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

<b>let </b><font color="Maroon">i</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">r</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> ;<br/>



<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b> <a href="rvsum_1.html#K15" target="_self">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="finseqop.html#K4">|-&gt;</a> <font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> <a href="binop_2.html#K11">*</a> <font color="Maroon">r</font>;<br/>
<a NAME="E1:79"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i><a class="ref" href="rvsum_1.html#T3" target="_self">Th3</a>, <a class="ref" href="finseq_2.html#T113">FINSEQ_2:113</a></i>;<br/>
<a NAME="E2:79"/><i><font color="Green">E51</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="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font> <a href="binop_2.html#K23">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:79_1"/><i><font color="Green">E60</font></i>: 
 <a href="rvsum_1.html#K15" target="_self">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="finseqop.html#K4">|-&gt;</a> <font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="binop_2.html#K11">*</a> <font color="Maroon">r</font>
 ;<br/>

<b>thus </b> <a href="rvsum_1.html#K15" target="_self">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="finseqop.html#K4">|-&gt;</a> <font color="Maroon">r</font></span>)</span> = 
 <a href="rvsum_1.html#K15" target="_self">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="finseqop.html#K4">|-&gt;</a> <font color="Maroon">r</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">r</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_2.html#T74">FINSEQ_2:74</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binop_2.html#K11">*</a> <font color="Maroon">r</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K11">*</a> <font color="Maroon">r</font></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T16" target="_self">Th16</a>, <a class="ref" href="rvsum_1.html#D2" target="_self">Def2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">r</font>

;<br/>


</div><b>end;</b></div>
<a NAME="E3:79"/>
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, <a class="txt" href="rvsum_1.html#E2:79"><i><font color="Green">E51</font></i></a>);<br/></i>
<b>hence </b><a NAME="E4:79"/>
 <a href="rvsum_1.html#K15" target="_self">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="finseqop.html#K4">|-&gt;</a> <font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="binop_2.html#K11">*</a> <font color="Maroon">r</font>
 ;<br/>


</div>
