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

<b>let </b><font color="Maroon">p1</font> be   <a href="finseq_1.html#NM1">FinSequence</a>, <font color="Maroon">p2</font> be   <a href="finseq_1.html#NM1">FinSequence</a>;<br/><b>let </b><font color="Maroon">q1</font> be   <a href="finseq_1.html#NM4">FinSubsequence</a>, <font color="Maroon">q2</font> be   <a href="finseq_1.html#NM4">FinSubsequence</a>;<br/>



<b>assume </b><a NAME="E1:118"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">q1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">p1</font> &amp; <font color="Maroon">q2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">p2</font> )
 ;<br/>

<b>then consider </b><font color="Maroon">ss</font> being   <a href="finseq_1.html#NM4">FinSubsequence</a><b> such that </b><br/><a NAME="E3:118"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">ss</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q1</font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font></span>)</span> <a href="pnproc_1.html#K14" target="_self">Shift</a> <font color="Maroon">q2</font></span>)</span>
 <b>and </b><br/><a NAME="E4:118"/><i><font color="Green">E35</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">ss</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q1</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q2</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="pnproc_1.html#D4" target="_self">Def4</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="118_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:118_1"/>
( ( <font color="Maroon">q1</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">q2</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) or ( <font color="Maroon">q1</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">q2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  ) or ( <font color="Maroon">q1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">q2</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) or ( <font color="Maroon">q1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">q2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="118_1_1"><b>suppose </b></a><a NAME="E1:118_1_1"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">q1</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">q2</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/><div class="add"><a NAME="E2:118_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font></span>)</span> <a href="pnproc_1.html#K14" target="_self">Shift</a> <font color="Maroon">q2</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="pnproc_1.html#T8" target="_self">Th8</a></i>;<br/><a NAME="E3:118_1_1"/><b>then </b><i><font color="Green">E38</font></i>: 
(  <a href="finseq_1.html#K15">Seq</a> <font color="Maroon">ss</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q1</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q2</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i><a class="ref" href="pnproc_1.html#T14" target="_self">Th14</a>, , </i>;<br/><a NAME="E4:118_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q1</font></span>)</span></span>)</span> <a href="pnproc_1.html#K14" target="_self">Shift</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="pnproc_1.html#T8" target="_self">Th8</a></i>;<br/><b>hence </b><a NAME="E5:118_1_1"/>
 ex <font color="Olive">ss</font> being  <a href="finseq_1.html#NM4">FinSubsequence</a> st <br/>( <font color="Olive">ss</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q1</font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font></span>)</span> <a href="pnproc_1.html#K14" target="_self">Shift</a> <font color="Maroon">q2</font></span>)</span> &amp;  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Olive">ss</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q1</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q2</font></span>)</span></span>)</span></span>)</span> &amp;  <a href="finseq_1.html#K15">Seq</a> <font color="Olive">ss</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q1</font></span>)</span></span>)</span> <a href="pnproc_1.html#K14" target="_self">Shift</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q2</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="pnproc_1.html#T14" target="_self">Th14</a>, <a class="ref" href="pnproc_1.html#D5" target="_self">Def5</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="118_1_2"><b>suppose </b></a><a NAME="E1:118_1_2"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">q1</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">q2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/><div class="add"><a NAME="E2:118_1_2"/><b>then </b><i><font color="Green">E43</font></i>: 
 <a href="finseq_1.html#K15">Seq</a> <font color="Maroon">ss</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q2</font>
 <b>by </b><i><a class="ref" href="pnproc_1.html#T14" target="_self">Th14</a>, </i>;<br/><a NAME="E3:118_1_2"/><i><font color="Green">E45</font></i>: 
 <a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q1</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, </i>;<br/><a NAME="E4:118_1_2"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q1</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/><a NAME="E5:118_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q1</font></span>)</span></span>)</span> <a href="pnproc_1.html#K14" target="_self">Shift</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q2</font>
 <b>by </b><i><a class="ref" href="pnproc_1.html#D6" target="_self">Def6</a>, </i>;<br/><b>hence </b><a NAME="E6:118_1_2"/>
