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<div class="add">

<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="finseq_1.html#NM1">FinSequence</a>] <b>means </b>for <font color="Olive">p</font>, <font color="Olive">q</font>, <font color="Olive">s</font> being  <a href="finseq_1.html#NM1">FinSequence</a>  st <font color="Olive">p</font> <a href="finseq_1.html#K7" target="_self">^</a> <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> <a href="finseq_1.html#K7" target="_self">^</a> <font color="Olive">s</font> &amp;  <a href="finseq_1.html#K3" target="_self">len</a> <font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">a<sub>1</sub></font> holds <br/> ex <font color="Olive">t</font> being  <a href="finseq_1.html#NM1">FinSequence</a> st <font color="Olive">p</font> <a href="finseq_1.html#K7" target="_self">^</a> <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E1:81"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="xboole_0.html#K1">{}</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be   <a href="finseq_1.html#NM1">FinSequence</a>;<br/><b>let </b><font color="Maroon">q</font> be   <a href="finseq_1.html#NM1">FinSequence</a>;<br/><b>let </b><font color="Maroon">s</font> be   <a href="finseq_1.html#NM1">FinSequence</a>;<br/>





<b>assume </b><a NAME="E1:81_1"/>
( <font color="Maroon">p</font> <a href="finseq_1.html#K7" target="_self">^</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <a href="xboole_0.html#K1">{}</a>  <a href="finseq_1.html#K7" target="_self">^</a> <font color="Maroon">s</font> &amp;  <a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3" target="_self">len</a> <a href="xboole_0.html#K1">{}</a>  )
 ;<br/>

<a NAME="E2:81_1"/><b>then </b>
 <a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E3:81_1"/><b>then </b><i><font color="Green">E33</font></i>: 
 <a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> 0
 
;<br/>
<b>take </b>
 <a href="xboole_0.html#K1">{}</a> 
;<br/>

<b>thus </b><font color="Maroon">p</font> <a href="finseq_1.html#K7" target="_self">^</a> <a href="xboole_0.html#K1">{}</a>  = 
<font color="Maroon">p</font>
<b>by </b><i><a class="txt" href="finseq_1.html#E3:81_1"><i><font color="Green">E33</font></i></a></i>
<br/>.= 
 <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i>, </i>
;<br/>


</div><b>end;</b></div>
<a NAME="E2:81"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">r</font> being  <a href="finseq_1.html#NM1">FinSequence</a><br/> for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">r</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">r</font> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p1"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Olive">x</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">r</font> be   <a href="finseq_1.html#NM1">FinSequence</a>;<br/><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E1:81_2"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">p</font>, <font color="Olive">q</font>, <font color="Olive">s</font> being  <a href="finseq_1.html#NM1">FinSequence</a>  st <font color="Olive">p</font> <a href="finseq_1.html#K7" target="_self">^</a> <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="finseq_1.html#K7" target="_self">^</a> <font color="Olive">s</font> &amp;  <a href="finseq_1.html#K3" target="_self">len</a> <font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">r</font> holds <br/> ex <font color="Olive">t</font> being  <a href="finseq_1.html#NM1">FinSequence</a> st <font color="Olive">p</font> <a href="finseq_1.html#K7" target="_self">^</a> <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font>
 ;<br/>

<b>let </b><font color="Maroon">p</font> be   <a href="finseq_1.html#NM1">FinSequence</a>;<br/><b>let </b><font color="Maroon">q</font> be   <a href="finseq_1.html#NM1">FinSequence</a>;<br/><b>let </b><font color="Maroon">s</font> be   <a href="finseq_1.html#NM1">FinSequence</a>;<br/>





<b>assume </b><a NAME="E2:81_2"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">p</font> <a href="finseq_1.html#K7" target="_self">^</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p2"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7" target="_self">^</a> <font color="Maroon">s</font> &amp;  <a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3" target="_self">len</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p2"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span> )
 ;<br/>

<div><i><font color="Green">E37</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="81_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:81_2_1"/>
 <a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3" target="_self">len</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p2"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span>
 ;<br/><b>then </b><i><font color="Green">E51</font></i>:  <a href="finseq_1.html#K4" target="_self">dom</a> <font color="Maroon">p</font> = 
 <a href="finseq_1.html#K2" target="_self">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p3"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="finseq_1.html#K4" target="_self">dom</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p2"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span>
<b>by </b><i/>
;<br/><a NAME="E3:81_2_1"/><i><font color="Green">E52</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="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4" target="_self">dom</a> <font color="Maroon">p</font> holds <br/><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p2"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_2_1_2"><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="E1:81_2_1_2"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4" target="_self">dom</a> <font color="Maroon">p</font>
 ;<br/>

