroon">i</font>,<font color="Maroon">f</font>,<font color="Maroon">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">f</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E5:32"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">k</font> being non <a href="realset1.html#V1">trivial</a>  <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font> <a href="nat_1.html#K1">+</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">k</font> be  non <a href="realset1.html#V1">trivial</a>  <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:32_2"/>
( <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">w</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 implies <span class="p1">(<span class="default"><a href="fsm_2.html#NK1" target="_self">GEN</a> <font color="Maroon">w</font>,the <a href="fsm_1.html#U2">InitS</a> of <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="fsm_2.html#K1" target="_self">-TwoStatesMooreSM</a> <font color="Maroon">i</font>,<font color="Maroon">f</font>,<font color="Maroon">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> )
 <b>and </b><br/><a NAME="E2:32_2"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">w</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 ;<br/>

<a NAME="E3:32_2"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">w</font> )
 
<b>by </b><i>, <a class="ref" href="nat_2.html#T21">NAT_2:21</a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<b>then consider </b><font color="Maroon">WI</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font>, <font color="Maroon">QI</font> being   <a href="fsm_1.html#NM1">State</a> of <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="fsm_2.html#K1" target="_self">-TwoStatesMooreSM</a> <font color="Maroon">i</font>,<font color="Maroon">f</font>,<font color="Maroon">o</font></span>)</span>, <font color="Maroon">QI1</font> being   <a href="fsm_1.html#NM1">State</a> of <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="fsm_2.html#K1" target="_self">-TwoStatesMooreSM</a> <font color="Maroon">i</font>,<font color="Maroon">f</font>,<font color="Maroon">o</font></span>)</span><b> such that </b><br/><a NAME="E5:32_2"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">WI</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> &amp; <font color="Maroon">QI</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="fsm_2.html#NK1" target="_self">GEN</a> <font color="Maroon">w</font>,the <a href="fsm_1.html#U2">InitS</a> of <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="fsm_2.html#K1" target="_self">-TwoStatesMooreSM</a> <font color="Maroon">i</font>,<font color="Maroon">f</font>,<font color="Maroon">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> &amp; <font color="Maroon">QI1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="fsm_2.html#NK1" target="_self">GEN</a> <font color="Maroon">w</font>,the <a href="fsm_1.html#U2">InitS</a> of <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="fsm_2.html#K1" target="_self">-TwoStatesMooreSM</a> <font color="Maroon">i</font>,<font color="Maroon">f</font>,<font color="Maroon">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">WI</font> <a href="fsm_1.html#K1">-succ_of</a> <font color="Maroon">QI</font> <a href="hidden.html#R1">=</a> <font color="Maroon">QI1</font> )
 <b>by </b><i><a class="ref" href="fsm_1.html#D2">FSM_1:def 2</a></i>;<br/>
<a NAME="E6:32_2"/>
<span class="p1">(<span class="default"><a href="fsm_2.html#NK1" target="_self">GEN</a> <font color="Maroon">w</font>,the <a href="fsm_1.html#U2">InitS</a> of <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="fsm_2.html#K1" target="_self">-TwoStatesMooreSM</a> <font color="Maroon">i</font>,<font color="Maroon">f</font>,<font color="Maroon">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font>
 
<b>by </b><i>, , , <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>
<b>hence </b><a NAME="E7:32_2"/>
<span class="p1">(<span class="default"><a href="fsm_2.html#NK1" target="_self">GEN</a> <font color="Maroon">w</font>,the <a href="fsm_1.html#U2">InitS</a> of <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="fsm_2.html#K1" target="_self">-TwoStatesMooreSM</a> <font color="Maroon">i</font>,<font color="Maroon">f</font>,<font color="Maroon">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E6:32"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">j</font> is  non <a href="realset1.html#V1">trivial</a>  <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i>, <a class="ref" href="nat_2.html#D1">NAT_2:def 1</a></i>;<br/>
<a NAME="E7:32"/>
for <font color="Olive">k</font> being non <a href="realset1.html#V1">trivial</a>  <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 
<b>from </b><i><a class="ref" href="nat_2.html#S2">NAT_2:sch 2</a>(, );<br/></i>
<b>hence </b><a NAME="E8:32"/>
(  not <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">w</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 or <span class="p1">(<span class="default"><a href="fsm_2.html#NK1" target="_self">GEN</a> <font color="Maroon">w</font>,the <a href="fsm_1.html#U2">InitS</a> of <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="fsm_2.html#K1" target="_self">-TwoStatesMooreSM</a> <font color="Maroon">i</font>,<font color="Maroon">f</font>,<font color="Maroon">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> )
 <b>by </b><i/>;<br/>


</div>
