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<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="E1:117"/><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/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="117_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:117_1"/>
( <font color="Maroon">q2</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <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="117_1_1"><b>suppose </b></a><a NAME="E1:117_1_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">q2</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>consider </b><font color="Maroon">ss</font> being   <a href="finseq_1.html#NM4">FinSubsequence</a><b> such that </b><br/><a NAME="E3:117_1_1"/><i><font color="Green">E35</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>by </b><i><a class="txt" href="pnproc_1.html#E22"><i><font color="Green">Lemma80</font></i></a>, </i>;<br/><a NAME="E4:117_1_1"/>
<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#T13" target="_self">Th13</a>, <a class="ref" href="pnproc_1.html#T8" target="_self">Th8</a></i>;<br/><a NAME="E5:117_1_1"/><b>then </b><i><font color="Green">E37</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"><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="pnproc_1.html#T14" target="_self">Th14</a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E6:117_1_1"/>
 <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#T13" target="_self">Th13</a>, </i>;<br/><a NAME="E7:117_1_1"/><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">q2</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/><b>hence </b><a NAME="E8:117_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> )
 <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="117_1_2"><b>suppose </b></a><a NAME="E1:117_1_2"/><i><font color="Green">E38</font></i>: 
<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"><b>consider </b><font color="Maroon">ss</font> being   <a href="finseq_1.html#NM4">FinSubsequence</a><b> such that </b><br/><a NAME="E3:117_1_2"/><i><font color="Green">E41</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>by </b><i><a class="txt" href="pnproc_1.html#E22"><i><font color="Green">Lemma80</font></i></a>, </i>;<br/><b>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="E5:117_1_2"/><i><font color="Green">E43</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">k1</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">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="E7:117_1_2"/><i><font color="Green">E45</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">k2</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D12">FINSEQ_1:def 12</a></i>;<br/><a NAME="E8:117_1_2"/><i><font color="Green">E123</font></i>: 
 <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">ss</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">ss</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T71">FINSEQ_1:71</a></i>;<br/><a NAME="E9:117_1_2"/>
 <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="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/><b>then </b><i><font color="Green">E124</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="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">ss</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>
<br/>.= 
 <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"><span class="p3">(<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="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>
<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/>.= 
 <a href="finseq_1.html#K4">dom</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="finseq_1.html#K8">^</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>
<b>by </b><i><a class="txt" href="pnproc_1.html#E22"><i><font color="Green">Lemma80</font></i></a>, <a class="ref" href="pnproc_1.html#T13" target="_self">Th13</a>, </i>
<br/>.= 
 <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#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></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#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></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
;<br/><a NAME="E11:117_1_2"/>
(  <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="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> &amp;  <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> <a href="hidden.html#R1">=</a>  <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> )
 <b>by </b><i><a class="ref" href="pnproc_1.html#D5" target="_self">Def5</a>, , <a class="ref" href="finseq_3.html#T44">FINSEQ_3:44</a></i>;<br/><a NAME="E12:117_1_2"/><b>then </b>
(  <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="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> &amp;  <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> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">q1</font> )
 <b>by </b><i><a class="ref" href="pre_circ.html#T21">PRE_CIRC:21</a></i>;<br/><a NAME="E13:117_1_2"/><b>then </b>
(  <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="card_1.html#K1">Card</a> <font color="Maroon">q2</font> &amp;  <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> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">q1</font> )
 <b>by </b><i><a class="ref" href="pnproc_1.html#T9" target="_self">Th9</a></i>;<br/><a NAME="E14:117_1_2"/><b>then </b>
(  <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> &amp;  <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> <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">q1</font></span>)</span> )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E15:117_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> )
 <b>by </b><i><a class="ref" href="pnproc_1.html#T14" target="_self">Th14</a>, <a class="ref" href="pnproc_1.html#D6" target="_self">Def6</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
