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

<b>set </b><font color="Maroon">seq1</font> =  <a href="funct_3.html#K6">incl</a> <a href="numbers.html#K5">NAT</a> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_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="E1:60_1"/>
( <span class="p1">(<span class="default"><a href="funct_3.html#K6">incl</a> <a href="numbers.html#K5">NAT</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp; <span class="p1">(<span class="default"><a href="funct_3.html#K6">incl</a> <a href="numbers.html#K5">NAT</a> </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="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>;<br/><b>hence </b><a NAME="E2:60_1"/>
<span class="p1">(<span class="default"><a href="funct_3.html#K6">incl</a> <a href="numbers.html#K5">NAT</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="funct_3.html#K6">incl</a> <a href="numbers.html#K5">NAT</a> </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>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/></div><b>end;</b></div>
<a NAME="E2:60"/><b>then </b><i><font color="Green">E31</font></i>: 
 <a href="funct_3.html#K6">incl</a> <a href="numbers.html#K5">NAT</a>  is <a href="seqm_3.html#V1" target="_self">increasing</a>
 
<b>by </b><i/>;<br/>
<a NAME="E3:60"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <span class="p1">(<span class="default"><a href="funct_3.html#K6">incl</a> <a href="numbers.html#K5">NAT</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">seq1</font> =  <a href="funct_3.html#K6">incl</a> <a href="numbers.html#K5">NAT</a>  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#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="60_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/><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> <font color="Maroon">n</font>
<b>by </b><i><a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:60"/>
<font color="Maroon">seq</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/>


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