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<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="E1:33_1_1"/>
for <font color="Olive">w1</font>, <font color="Olive">w2</font> being   <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">I</font>  st <font color="Olive">w1</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Olive">w2</font> <a href="funct_1.html#K1">.</a> 1 &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">w1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Olive">w2</font> holds <br/> <a href="fsm_2.html#NK1" target="_self">GEN</a> <font color="Olive">w1</font>,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>  <a href="fsm_2.html#NK1" target="_self">GEN</a> <font color="Olive">w2</font>,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>
 
<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">w1</font> be    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">I</font>;<br/><b>let </b><font color="Maroon">w2</font> be    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">I</font>;<br/>



<b>assume </b><a NAME="E1:33_1_1_1"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">w1</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">w2</font> <a href="funct_1.html#K1">.</a> 1 &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">w1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">w2</font> )
 ;<br/>

<b>reconsider </b><font color="Maroon">p</font> =  <a href="fsm_2.html#NK1" target="_self">GEN</a> <font color="Maroon">w1</font>,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>, <font color="Maroon">q</font> =  <a href="fsm_2.html#NK1" target="_self">GEN</a> <font color="Maroon">w2</font>,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> as   <a href="finseq_1.html#NM1">FinSequence</a> ;<br/>
<i><font color="Green">E39</font></i>: <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> 1 = 
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>
<b>by </b><i><a class="ref" href="fsm_1.html#D2">FSM_1:def 2</a></i>
<br/>.= 
<font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> 1
<b>by </b><i><a class="ref" href="fsm_1.html#D2">FSM_1:def 2</a></i>
;<br/>
<a NAME="E4:33_1_1_1"/><i><font color="Green">E40</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">w1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">w1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 
<b>by </b><i>, <a class="ref" href="fsm_1.html#D2">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">j</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:33_1_1_1_2"/><i><font color="Green">E44</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font>
 ;<br/><a NAME="E2:33_1_1_1_2"/><b>then </b><i><font color="Green">E46</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><a NAME="E3:33_1_1_1_2"/>
( <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">j</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 )
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E4:33_1_1_1_2"/><b>then </b>
not <!--HUMANRECFAILED-->( not <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> &amp; <font color="Maroon">q</font> <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>, , , , </i>;<br/><b>hence </b><a NAME="E5:33_1_1_1_2"/>
( <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> implies <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> )
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:33_1_1_1"/>
 <a href="fsm_2.html#NK1" target="_self">GEN</a> <font color="Maroon">w1</font>,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>  <a href="fsm_2.html#NK1" target="_self">GEN</a> <font color="Maroon">w2</font>,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>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T18">FINSEQ_1:18</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:33_1_1"/>
<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> is <a href="fsm_2.html#V1" target="_self">calculating_type</a>
 <b>by </b><i/>;<br/>


</div>
