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

<b>let </b><font color="Maroon">r</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">j</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">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:8"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>and </b><br/><a NAME="E2:8"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font>
 ;<br/>

<a NAME="E3:8"/><i><font color="Green">E58</font></i>: 
 <a href="rvsum_1.html#K16">Product</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="newton.html#K1">|-&gt;</a> <font color="Maroon">r</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i>, </i>;<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>( <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a<sub>1</sub></font> implies  <a href="rvsum_1.html#K16">Product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font></span>)</span> <a href="newton.html#K1">|-&gt;</a> <font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K16">Product</a> <span class="p2">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="newton.html#K1">|-&gt;</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K16">Product</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="newton.html#K1">|-&gt;</a> <font color="Maroon">r</font></span>)</span></span>)</span> );<br/>
<a NAME="E4:8"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:8_1"/>
<font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 ;<br/>

<a NAME="E2:8_1"/><b>then </b><i><font color="Green">E66</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="nat_1.html#T19">NAT_1:19</a></i>;<br/>
<b>hence </b> <a href="rvsum_1.html#K16">Product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">0 <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font></span>)</span> <a href="newton.html#K1">|-&gt;</a> <font color="Maroon">r</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K16">Product</a> <span class="p2">(<span class="default">0 <a href="newton.html#K1">|-&gt;</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K16">Product</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K6">&lt;*&gt;</a> <a href="numbers.html#K1">REAL</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="jordan3.html#T2">JORDAN3:2</a>, <a class="ref" href="rvsum_1.html#T124">RVSUM_1:124</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K16">Product</a> <span class="p2">(<span class="default">0 <a href="newton.html#K1">|-&gt;</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K16">Product</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="newton.html#K1">|-&gt;</a> <font color="Maroon">r</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="finseq_2.html#T72">FINSEQ_2:72</a></i>
;<br/><br/>


</div><b>end;</b></div>
<a NAME="E5:8"/><i><font color="Green">E67</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="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_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><b>that </b><br/><a NAME="E1:8_2"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>]
 <b>and </b><br/><a NAME="E2:8_2"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:8_2_1"/>
( <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> or <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 )
 </span><b>by </b><i>, <a class="ref" href="nat_1.html#T26">NAT_1:26</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_2_1_1"><b>suppose </b></a><a NAME="E1:8_2_1_1"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 ;<br/><div class="add"> <a href="rvsum_1.html#K16">Product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="newton.html#K1">|-&gt;</a> <font color="Maroon">r</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K16">Product</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font></span>)</span> <a href="newton.html#K1">|-&gt;</a> <font color="Maroon">r</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> <span class="p2">(<span class="default">1 <a href="newton.html#K1">|-&gt;</a> <font color="Maroon">r</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T134">RVSUM_1:134</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K16">Product</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K1">|-&gt;</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K16">Product</a> <span class="p3">(<span class="default"><font color="Maroon">j</font> <a href="newton.html#K1">|-&gt;</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font>
<b>by </b><i>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K16">Product</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K1">|-&gt;</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K16">Product</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="newton.html#K1">|-&gt;</a> <font color="Maroon">r</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T75">XCMPLX_1:75</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K16">Product</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="newton.html#K1">|-&gt;</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K16">Product</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="newton.html#K1">|-&gt;</a> <font color="Maroon">r</font></span>)</span></span>)</span>
<b>by </b><i/>
;<br/><b>hence </b><a NAME="E3:8_2_1_1"/>
 <a href="rvsum_1.html#K16">Product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font></span>)</span> <a href="newton.html#K1">|-&gt;</a> <font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K16">Product</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="newton.html#K1">|-&gt;</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K16">Product</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="newton.html#K1">|-&gt;</a> <font color="Maroon">r</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="jordan4.html#T3">JORDAN4:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_2_1_2"><b>suppose </b></a><a NAME="E1:8_2_1_2"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><b>hence </b> <a href="rvsum_1.html#K16">Product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font></span>)</span> <a href="newton.html#K1">|-&gt;</a> <font color="Maroon">r</font></span>)</span> = 
 <a href="rvsum_1.html#K16">Product</a> <span class="p1">(<span class="default">0 <a href="newton.html#K1">|-&gt;</a> <font color="Maroon">r</font></span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#T51">BINARITH:51</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="finseq_2.html#T72">FINSEQ_2:72</a>, <a class="ref" href="rvsum_1.html#T124">RVSUM_1:124</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K16">Product</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="newton.html#K1">|-&gt;</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K16">Product</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="newton.html#K1">|-&gt;</a> <font color="Maroon">r</font></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="xcmplx_1.html#T60">XCMPLX_1:60</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E6:8"/>
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="E7:8"/>
 <a href="rvsum_1.html#K16">Product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font></span>)</span> <a href="newton.html#K1">|-&gt;</a> <font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K16">Product</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="newton.html#K1">|-&gt;</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K16">Product</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="newton.html#K1">|-&gt;</a> <font color="Maroon">r</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/>


</div>