<b>hence </b><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p3"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7" target="_self">^</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p2"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
<b>by </b><i><a class="txt" href="finseq_1.html#E3:81_2_1"><i><font color="Green">E52</font></i></a>, , </i>
;<br/><br/>


</div><b>end;</b></div><font color="Maroon">p</font> <a href="finseq_1.html#K7" target="_self">^</a> <a href="xboole_0.html#K1">{}</a>  = 
<font color="Maroon">p</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">r</font> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p1"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span>
<b>by </b><i><a class="txt" href="finseq_1.html#E3:81_2_1"><i><font color="Green">E52</font></i></a>, , </i>
;<br/><b>hence </b><a NAME="E5:81_2_1"/>
 ex <font color="Olive">t</font> being  <a href="finseq_1.html#NM1">FinSequence</a> st <font color="Maroon">p</font> <a href="finseq_1.html#K7" target="_self">^</a> <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p1"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span>
 ;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:81_2_2"/>
 <a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3" target="_self">len</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p2"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span>
 ;<br/><a NAME="E2:81_2_2"/><b>then </b>
 <a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">r</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <span class="p2"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E3:81_2_2"/><b>then </b><i><font color="Green">E54</font></i>: 
 <a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">r</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="finseq_1.html#T8" target="_self">Th8</a></i>;<br/><a NAME="E4:81_2_2"/>
 <a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">r</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <span class="p2"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E5:81_2_2"/><b>then </b>
 <a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">r</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="finseq_1.html#T8" target="_self">Th8</a></i>;<br/><a NAME="E6:81_2_2"/><b>then </b><i><font color="Green">E55</font></i>: 
 <a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">r</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T26">NAT_1:26</a></i>;<br/><a NAME="E7:81_2_2"/>
<font color="Maroon">p</font> <a href="finseq_1.html#K7" target="_self">^</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span> <a href="finseq_1.html#K7" target="_self">^</a> <font color="Maroon">s</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T4" target="_self">Th4</a></i>;<br/><b>then consider </b><font color="Maroon">t</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E9:81_2_2"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">p</font> <a href="finseq_1.html#K7" target="_self">^</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font>
 <b>by </b><i>, </i>;<br/><a NAME="E10:81_2_2"/>
<font color="Maroon">p</font> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p2"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p1"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="finseq_1.html#E3:81_2_2"><i><font color="Green">E54</font></i></a>, <a class="ref" href="finseq_1.html#T4" target="_self">Th4</a></i>;<br/><b>hence </b><a NAME="E11:81_2_2"/>
 ex <font color="Olive">t</font> being  <a href="finseq_1.html#NM1">FinSequence</a> st <font color="Maroon">p</font> <a href="finseq_1.html#K7" target="_self">^</a> <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p1"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:81_2"/>
 ex <font color="Olive">t</font> being  <a href="finseq_1.html#NM1">FinSequence</a> st <font color="Maroon">p</font> <a href="finseq_1.html#K7" target="_self">^</a> <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p1"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E3:81"/>
for <font color="Olive">r</font> being  <a href="finseq_1.html#NM1">FinSequence</a> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">r</font>]
 
<b>from </b><i><a class="ref" href="finseq_1.html#S3" target="_self">FINSEQ_1:sch 3</a>(, );<br/></i>
<b>hence </b><a NAME="E4:81"/>
for <font color="Olive">p</font>, <font color="Olive">q</font>, <font color="Olive">r</font>, <font color="Olive">s</font> being  <a href="finseq_1.html#NM1">FinSequence</a>  st <font color="Olive">p</font> <a href="finseq_1.html#K7" target="_self">^</a> <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <font color="Olive">r</font> <a href="finseq_1.html#K7" target="_self">^</a> <font color="Olive">s</font> &amp;  <a href="finseq_1.html#K3" target="_self">len</a> <font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3" target="_self">len</a> <font color="Olive">r</font> holds <br/> ex <font color="Olive">t</font> being  <a href="finseq_1.html#NM1">FinSequence</a> st <font color="Olive">p</font> <a href="finseq_1.html#K7" target="_self">^</a> <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Olive">r</font>
 ;<br/>


</div>