 ex <font color="Olive">ss</font> being  <a href="finseq_1.html#NM4">FinSubsequence</a> st <br/>( <font color="Olive">ss</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q1</font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font></span>)</span> <a href="pnproc_1.html#K14" target="_self">Shift</a> <font color="Maroon">q2</font></span>)</span> &amp;  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Olive">ss</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q1</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q2</font></span>)</span></span>)</span></span>)</span> &amp;  <a href="finseq_1.html#K15">Seq</a> <font color="Olive">ss</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q1</font></span>)</span></span>)</span> <a href="pnproc_1.html#K14" target="_self">Shift</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q2</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="pnproc_1.html#T14" target="_self">Th14</a>, <a class="ref" href="pnproc_1.html#D5" target="_self">Def5</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="118_1_3"><b>suppose </b></a><a NAME="E1:118_1_3"/><i><font color="Green">E123</font></i>: 
( <font color="Maroon">q1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">q2</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/><div class="add"><a NAME="E2:118_1_3"/><b>then </b><i><font color="Green">E124</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font></span>)</span> <a href="pnproc_1.html#K14" target="_self">Shift</a> <font color="Maroon">q2</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="pnproc_1.html#T8" target="_self">Th8</a></i>;<br/><a NAME="E3:118_1_3"/>
 <a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q2</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, </i>;<br/><a NAME="E4:118_1_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q1</font></span>)</span></span>)</span> <a href="pnproc_1.html#K14" target="_self">Shift</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="pnproc_1.html#T8" target="_self">Th8</a></i>;<br/><b>hence </b><a NAME="E5:118_1_3"/>
 ex <font color="Olive">ss</font> being  <a href="finseq_1.html#NM4">FinSubsequence</a> st <br/>( <font color="Olive">ss</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q1</font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font></span>)</span> <a href="pnproc_1.html#K14" target="_self">Shift</a> <font color="Maroon">q2</font></span>)</span> &amp;  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Olive">ss</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q1</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q2</font></span>)</span></span>)</span></span>)</span> &amp;  <a href="finseq_1.html#K15">Seq</a> <font color="Olive">ss</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q1</font></span>)</span></span>)</span> <a href="pnproc_1.html#K14" target="_self">Shift</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q2</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="pnproc_1.html#T14" target="_self">Th14</a>, <a class="ref" href="pnproc_1.html#D5" target="_self">Def5</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="118_1_4"><b>suppose </b></a><a NAME="E1:118_1_4"/><i><font color="Green">E125</font></i>: 
( <font color="Maroon">q1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">q2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/><div class="add"><a NAME="E2:118_1_4"/><i><font color="Green">E126</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q1</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q2</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q1</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q2</font></span>)</span></span>)</span> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="finseq_1.html#D1">FINSEQ_1:def 1</a></i>;<br/><i><font color="Green">E130</font></i>:  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q1</font></span>)</span> = 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q1</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
<span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q1</font></span>)</span> ) </span>}</span> 
<b>by </b><i><a class="ref" href="finseq_1.html#D1">FINSEQ_1:def 1</a></i>
;<br/><a NAME="E4:118_1_4"/>
 <a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, </i>;<br/><a NAME="E5:118_1_4"/><b>then </b><i><font color="Green">E131</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q1</font></span>)</span></span>)</span> <a href="pnproc_1.html#K14" target="_self">Shift</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q1</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q1</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q2</font></span>)</span></span>)</span> ) </span>}</span> 
 <b>by </b><i/>;<br/><a NAME="E6:118_1_4"/><i><font color="Green">E132</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q1</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q2</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q1</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q1</font></span>)</span></span>)</span> <a href="pnproc_1.html#K14" target="_self">Shift</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q2</font></span>)</span></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="118_1_4_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:118_1_4_2"/>
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q1</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q2</font></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q1</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q1</font></span>)</span></span>)</span> <a href="pnproc_1.html#K14" target="_self">Shift</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q2</font></span>)</span></span>)</span></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="118_1_4_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:118_1_4_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q1</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q2</font></span>)</span></span>)</span></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">k1</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:118_1_4_2_1"/><i><font color="Green">E133</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k1</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k1</font> &amp; <font color="Maroon">k1</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q1</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q2</font></span>)</span></span>)</span> )
