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<b>let </b><font color="Maroon">F</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> ;<br/>

<b>assume </b><a NAME="E1:83"/><i><font color="Green">E48</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K3">dom</a> <font color="Maroon">F</font> holds <br/>0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font>
 ;<br/>

<b>set </b><font color="Maroon">i</font> =  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font>;<br/>
<b>set </b><font color="Maroon">R1</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> <a href="finseqop.html#K4">|-&gt;</a> 0;<br/>
<b>reconsider </b><font color="Maroon">R2</font> = <font color="Maroon">F</font> as    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="finseq_2.html#T110">FINSEQ_2:110</a></i>;<br/>
<a NAME="E3:83"/><i><font color="Green">E51</font></i>: 
 <a href="setwop_2.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="setwop_2.html#K3">dom</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:83_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span>
 ;<br/><a NAME="E2:83_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> <a href="finseqop.html#K4">|-&gt;</a> 0</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/><b>hence </b><a NAME="E3:83_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> <a href="finseqop.html#K4">|-&gt;</a> 0</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">R2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">j</font>
 <b>by </b><i>, <a class="txt" href="rvsum_1.html#E24"><i><font color="Green">Lemma55</font></i></a>, <a class="ref" href="rvsum_1.html#T20" target="_self">Th20</a></i>;<br/></div><b>end;</b></div>
<a NAME="E5:83"/><b>then </b>
 <a href="rvsum_1.html#K15" target="_self">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> <a href="finseqop.html#K4">|-&gt;</a> 0</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rvsum_1.html#K15" target="_self">Sum</a> <font color="Maroon">R2</font>
 
<b>by </b><i><a class="txt" href="rvsum_1.html#E3:83"><i><font color="Green">E51</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:83"/>
0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rvsum_1.html#K15" target="_self">Sum</a> <font color="Maroon">F</font>
 <b>by </b><i/>;<br/>


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