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<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>for <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 holds <br/> <a href="rvsum_1.html#K16">Product</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="finseqop.html#K4">|-&gt;</a> <font color="Olive">r</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">r</font> <a href="power.html#K4">to_power</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E1:65"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">r</font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E1:65_1"/>
<font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<a NAME="E2:65_1"/>
 <a href="rvsum_1.html#K16">Product</a> <span class="p1">(<span class="default">0 <a href="finseqop.html#K4">|-&gt;</a> <font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="rvsum_1.html#T124">RVSUM_1:124</a>, <a class="ref" href="finseq_2.html#T72">FINSEQ_2:72</a></i>;<br/>
<b>hence </b><a NAME="E3:65_1"/>
 <a href="rvsum_1.html#K16">Product</a> <span class="p1">(<span class="default">0 <a href="finseqop.html#K4">|-&gt;</a> <font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="power.html#K4">to_power</a> 0
 <b>by </b><i><a class="ref" href="power.html#T29">POWER:29</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:65"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">n</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">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font> <a href="binop_2.html#K23">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:65_2"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>]
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">r</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>assume </b><a NAME="E1:65_2_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/> <a href="rvsum_1.html#K16">Product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</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#K16">Product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</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"><a href="rvsum_1.html#K16">Product</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="finseqop.html#K4">|-&gt;</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font>
<b>by </b><i><a class="ref" href="rvsum_1.html#T126">RVSUM_1:126</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="power.html#K4">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">r</font>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="power.html#K4">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="power.html#K4">to_power</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="power.html#T30">POWER:30</a></i>
;<br/><b>hence </b><a NAME="E3:65_2_1"/>
 <a href="rvsum_1.html#K16">Product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</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="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="power.html#T32">POWER:32</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:65_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:65"/>
for <font color="Olive">n</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">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<b>hence </b><a NAME="E4:65"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 holds <br/> <a href="rvsum_1.html#K16">Product</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="finseqop.html#K4">|-&gt;</a> <font color="Olive">r</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">r</font> <a href="power.html#K4">to_power</a> <font color="Olive">n</font>
 ;<br/>


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