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

<b>let </b><font color="Maroon" title="c2">seq</font>, <font color="Maroon" title="c3">seq1</font>, <font color="Maroon" title="c4">seq2</font> be   <a href="comseq_1.html#NM1" title="COMSEQ_1:NM.1">Complex_Sequence</a>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:64"/><i><font color="Green" title="E41">A1</font></i>: 
<font color="Maroon" title="c2">seq</font> is    <a href="comseq_3.html#M1" title="COMSEQ_3:mode.1">subsequence</a> of <font color="Maroon" title="c3">seq1</font>
 <b>and </b><br/><a NAME="E2:64"/><i><font color="Green" title="E42">A2</font></i>: 
<font color="Maroon" title="c3">seq1</font> is    <a href="comseq_3.html#M1" title="COMSEQ_3:mode.1">subsequence</a> of <font color="Maroon" title="c4">seq2</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c5">Nseq</font> being  <a href="seqm_3.html#V1" title="SEQM_3:attr.1">increasing</a> <a href="seqm_3.html#NM1" title="SEQM_3:NM.1">Seq_of_Nat</a><b> such that </b><br/><a NAME="E4:64"/><i><font color="Green" title="E43">A3</font></i>: 
<font color="Maroon" title="c2">seq</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">seq1</font> <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">*</a> <font color="Maroon" title="c5">Nseq</font>
 <b>by </b><i><a class="txt" href="comseq_3.html#E1:64"><i><font color="Green" title="E41">A1</font></i></a>, <a class="ref" href="comseq_3.html#D9" target="_self" title="COMSEQ_3:def.9">Def9</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">Nseq1</font> being  <a href="seqm_3.html#V1" title="SEQM_3:attr.1">increasing</a> <a href="seqm_3.html#NM1" title="SEQM_3:NM.1">Seq_of_Nat</a><b> such that </b><br/><a NAME="E6:64"/><i><font color="Green" title="E44">A4</font></i>: 
<font color="Maroon" title="c3">seq1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c4">seq2</font> <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">*</a> <font color="Maroon" title="c6">Nseq1</font>
 <b>by </b><i><a class="txt" href="comseq_3.html#E2:64"><i><font color="Green" title="E42">A2</font></i></a>, <a class="ref" href="comseq_3.html#D9" target="_self" title="COMSEQ_3:def.9">Def9</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c4">seq2</font> <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">Nseq1</font> <a href="seqm_3.html#K3" title="SEQM_3:func.3">*</a> <font color="Maroon" title="c5">Nseq</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">n</font> = 
<font color="Maroon" title="c4">seq2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">Nseq1</font> <a href="seqm_3.html#K3" title="SEQM_3:func.3">*</a> <font color="Maroon" title="c5">Nseq</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>
<br/>.= 
<font color="Maroon" title="c4">seq2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seqm_3.html#T31" title="SEQM_3:th.31">SEQM_3:31</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">seq2</font> <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">*</a> <font color="Maroon" title="c6">Nseq1</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>
<br/>.= 
<font color="Maroon" title="c2">seq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">n</font>
<b>by </b><i><a class="txt" href="comseq_3.html#E4:64"><i><font color="Green" title="E43">A3</font></i></a>, <a class="txt" href="comseq_3.html#E6:64"><i><font color="Green" title="E44">A4</font></i></a>, <a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E8:64"/><b>then </b>
<font color="Maroon" title="c2">seq</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c4">seq2</font> <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">Nseq1</font> <a href="seqm_3.html#K3" title="SEQM_3:func.3">*</a> <font color="Maroon" title="c5">Nseq</font></span>)</span>
 
<b>by </b><i><a class="ref" href="comseq_1.html#T6" title="COMSEQ_1:th.6">COMSEQ_1:6</a></i>;<br/>
<b>hence </b><a NAME="E9:64"/>
<font color="Maroon" title="c2">seq</font> is    <a href="comseq_3.html#M1" title="COMSEQ_3:mode.1">subsequence</a> of <font color="Maroon" title="c4">seq2</font>
 <b>by </b><i><a class="ref" href="comseq_3.html#D9" target="_self" title="COMSEQ_3:def.9">Def9</a></i>;<br/>


</div>
