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

<b>let </b><font color="Maroon" title="c1">F</font> be  <a href="seq_1.html#NV1" title="SEQ_1:NV.1">real-yielding</a> <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>;<br/>

<b>assume </b><a NAME="E1:103"/><i><font color="Green" title="E47">A1</font></i>: 
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">F</font> holds <br/> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">F</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font>
 ;<br/>

<b>set </b><font color="Maroon" title="c2">i</font> =  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">F</font>;<br/>
<b>set </b><font color="Maroon" title="c3">R1</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ;<br/>
<b>reconsider </b><font color="Maroon" title="c4">F1</font> = <font color="Maroon" title="c1">F</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <b>by </b><i><a class="txt" href="rvsum_1.html#E31"><i><font color="Green" title="E10">Lm4</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">R2</font> = <font color="Maroon" title="c4">F1</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <b>by </b><i><a class="ref" href="finseq_2.html#T110" title="FINSEQ_2:th.110">FINSEQ_2:110</a></i>;<br/>
<a NAME="E4:103"/><i><font color="Green" title="E48">A2</font></i>: 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">F</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<div><i><font color="Green" title="E49">A3</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="103_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">j</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>assume </b><a NAME="E1:103_1"/><i><font color="Green" title="E50">A4</font></i>: 
<font color="Maroon" title="c6">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">F</font></span>)</span>
 ;<br/><a NAME="E2:103_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/><b>hence </b><a NAME="E3:103_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">R2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">j</font>
 <b>by </b><i><a class="txt" href="rvsum_1.html#E1:103"><i><font color="Green" title="E47">A1</font></i></a>, <a class="txt" href="rvsum_1.html#E4:103"><i><font color="Green" title="E48">A2</font></i></a>, <a class="txt" href="rvsum_1.html#E1:103_1"><i><font color="Green" title="E50">A4</font></i></a></i>;<br/></div><b>end;</b></div>
<b>given </b><font color="Maroon" title="c6">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><a NAME="E7:103"/><i><font color="Green" title="E50">A5</font></i>: 
<font color="Maroon" title="c6">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">F</font>
 <b>and </b><br/><a NAME="E8:103"/><i><font color="Green" title="E51">A6</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">F</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">j</font>
 ;<br/>

<a NAME="E9:103"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="rvsum_1.html#E4:103"><i><font color="Green" title="E48">A2</font></i></a>, <a class="txt" href="rvsum_1.html#E7:103"><i><font color="Green" title="E50">A5</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/>
<a NAME="E10:103"/><b>then </b>
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c5">R2</font>
 
<b>by </b><i><a class="txt" href="rvsum_1.html#E4:103"><i><font color="Green" title="E48">A2</font></i></a>, <a class="txt" href="rvsum_1.html#E5:103"><i><font color="Green" title="E49">A3</font></i></a>, <a class="txt" href="rvsum_1.html#E7:103"><i><font color="Green" title="E50">A5</font></i></a>, <a class="txt" href="rvsum_1.html#E8:103"><i><font color="Green" title="E51">A6</font></i></a>, <a class="ref" href="rvsum_1.html#T113" target="_self" title="RVSUM_1:th.113">Th113</a></i>;<br/>
<b>hence </b><a NAME="E11:103"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="rvsum_1.html#K16" title="RVSUM_1:func.16">Sum</a> <font color="Maroon" title="c1">F</font>
 <b>by </b><i><a class="ref" href="rvsum_1.html#T111" target="_self" title="RVSUM_1:th.111">Th111</a></i>;<br/>


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