 <b>by </b><i/>;<br/>
<a NAME="E4:118_1_4_2_1"/><i><font color="Green">E134</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k1</font> &amp; <font color="Maroon">k1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q1</font></span>)</span> implies <font color="Maroon">k1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q1</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="pnproc_1.html#D6" target="_self">Def6</a></i>;<br/>
<a NAME="E5:118_1_4_2_1"/>
( <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q1</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k1</font> &amp; <font color="Maroon">k1</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q1</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q2</font></span>)</span></span>)</span> implies <font color="Maroon">k1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q1</font></span>)</span></span>)</span> <a href="pnproc_1.html#K14" target="_self">Shift</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q2</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="pnproc_1.html#T16" target="_self">Th16</a></i>;<br/>
<b>hence </b><a NAME="E6:118_1_4_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q1</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q1</font></span>)</span></span>)</span> <a href="pnproc_1.html#K14" target="_self">Shift</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q2</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="pnproc_1.html#T20" target="_self">Th20</a>, <a class="ref" href="pnproc_1.html#T21" target="_self">Th21</a>, <a class="ref" href="int_1.html#T20">INT_1:20</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:118_1_4_2"/><i><font color="Green">E135</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q1</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q1</font></span>)</span></span>)</span> <a href="pnproc_1.html#K14" target="_self">Shift</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q2</font></span>)</span></span>)</span></span>)</span>
 ;<br/>

<a NAME="E3:118_1_4_2"/><i><font color="Green">E137</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q1</font></span>)</span> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q2</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/>
<div><i><font color="Green">E138</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="118_1_4_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:118_1_4_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q1</font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">k1</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:118_1_4_2_2"/><i><font color="Green">E139</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k1</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k1</font> &amp; <font color="Maroon">k1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q1</font></span>)</span> )
 <b>by </b><i><a class="ref" href="pnproc_1.html#D6" target="_self">Def6</a></i>;<br/><a NAME="E4:118_1_4_2_2"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q1</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q1</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q2</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="pnproc_1.html#T21" target="_self">Th21</a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E5:118_1_4_2_2"/><b>then </b>
<font color="Maroon">k1</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q1</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q2</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="pnproc_1.html#T26" target="_self">Th26</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E6:118_1_4_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q1</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q2</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="pnproc_1.html#T26" target="_self">Th26</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="118_1_4_2_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:118_1_4_2_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q1</font></span>)</span></span>)</span> <a href="pnproc_1.html#K14" target="_self">Shift</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q2</font></span>)</span></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">k2</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:118_1_4_2_3"/><i><font color="Green">E140</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k2</font> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q1</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k2</font> &amp; <font color="Maroon">k2</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q1</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q2</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="pnproc_1.html#T16" target="_self">Th16</a></i>;<br/><a NAME="E4:118_1_4_2_3"/>
0 <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> <span class="p2">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q1</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="pnproc_1.html#T21" target="_self">Th21</a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E5:118_1_4_2_3"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k2</font>
