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

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">seq</font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/>



<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  = <font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font>;<br/>
<b>consider </b><font color="Maroon">seq1</font> being   <a href="seq_1.html#NM1">Real_Sequence</a><b> such that </b><br/><a NAME="E2:61"/><i><font color="Green">E31</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">seq1</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">n</font>)
 <b>from </b><i><a class="ref" href="seq_1.html#S1">SEQ_1:sch 1</a>();<br/></i>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1"><b>now </b></a><div class="add"><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> ;<br/><a NAME="E1:61_1"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">seq1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font> &amp; <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="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font> )
 <b>by </b><i/>;<br/><a NAME="E2:61_1"/>
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>hence </b><a NAME="E3:61_1"/>
<font color="Maroon">seq1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <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>
 <b>by </b><i><a class="txt" href="seqm_3.html#E5"><i><font color="Green">Lemma43</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E4:61"/><b>then </b><i><font color="Green">E34</font></i>: 
<font color="Maroon">seq1</font> is <a href="seqm_3.html#V1" target="_self">increasing</a>
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_2"><b>now </b></a><div class="add"><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> ;<br/><a NAME="E1:61_2"/>
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> 
 ;<br/><b>hence </b><a NAME="E2:61_2"/>
<font color="Maroon">seq1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">seq1</font> = <font color="Maroon">seq1</font> as  <a href="seqm_3.html#V1" target="_self">increasing</a> <a href="seqm_3.html#NM1">Seq_of_Nat</a> <b>by </b><i><a class="txt" href="seqm_3.html#E5"><i><font color="Green">Lemma43</font></i></a>, <a class="txt" href="seqm_3.html#E2"><i><font color="Green">Lemma35</font></i></a></i>;<br/>
<b>take </b><font color="Maroon">Nseq</font> = <font color="Maroon">seq1</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="seqm_3.html#D10" target="_self">SEQM_3:def 10</a><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_3"><b>now </b></a><div class="add"><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> ;<br/><b>thus </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> <font color="Maroon">n</font> = 
<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/>.= 
<font color="Maroon">seq</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> <font color="Maroon">k</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1" target="_self">^\</a> <font color="Maroon">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
<b>by </b><i><a class="txt" href="seqm_3.html#E4:61"><i><font color="Green">E34</font></i></a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:61"/>
<font color="Maroon">seq</font> <a href="seqm_3.html#K1" target="_self">^\</a> <font color="Maroon">k</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="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/>


</div>
