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<b>let </b><font color="Maroon">qs1</font> be    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">fsm</font>, <font color="Maroon">qs2</font> be    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">fsm</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:18_1_2"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">qs1</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">qs1</font> <a href="hidden.html#R1">=</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 &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">w</font> holds <br/> ex <font color="Olive">wi</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">IAlph</font> ex <font color="Olive">qs1i</font>, <font color="Olive">qs1i1</font> being  <a href="fsm_1.html#NM1">State</a> of <font color="Maroon">fsm</font> st <br/>( <font color="Olive">wi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Olive">qs1i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">qs1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Olive">qs1i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">qs1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Olive">wi</font> <a href="fsm_1.html#K1" target="_self">-succ_of</a> <font color="Olive">qs1i</font> <a href="hidden.html#R1">=</a> <font color="Olive">qs1i1</font> ) ) )
 <b>and </b><br/><a NAME="E2:18_1_2"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">qs2</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">qs2</font> <a href="hidden.html#R1">=</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 &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">w</font> holds <br/> ex <font color="Olive">wi</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">IAlph</font> ex <font color="Olive">qs2i</font>, <font color="Olive">qs2i1</font> being  <a href="fsm_1.html#NM1">State</a> of <font color="Maroon">fsm</font> st <br/>( <font color="Olive">wi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Olive">qs2i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">qs2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Olive">qs2i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">qs2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Olive">wi</font> <a href="fsm_1.html#K1" target="_self">-succ_of</a> <font color="Olive">qs2i</font> <a href="hidden.html#R1">=</a> <font color="Olive">qs2i1</font> ) ) )
 ;<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>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">qs1</font> implies <font color="Maroon">qs1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">qs2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> );<br/>
<a NAME="E3:18_1_2"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
;<br/>
<div><i><font color="Green">E44</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_2_1"><b>now </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="E1:18_1_2_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>]
 ;<br/><b>thus </b><a NAME="E2:18_1_2_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_2_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:18_1_2_1_1"/><i><font color="Green">E46</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>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">qs1</font> )
 ;<br/>

<a NAME="E2:18_1_2_1_1"/><i><font color="Green">E47</font></i>: 
( 0 <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> or ( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> &amp; 0 <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> 1 ) )
 
;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:18_1_2_1_1_1"/>
( 0 <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> or 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> )
 </span><b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:18_1_2_1_1_1_1"/>
0 <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:18_1_2_1_1_1_1"/>
<font color="Maroon">qs1</font> <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">qs2</font> <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/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:18_1_2_1_1_1_2"/><i><font color="Green">E79</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 ;<br/><div class="add"><a NAME="E2:18_1_2_1_1_1_2"/><i><font color="Green">E81</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">w</font>
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><b>then consider </b><font color="Maroon">i1</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">IAlph</font>, <font color="Maroon">qs1i</font> being   <a href="fsm_1.html#NM1">State</a> of <font color="Maroon">fsm</font>, <font color="Maroon">qs1i1</font> being   <a href="fsm_1.html#NM1">State</a> of <font color="Maroon">fsm</font><b> such that </b><br/><a NAME="E4:18_1_2_1_1_1_2"/><i><font color="Green">E82</font></i>: 
( <font color="Maroon">i1</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">qs1i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">qs1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> &amp; <font color="Maroon">qs1i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">qs1</font> <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">i1</font> <a href="fsm_1.html#K1" target="_self">-succ_of</a> <font color="Maroon">qs1i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">qs1i1</font> )
 <b>by </b><i>, </i>;<br/><b>consider </b><font color="Maroon">i2</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">IAlph</font>, <font color="Maroon">qs2i</font> being   <a href="fsm_1.html#NM1">State</a> of <font color="Maroon">fsm</font>, <font color="Maroon">qs2i1</font> being   <a href="fsm_1.html#NM1">State</a> of <font color="Maroon">fsm</font><b> such that </b><br/><a NAME="E6:18_1_2_1_1_1_2"/><i><font color="Green">E83</font></i>: 
( <font color="Maroon">i2</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">qs2i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">qs2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> &amp; <font color="Maroon">qs2i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">qs2</font> <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">i2</font> <a href="fsm_1.html#K1" target="_self">-succ_of</a> <font color="Maroon">qs2i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">qs2i1</font> )
 <b>by </b><i>, , </i>;<br/><b>thus </b><a NAME="E7:18_1_2_1_1_1_2"/>
<font color="Maroon">qs1</font> <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">qs2</font> <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="ref" href="fsm_1.html#T1" target="_self">Th1</a>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E5:18_1_2"/>
for <font color="Olive">k</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">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<b>hence </b><a NAME="E6:18_1_2"/>
<font color="Maroon">qs1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">qs2</font>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T18">FINSEQ_1:18</a></i>;<br/>


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