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<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">R</font> be    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="supinf_1.html#K3">REAL</a> ;<br/>



<i><font color="Green">E38</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="finseqop.html#K4">|-&gt;</a> 0</span>)</span> = 
<font color="Maroon">n</font>
<b>by </b><i><a class="ref" href="finseq_2.html#T69">FINSEQ_2:69</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">R</font>
<b>by </b><i><a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>
;<br/>
<i><font color="Green">E39</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="finseqop.html#K4">|-&gt;</a> 0</span>)</span>,<font color="Maroon">R</font></span>)</span> = 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="binop_2.html#K35">multreal</a>  <a href="finseqop.html#K1">.:</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="finseqop.html#K4">|-&gt;</a> 0</span>)</span>,<font color="Maroon">R</font></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#D9">RVSUM_1:def 9</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="finseqop.html#K4">|-&gt;</a> 0</span>)</span>
<b>by </b><i>, <a class="ref" href="finseq_2.html#T86">FINSEQ_2:86</a></i>
<br/>.= 
<font color="Maroon">n</font>
<b>by </b><i><a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>
;<br/>
<a NAME="E3:5"/><i><font color="Green">E40</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="finseqop.html#K4">|-&gt;</a> 0</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T69">FINSEQ_2:69</a></i>;<br/>
<a NAME="E4:5"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="finseqop.html#K4">|-&gt;</a> 0</span>)</span>,<font color="Maroon">R</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="finseqop.html#K4">|-&gt;</a> 0</span>)</span>,<font color="Maroon">R</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="finseqop.html#K4">|-&gt;</a> 0</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:5_3"/><i><font color="Green">E41</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="finseqop.html#K4">|-&gt;</a> 0</span>)</span>,<font color="Maroon">R</font></span>)</span> )
 ;<br/>

<a NAME="E2:5_3"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</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="rvsum_1.html#K14">mlt</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="finseqop.html#K4">|-&gt;</a> 0</span>)</span>,<font color="Maroon">R</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<span class="p1">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="finseqop.html#K4">|-&gt;</a> 0</span>)</span>,<font color="Maroon">R</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="finseqop.html#K4">|-&gt;</a> 0</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T87">RVSUM_1:87</a></i>
<br/>.= 
0 <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="finseq_2.html#T70">FINSEQ_2:70</a></i>
;<br/>
<b>hence </b><a NAME="E4:5_3"/>
<span class="p1">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="finseqop.html#K4">|-&gt;</a> 0</span>)</span>,<font color="Maroon">R</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="finseqop.html#K4">|-&gt;</a> 0</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font>
 <b>by </b><i>, , <a class="ref" href="finseq_2.html#T70">FINSEQ_2:70</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:5"/>
 <a href="rvsum_1.html#K14">mlt</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="finseqop.html#K4">|-&gt;</a> 0</span>)</span>,<font color="Maroon">R</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="finseqop.html#K4">|-&gt;</a> 0
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T18">FINSEQ_1:18</a></i>;<br/>


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