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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></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="nat_1.html#NM1">Nat</a>] means  <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">c<sub>2</sub></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">c<sub>2</sub></font>;<br/>
<a NAME="E1:79"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i><a class="ref" href="rvsum_1.html#T102" target="_self">Th102</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">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></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">c<sub>3</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:79_1"/><i><font color="Green">E52</font></i>: 
 <a href="rvsum_1.html#K15" target="_self">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finseqop.html#K4">|-&gt;</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="finseqop.html#K4">|-&gt;</a> <font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></font> <a href="finseqop.html#K4">|-&gt;</a> <font color="Maroon">c<sub>2</sub></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">c<sub>2</sub></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">c<sub>3</sub></font> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>2</sub></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">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="rvsum_1.html#E1:79_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="rvsum_1.html#T104" target="_self">Th104</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>2</sub></font>

;<br/>


</div><b>end;</b></div>
<a NAME="E3:79"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></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#E1:79"><i><font color="Green">E50</font></i></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">c<sub>1</sub></font> <a href="finseqop.html#K4">|-&gt;</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>


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