 <b>by </b><i><a class="ref" href="pnproc_1.html#T26" target="_self">Th26</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E6:118_1_4_2_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q1</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q2</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="pnproc_1.html#T26" target="_self">Th26</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:118_1_4_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q1</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q2</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="pnproc_1.html#T20" target="_self">Th20</a>, <a class="ref" href="pnproc_1.html#T23" target="_self">Th23</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div><a NAME="E7:118_1_4"/><i><font color="Green">E141</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q1</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q1</font></span>)</span></span>)</span> <a href="pnproc_1.html#K14" target="_self">Shift</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q2</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q1</font></span>)</span></span>)</span> <a href="pnproc_1.html#K14" target="_self">Shift</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q2</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="relat_1.html#T13">RELAT_1:13</a></i>;<br/><a NAME="E8:118_1_4"/>
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q1</font></span>)</span> <a href="xboole_0.html#R1">misses</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q1</font></span>)</span></span>)</span> <a href="pnproc_1.html#K14" target="_self">Shift</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q2</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/><b>then reconsider </b><font color="Maroon">ss1</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q1</font></span>)</span></span>)</span> <a href="pnproc_1.html#K14" target="_self">Shift</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q2</font></span>)</span></span>)</span> as   <a href="funct_1.html#NM1">Function</a> <b>by </b><i><a class="ref" href="grfunc_1.html#T31">GRFUNC_1:31</a></i>;<br/><a NAME="E10:118_1_4"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">ss</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">ss</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ss1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="118_1_4_3"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:118_1_4_3"/><i><font color="Green">E142</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">ss</font></span>)</span>
 ;<br/>

<a NAME="E2:118_1_4_3"/><b>then </b><i><font color="Green">E143</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">ss</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">ss</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D14">FINSEQ_1:def 14</a></i>;<br/>
<i><font color="Green">E144</font></i>: <span class="p1">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">ss</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon">ss</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">ss</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D14">FINSEQ_1:def 14</a></i>
<br/>.= 
<font color="Maroon">ss</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">ss</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="pnproc_1.html#T23" target="_self">Th23</a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
<br/>.= 
<font color="Maroon">ss</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">q1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p4">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font></span>)</span> <a href="pnproc_1.html#K14" target="_self">Shift</a> <font color="Maroon">q2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="pnproc_1.html#T14" target="_self">Th14</a>, <a class="ref" href="relat_1.html#T13">RELAT_1:13</a></i>
<br/>.= 
<font color="Maroon">ss</font> <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#K14">Sgm</a> <span class="p4">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">q1</font></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p4">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font></span>)</span> <a href="pnproc_1.html#K14" target="_self">Shift</a> <font color="Maroon">q2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="pnproc_1.html#T13" target="_self">Th13</a>, , </i>
;<br/>
<div><i><font color="Green">E145</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="118_1_4_3_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:118_1_4_3_2"/><i><font color="Green">E201</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q1</font></span>)</span>
 ;<br/><a NAME="E2:118_1_4_3_2"/><b>then </b><i><font color="Green">E202</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">q1</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E3:118_1_4_3_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">q1</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">q1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E4:118_1_4_3_2"/><b>then </b><i><font color="Green">E203</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">q1</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">q1</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T71">FINSEQ_1:71</a></i>;<br/><span class="p1">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">ss</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<font color="Maroon">ss</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">q1</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="pnproc_1.html#T26" target="_self">Th26</a>, <a class="ref" href="pnproc_1.html#T29" target="_self">Th29</a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
<br/>.= 
<font color="Maroon">q1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">q1</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="pnproc_1.html#T14" target="_self">Th14</a>, <a class="ref" href="pnproc_1.html#T31" target="_self">Th31</a>, <a class="ref" href="grfunc_1.html#T34">GRFUNC_1:34</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">q1</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">q1</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="pnproc_1.html#T29" target="_self">Th29</a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D14">FINSEQ_1:def 14</a></i>
;<br/><b>hence </b><a NAME="E6:118_1_4_3_2"/>
