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

<b>set </b><font color="Maroon" title="c6">M</font> = <font color="Maroon" title="c1">I</font> <a href="fsm_2.html#K1" title="FSM_2:func.1">-TwoStatesMooreSM</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">o</font>;<br/>
<a NAME="E1:33_1_1"/>
for <font color="Olive" title="b1">w1</font>, <font color="Olive" title="b2">w2</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">I</font>  st <font color="Olive" title="b1">w1</font> <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="Olive" title="b2">w2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">w2</font> holds <br/> <a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Olive" title="b1">w1</font>,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="c1">I</font> <a href="fsm_2.html#K1" title="FSM_2:func.1">-TwoStatesMooreSM</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">o</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Olive" title="b2">w2</font>,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="c1">I</font> <a href="fsm_2.html#K1" title="FSM_2:func.1">-TwoStatesMooreSM</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">o</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">w1</font>, <font color="Maroon" title="c8">w2</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">I</font>;<br/>



<b>assume </b><a NAME="E1:33_1_1_1"/><i><font color="Green" title="E17">A1</font></i>: 
( <font color="Maroon" title="c7">w1</font> <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="c8">w2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">w2</font> )
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c9">p</font> =  <a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Maroon" title="c7">w1</font>,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="c1">I</font> <a href="fsm_2.html#K1" title="FSM_2:func.1">-TwoStatesMooreSM</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">o</font></span>)</span>, <font color="Maroon" title="c10">q</font> =  <a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Maroon" title="c8">w2</font>,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="c1">I</font> <a href="fsm_2.html#K1" title="FSM_2:func.1">-TwoStatesMooreSM</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">o</font></span>)</span> as   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> ;<br/>
<i><font color="Green" title="E18">A2</font></i>: <font color="Maroon" title="c9">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 = 
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="c1">I</font> <a href="fsm_2.html#K1" title="FSM_2:func.1">-TwoStatesMooreSM</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">o</font></span>)</span>
<b>by </b><i><a class="ref" href="fsm_1.html#D2" title="FSM_1:def.2">FSM_1:def 2</a></i>
<br/>.= 
<font color="Maroon" title="c10">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<b>by </b><i><a class="ref" href="fsm_1.html#D2" title="FSM_1:def.2">FSM_1:def 2</a></i>
;<br/>
<a NAME="E4:33_1_1_1"/><i><font color="Green" title="E19">A3</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">w1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">w1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 
<b>by </b><i><a class="txt" href="fsm_2.html#E1:33_1_1_1"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="fsm_1.html#D2" title="FSM_1:def.2">FSM_1:def 2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c11">j</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> ;<br/><b>assume </b><a NAME="E1:33_1_1_1_2"/><i><font color="Green" title="E20">A4</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">j</font>
 ;<br/><a NAME="E2:33_1_1_1_2"/><b>then </b><i><font color="Green" title="E21">A5</font></i>: 
<font color="Maroon" title="c11">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 ;<br/><a NAME="E3:33_1_1_1_2"/>
( <font color="Maroon" title="c11">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c11">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 )
 <b>by </b><i><a class="txt" href="fsm_2.html#E1:33_1_1_1_2"><i><font color="Green" title="E20">A4</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E4:33_1_1_1_2"/><b>then </b>
not <!--HUMANRECFAILED-->( not <font color="Maroon" title="c9">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">j</font> &amp; <font color="Maroon" title="c11">j</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="c9">p</font> &amp; not <!--HUMANRECFAILED-->( <font color="Maroon" title="c9">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> &amp; <font color="Maroon" title="c10">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> ) )
 <b>by </b><i><a class="txt" href="fsm_2.html#E1:33_1_1_1"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="fsm_2.html#E3:33_1_1_1"><i><font color="Green" title="E18">A2</font></i></a>, <a class="txt" href="fsm_2.html#E4:33_1_1_1"><i><font color="Green" title="E19">A3</font></i></a>, <a class="txt" href="fsm_2.html#E2:33_1_1_1_2"><i><font color="Green" title="E21">A5</font></i></a>, <a class="ref" href="fsm_2.html#T17" target="_self" title="FSM_2:th.17">Th17</a></i>;<br/><b>hence </b><a NAME="E5:33_1_1_1_2"/>
( <font color="Maroon" title="c11">j</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="c9">p</font> implies <font color="Maroon" title="c9">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">j</font> )
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:33_1_1_1"/>
 <a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Maroon" title="c7">w1</font>,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="c1">I</font> <a href="fsm_2.html#K1" title="FSM_2:func.1">-TwoStatesMooreSM</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">o</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Maroon" title="c8">w2</font>,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="c1">I</font> <a href="fsm_2.html#K1" title="FSM_2:func.1">-TwoStatesMooreSM</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">o</font></span>)</span>
 <b>by </b><i><a class="txt" href="fsm_2.html#E4:33_1_1_1"><i><font color="Green" title="E19">A3</font></i></a>, <a class="ref" href="finseq_1.html#T18" title="FINSEQ_1:th.18">FINSEQ_1:18</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:33_1_1"/>
<font color="Maroon" title="c1">I</font> <a href="fsm_2.html#K1" title="FSM_2:func.1">-TwoStatesMooreSM</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">o</font> is <a href="fsm_2.html#V1" title="FSM_2:attr.1">calculating_type</a>
 <b>by </b><i><a class="ref" href="fsm_2.html#T7" target="_self" title="FSM_2:th.7">Th7</a></i>;<br/>


</div>
