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<b>let </b><font color="Maroon">IAlph</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">fsm</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="fsm_1.html#L1" target="_self">FSM</a> of <font color="Maroon">IAlph</font>;<br/><b>let </b><font color="Maroon">w1</font> be    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">IAlph</font>;<br/><b>let </b><font color="Maroon">w2</font> be    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">IAlph</font>;<br/><b>let </b><font color="Maroon">q1</font> be   <a href="fsm_1.html#NM1">State</a> of <font color="Maroon">fsm</font>;<br/><b>let </b><font color="Maroon">q2</font> be   <a href="fsm_1.html#NM1">State</a> of <font color="Maroon">fsm</font>;<br/>











<b>assume </b><a NAME="E1:28"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">q1</font>,<font color="Maroon">w1</font> <a href="fsm_1.html#R1" target="_self">-leads_to</a> <font color="Maroon">q2</font>
 ;<br/>

<b>set </b><font color="Maroon">q1w</font> = <font color="Maroon">q1</font>,<span class="p1">(<span class="default"><font color="Maroon">w1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">w2</font></span>)</span> <a href="fsm_1.html#K2" target="_self">-admissible</a> ;<br/>
<b>set </b><font color="Maroon">q2w2</font> = <font color="Maroon">q2</font>,<font color="Maroon">w2</font> <a href="fsm_1.html#K2" target="_self">-admissible</a> ;<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>( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <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">w2</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 implies <span class="p1">(<span class="default"><font color="Maroon">q1</font>,<span class="p2">(<span class="default"><font color="Maroon">w1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">w2</font></span>)</span> <a href="fsm_1.html#K2" target="_self">-admissible</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<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> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">q2</font>,<font color="Maroon">w2</font> <a href="fsm_1.html#K2" target="_self">-admissible</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> );<br/>
<a NAME="E2:28"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
;<br/>
<a NAME="E3:28"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">k</font> being   <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="28_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:28_1"/><i><font color="Green">E39</font></i>: 
( 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> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">w2</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 implies <span class="p1">(<span class="default"><font color="Maroon">q1</font>,<span class="p2">(<span class="default"><font color="Maroon">w1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">w2</font></span>)</span> <a href="fsm_1.html#K2" target="_self">-admissible</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<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> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">q2</font>,<font color="Maroon">w2</font> <a href="fsm_1.html#K2" target="_self">-admissible</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> )
 ;<br/>

<b>assume </b><a NAME="E2:28_1"/><i><font color="Green">E40</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 &amp; <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">w2</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:28_1_1"/>
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0 or 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1_1"><b>suppose </b></a><a NAME="E1:28_1_1_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><font color="Maroon">q1</font>,<span class="p2">(<span class="default"><font color="Maroon">w1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">w2</font></span>)</span> <a href="fsm_1.html#K2" target="_self">-admissible</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<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> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> = 
<font color="Maroon">q2</font>
<b>by </b><i><a class="ref" href="fsm_1.html#T1" target="_self">Th1</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">q2</font>,<font color="Maroon">w2</font> <a href="fsm_1.html#K2" target="_self">-admissible</a> </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>
<b>by </b><i>, </i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1_2"><b>suppose </b></a><a NAME="E1:28_1_1_2"/><i><font color="Green">E43</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font>
 ;<br/><div class="add"><a NAME="E2:28_1_1_2"/><i><font color="Green">E44</font></i>: 
0 <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> 1
 ;<br/><a NAME="E3:28_1_1_2"/><b>then </b><i><font color="Green">E45</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:28_1_1_2"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">w2</font>
 <b>by </b><i><a class="ref" href="fsm_1.html#T7" target="_self">Th7</a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E5:28_1_1_2"/><i><font color="Green">E47</font></i>: 
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">w1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E6:28_1_1_2"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">w1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">w2</font></span>)</span> <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> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">w2</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/><a NAME="E7:28_1_1_2"/><b>then </b>
<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> <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">w1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">w2</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</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">IAlph</font>, <font color="Maroon">qi</font> being   <a href="fsm_1.html#NM1">State</a> of <font color="Maroon">fsm</font>, <font color="Maroon">qi1</font> being   <a href="fsm_1.html#NM1">State</a> of <font color="Maroon">fsm</font><b> such that </b><br/><a NAME="E9:28_1_1_2"/><i><font color="Green">E79</font></i>: 
( <font color="Maroon">wi</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">w1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">w2</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<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> <font color="Maroon">k</font></span>)</span> &amp; <font color="Maroon">qi</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">q1</font>,<span class="p2">(<span class="default"><font color="Maroon">w1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">w2</font></span>)</span> <a href="fsm_1.html#K2" target="_self">-admissible</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<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> <font color="Maroon">k</font></span>)</span> &amp; <font color="Maroon">qi1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">q1</font>,<span class="p2">(<span class="default"><font color="Maroon">w1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">w2</font></span>)</span> <a href="fsm_1.html#K2" target="_self">-admissible</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">wi</font> <a href="fsm_1.html#K1" target="_self">-succ_of</a> <font color="Maroon">qi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">qi1</font> )
 <b>by </b><i>, </i>;<br/><b>consider </b><font color="Maroon">w2i</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">IAlph</font>, <font color="Maroon">q2i</font> being   <a href="fsm_1.html#NM1">State</a> of <font color="Maroon">fsm</font>, <font color="Maroon">q2i1</font> being   <a href="fsm_1.html#NM1">State</a> of <font color="Maroon">fsm</font><b> such that </b><br/><a NAME="E11:28_1_1_2"/><i><font color="Green">E81</font></i>: 
( <font color="Maroon">w2i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> &amp; <font color="Maroon">q2i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">q2</font>,<font color="Maroon">w2</font> <a href="fsm_1.html#K2" target="_self">-admissible</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> &amp; <font color="Maroon">q2i1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">q2</font>,<font color="Maroon">w2</font> <a href="fsm_1.html#K2" target="_self">-admissible</a> </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">w2i</font> <a href="fsm_1.html#K1" target="_self">-succ_of</a> <font color="Maroon">q2i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q2i1</font> )
 <b>by </b><i>, , </i>;<br/><a NAME="E12:28_1_1_2"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">w2</font>
 <b>by </b><i>, , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>hence </b><a NAME="E13:28_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">q1</font>,<span class="p2">(<span class="default"><font color="Maroon">w1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">w2</font></span>)</span> <a href="fsm_1.html#K2" target="_self">-admissible</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<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> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">q2</font>,<font color="Maroon">w2</font> <a href="fsm_1.html#K2" target="_self">-admissible</a> </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>
 <b>by </b><i><a class="txt" href="fsm_1.html#E2:28"><i><font color="Green">E23</font></i></a>, <a class="ref" href="fsm_1.html#T7" target="_self">Th7</a>, , , , <a class="ref" href="fsm_1.html#T8" target="_self">Th8</a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>thus </b><a NAME="E4:28"/>
for <font color="Olive">n</font> being   <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">n</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, <a class="txt" href="fsm_1.html#E1:28"><i><font color="Green">E22</font></i></a>);</i><br/>


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