<font color="Maroon">ss1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">ss</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="pnproc_1.html#T28" target="_self">Th28</a>, <a class="ref" href="grfunc_1.html#T34">GRFUNC_1:34</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="118_1_4_3_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:118_1_4_3_3"/><i><font color="Green">E204</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q1</font></span>)</span></span>)</span> <a href="pnproc_1.html#K14" target="_self">Shift</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q2</font></span>)</span></span>)</span>
 ;<br/><a NAME="E2:118_1_4_3_3"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q1</font></span>)</span></span>)</span> <a href="pnproc_1.html#K14" target="_self">Shift</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</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> <span class="p4">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q1</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Olive">j</font></span>)</span> where <font color="Olive">j</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q2</font></span>)</span> </span>}</span> 
 <b>by </b><i><a class="txt" href="pnproc_1.html#E10"><i><font color="Green">Lemma50</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">j</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:118_1_4_3_3"/><i><font color="Green">E206</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q1</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q2</font></span>)</span> )
 <b>by </b><i><a class="ref" href="pnproc_1.html#T28" target="_self">Th28</a></i>;<br/><i><font color="Green">E207</font></i>: <font color="Maroon">ss1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q1</font></span>)</span></span>)</span> <a href="pnproc_1.html#K14" target="_self">Shift</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q2</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="pnproc_1.html#T28" target="_self">Th28</a>, <a class="ref" href="grfunc_1.html#T34">GRFUNC_1:34</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
<b>by </b><i><a class="ref" href="pnproc_1.html#T29" target="_self">Th29</a>, <a class="txt" href="pnproc_1.html#E10"><i><font color="Green">Lemma50</font></i></a></i>
;<br/><b>consider </b><font color="Maroon">l1</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E7:118_1_4_3_3"/><i><font color="Green">E208</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">q1</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">l1</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D12">FINSEQ_1:def 12</a></i>;<br/><b>consider </b><font color="Maroon">l2</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E9:118_1_4_3_3"/><i><font color="Green">E209</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font></span>)</span> <a href="pnproc_1.html#K14" target="_self">Shift</a> <font color="Maroon">q2</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">l2</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D12">FINSEQ_1:def 12</a></i>;<br/><a NAME="E10:118_1_4_3_3"/>
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">q1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">q1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="pnproc_1.html#T32" target="_self">Th32</a>, <a class="ref" href="finseq_3.html#T44">FINSEQ_3:44</a></i>;<br/><a NAME="E11:118_1_4_3_3"/><b>then </b><i><font color="Green">E210</font></i>: 
 <a href="card_1.html#K1">Card</a> <font color="Maroon">q1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">q1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="pre_circ.html#T21">PRE_CIRC:21</a></i>;<br/><a NAME="E12:118_1_4_3_3"/><i><font color="Green">E211</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">q1</font>
 <b>by </b><i/>;<br/><a NAME="E13:118_1_4_3_3"/><i><font color="Green">E212</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q2</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E14:118_1_4_3_3"/>
 <a href="card_1.html#K1">Card</a> <font color="Maroon">q2</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q2</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E15:118_1_4_3_3"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font></span>)</span> <a href="pnproc_1.html#K14" target="_self">Shift</a> <font color="Maroon">q2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q2</font></span>)</span>
 <b>by </b><i><a class="ref" href="pnproc_1.html#T9" target="_self">Th9</a></i>;<br/><a NAME="E16:118_1_4_3_3"/><b>then </b><i><font color="Green">E213</font></i>: 
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font></span>)</span> <a href="pnproc_1.html#K14" target="_self">Shift</a> <font color="Maroon">q2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q2</font></span>)</span>
 <b>by </b><i><a class="ref" href="pre_circ.html#T21">PRE_CIRC:21</a></i>;<br/><a NAME="E17:118_1_4_3_3"/><i><font color="Green">E218</font></i>: 
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font></span>)</span> <a href="pnproc_1.html#K14" target="_self">Shift</a> <font color="Maroon">q2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font></span>)</span> <a href="pnproc_1.html#K14" target="_self">Shift</a> <font color="Maroon">q2</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="pnproc_1.html#D7" target="_self">Def7</a>, <a class="ref" href="finseq_3.html#T44">FINSEQ_3:44</a></i>;<br/><a NAME="E18:118_1_4_3_3"/><i><font color="Green">E219</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font></span>)</span> <a href="pnproc_1.html#K14" target="_self">Shift</a> <font color="Maroon">q2</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q2</font></span>)</span>
