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

<b>let </b><font color="Maroon">seq</font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/><b>let </b><font color="Maroon">seq1</font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:64"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">seq</font> is <a href="seqm_3.html#V2" target="_self">decreasing</a>
 <b>and </b><br/><a NAME="E2:64"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">seq1</font> is    <a href="seqm_3.html#M1" target="_self">subsequence</a> of <font color="Maroon">seq</font>
 ;<br/>

<b>consider </b><font color="Maroon">Nseq</font> being  <a href="seqm_3.html#V1" target="_self">increasing</a> <a href="seqm_3.html#NM1">Seq_of_Nat</a><b> such that </b><br/><a NAME="E4:64"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">seq1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">seq</font> <a href="seqm_3.html#K3" target="_self">*</a> <font color="Maroon">Nseq</font>
 <b>by </b><i><a class="txt" href="seqm_3.html#E5"><i><font color="Green">Lemma43</font></i></a>, </i>;<br/>
<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="seqm_3.html#D12" target="_self">SEQM_3:def 12</a><br/>

<a NAME="E5:64"/>
<font color="Maroon">Nseq</font> <a href="seqm_3.html#K2" target="_self">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">Nseq</font> <a href="seqm_3.html#K2" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E6:64"/><b>then </b>
<font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">Nseq</font> <a href="seqm_3.html#K2" target="_self">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">Nseq</font> <a href="seqm_3.html#K2" target="_self">.</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:64"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K3" target="_self">*</a> <font color="Maroon">Nseq</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">Nseq</font> <a href="seqm_3.html#K2" target="_self">.</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E8:64"/>
<font color="Maroon">seq1</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">seq1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="seqm_3.html#D1" target="_self">Def1</a>, </i>;<br/>


</div>
