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<b>let </b><font color="Maroon">a</font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><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>defpred </b><font color="Maroon">S<sub>1</sub></font>[ <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ] <b>means </b> <a href="rvsum_1.html#K16">Product</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="nat_3.html#K2" target="_self">|^</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K16">Product</a> <font color="Maroon">f</font></span>)</span> <a href="newton.html#K2">|^</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E1:25"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1"><b>proof </b></a><div class="add">

<b>thus </b> <a href="rvsum_1.html#K16">Product</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="nat_3.html#K2" target="_self">|^</a> 0</span>)</span> = 
 <a href="rvsum_1.html#K16">Product</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="newton.html#K1">|-&gt;</a> 1</span>)</span>
<b>by </b><i/>
<br/>.= 
1
<b>by </b><i><a class="ref" href="rvsum_1.html#T132">RVSUM_1:132</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K16">Product</a> <font color="Maroon">f</font></span>)</span> <a href="newton.html#K2">|^</a> 0
<b>by </b><i><a class="ref" href="newton.html#T9">NEWTON:9</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E2:25"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">b</font> being <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b</font> <a href="xcmplx_0.html#K2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">b</font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E1:25_2"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">b</font>]
 ;<br/>

<b>thus </b> <a href="rvsum_1.html#K16">Product</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="nat_3.html#K2" target="_self">|^</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K16">Product</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="nat_3.html#K2" target="_self">|^</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K16">Product</a> <font color="Maroon">f</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K16">Product</a> <font color="Maroon">f</font></span>)</span> <a href="newton.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
<b>by </b><i>, <a class="ref" href="newton.html#T11">NEWTON:11</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E3:25"/>
for <font color="Olive">b</font> being <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S2">NAT_1:sch 2</a>(, );<br/></i>
<b>hence </b><a NAME="E4:25"/>
 <a href="rvsum_1.html#K16">Product</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="nat_3.html#K2" target="_self">|^</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K16">Product</a> <font color="Maroon">f</font></span>)</span> <a href="newton.html#K2">|^</a> <font color="Maroon">a</font>
 ;<br/>


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