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

<span class="kw">set </span><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"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="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> implies  <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>) )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:33_1_1_1"/>
<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
 <span class="kw">and </span><br/><a NAME="E2:33_1_1_1"/><span class="lab"><font color="Green" title="E17">A1</font></span>: 
 <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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <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>)</span><br/>

<span class="kw">reconsider </span><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/>
<span class="lab"><font color="Green" title="E18">A2</font></span>: <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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="fsm_1.html#D2" title="FSM_1:def.2">FSM_1:def 2</a></span>
<br/>.= 
<font color="Maroon" title="c10">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="fsm_1.html#D2" title="FSM_1:def.2">FSM_1:def 2</a></span>
;<br/>
<a NAME="E5:33_1_1_1"/><span class="lab"><font color="Green" title="E19">A3</font></span>: 
 <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#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fsm_1.html#D2" title="FSM_1:def.2">FSM_1:def 2</a></span>;<br/>
<a NAME="E6:33_1_1_1"/><span class="lab"><font color="Green" title="E20">A4</font></span>: 
 <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#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:33_1_1_1"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="ref" href="fsm_1.html#D2" title="FSM_1:def.2">FSM_1:def 2</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c11">j</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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> 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> )</span><br/><span class="kw">assume </span><a NAME="E1:33_1_1_1_2"/><span class="lab"><font color="Green" title="E21">A5</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="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> )</span><br/><a NAME="E2:33_1_1_1_2"/><span class="lab"><font color="Green" title="E22">A6</font></span>: 
<font color="Maroon" title="c11">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/><a NAME="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 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:33_1_1_1_2"><span class="lab"><font color="Green" title="E21">A5</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E4:33_1_1_1_2"/><span class="kw">then </span>
 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> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:33_1_1_1"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="txt" href="#E4:33_1_1_1"><span class="lab"><font color="Green" title="E18">A2</font></span></a>, <a class="txt" href="#E5:33_1_1_1"><span class="lab"><font color="Green" title="E19">A3</font></span></a>, <a class="txt" href="#E2:33_1_1_1_2"><span class="lab"><font color="Green" title="E22">A6</font></span></a>, <a class="ref" href="fsm_2.html#T17" target="_self" title="FSM_2:th.17">Th17</a></span>;<br/><span class="kw">hence </span><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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E8:33_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>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:33_1_1_1"><span class="lab"><font color="Green" title="E19">A3</font></span></a>, <a class="txt" href="#E6:33_1_1_1"><span class="lab"><font color="Green" title="E20">A4</font></span></a>, <a class="ref" href="finseq_1.html#T14" title="FINSEQ_1:th.14">FINSEQ_1:14</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fsm_2.html#T7" target="_self" title="FSM_2:th.7">Th7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