 <b>by </b><i><a class="ref" href="pnproc_1.html#D7" target="_self">Def7</a>, <a class="ref" href="pnproc_1.html#D8" target="_self">Def8</a>, <a class="ref" href="finseq_3.html#T44">FINSEQ_3:44</a></i>;<br/><a NAME="E19:118_1_4_3_3"/><i><font color="Green">E220</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font></span>)</span> <a href="pnproc_1.html#K14" target="_self">Shift</a> <font color="Maroon">q2</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="pnproc_1.html#D8" target="_self">Def8</a>, <a class="ref" href="pnproc_1.html#D9" target="_self">Def9</a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E20:118_1_4_3_3"/><i><font color="Green">E221</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font></span>)</span> <a href="pnproc_1.html#K14" target="_self">Shift</a> <font color="Maroon">q2</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="pnproc_1.html#T29" target="_self">Th29</a>, , , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E21:118_1_4_3_3"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font></span>)</span> <a href="pnproc_1.html#K14" target="_self">Shift</a> <font color="Maroon">q2</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font></span>)</span> <a href="pnproc_1.html#K14" target="_self">Shift</a> <font color="Maroon">q2</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="pnproc_1.html#T29" target="_self">Th29</a>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E22:118_1_4_3_3"/><b>then </b><i><font color="Green">E222</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font></span>)</span> <a href="pnproc_1.html#K14" target="_self">Shift</a> <font color="Maroon">q2</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font></span>)</span> <a href="pnproc_1.html#K14" target="_self">Shift</a> <font color="Maroon">q2</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T71">FINSEQ_1:71</a></i>;<br/><span class="p1">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">ss</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<font color="Maroon">ss</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font></span>)</span> <a href="pnproc_1.html#K14" target="_self">Shift</a> <font color="Maroon">q2</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span>
<b>by </b><i><a class="ref" href="pnproc_1.html#T26" target="_self">Th26</a>, <a class="ref" href="pnproc_1.html#T29" target="_self">Th29</a>, , , <a class="ref" href="pnproc_1.html#T34" target="_self">Th34</a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font></span>)</span> <a href="pnproc_1.html#K14" target="_self">Shift</a> <font color="Maroon">q2</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#K14">Sgm</a> <span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font></span>)</span> <a href="pnproc_1.html#K14" target="_self">Shift</a> <font color="Maroon">q2</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span>
<b>by </b><i><a class="ref" href="pnproc_1.html#T14" target="_self">Th14</a>, <a class="ref" href="pnproc_1.html#D10" target="_self">Def10</a>, <a class="ref" href="grfunc_1.html#T34">GRFUNC_1:34</a></i>
<br/>.= 
<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">p1</font></span>)</span> <a href="pnproc_1.html#K14" target="_self">Shift</a> <font color="Maroon">q2</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font></span>)</span> <a href="pnproc_1.html#K14" target="_self">Shift</a> <font color="Maroon">q2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
<b>by </b><i><a class="ref" href="pnproc_1.html#T29" target="_self">Th29</a>, , <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font></span>)</span> <a href="pnproc_1.html#K14" target="_self">Shift</a> <font color="Maroon">q2</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D14">FINSEQ_1:def 14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
<b>by </b><i><a class="ref" href="pnproc_1.html#T29" target="_self">Th29</a>, </i>
;<br/><b>hence </b><a NAME="E24:118_1_4_3_3"/>
<font color="Maroon">ss1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">ss</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="pnproc_1.html#T31" target="_self">Th31</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:118_1_4_3"/>
<font color="Maroon">ss1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">ss</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="pnproc_1.html#D5" target="_self">Def5</a>, <a class="ref" href="pnproc_1.html#T19" target="_self">Th19</a>, <a class="ref" href="pnproc_1.html#T21" target="_self">Th21</a>, <a class="ref" href="pnproc_1.html#T27" target="_self">Th27</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div><a NAME="E11:118_1_4"/><b>then </b>
 <a href="finseq_1.html#K15">Seq</a> <font color="Maroon">ss</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ss1</font>
 <b>by </b><i><a class="ref" href="pnproc_1.html#D5" target="_self">Def5</a>, <a class="ref" href="pnproc_1.html#T19" target="_self">Th19</a>, <a class="ref" href="pnproc_1.html#T20" target="_self">Th20</a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/><b>hence </b><a NAME="E12:118_1_4"/>
 ex <font color="Olive">ss</font> being  <a href="finseq_1.html#NM4">FinSubsequence</a> st <br/>( <font color="Olive">ss</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q1</font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font></span>)</span> <a href="pnproc_1.html#K14" target="_self">Shift</a> <font color="Maroon">q2</font></span>)</span> &amp;  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Olive">ss</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q1</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q2</font></span>)</span></span>)</span></span>)</span> &amp;  <a href="finseq_1.html#K15">Seq</a> <font color="Olive">ss</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q1</font></span>)</span></span>)</span> <a href="pnproc_1.html#K14" target="_self">Shift</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q2</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="pnproc_1.html#T14" target="_self">Th14</a>, <a class="ref" href="pnproc_1.html#D5" target="_self">Def5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
