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<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">j</font>, <font color="Olive" title="b2">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c1">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">i</font> holds <br/> <a href="rvsum_1.html#K21" title="RVSUM_1:func.21">Product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Olive" title="b1">j</font></span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|-&gt;</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K21" title="RVSUM_1:func.21">Product</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">i</font> <a href="newton.html#K1" title="NEWTON:func.1">|-&gt;</a> <font color="Maroon" title="c1">r</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K21" title="RVSUM_1:func.21">Product</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">j</font> <a href="newton.html#K1" title="NEWTON:func.1">|-&gt;</a> <font color="Maroon" title="c1">r</font></span>)</span></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">j</font>, <font color="Maroon" title="c3">i</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c2">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">i</font> implies  <a href="rvsum_1.html#K21" title="RVSUM_1:func.21">Product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c2">j</font></span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|-&gt;</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K21" title="RVSUM_1:func.21">Product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="newton.html#K1" title="NEWTON:func.1">|-&gt;</a> <font color="Maroon" title="c1">r</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K21" title="RVSUM_1:func.21">Product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">j</font> <a href="newton.html#K1" title="NEWTON:func.1">|-&gt;</a> <font color="Maroon" title="c1">r</font></span>)</span></span>)</span> )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:13"/><span class="lab"><font color="Green" title="E8">A1</font></span>: 
<font color="Maroon" title="c1">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">and </span><br/><a NAME="E2:13"/><span class="lab"><font color="Green" title="E9">A2</font></span>: 
<font color="Maroon" title="c2">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="rvsum_1.html#K21" title="RVSUM_1:func.21">Product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c2">j</font></span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|-&gt;</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K21" title="RVSUM_1:func.21">Product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="newton.html#K1" title="NEWTON:func.1">|-&gt;</a> <font color="Maroon" title="c1">r</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K21" title="RVSUM_1:func.21">Product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">j</font> <a href="newton.html#K1" title="NEWTON:func.1">|-&gt;</a> <font color="Maroon" title="c1">r</font></span>)</span></span>)</span></span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <span class="kw">means </span>( <font color="Maroon" title="c2">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> $1 implies  <a href="rvsum_1.html#K21" title="RVSUM_1:func.21">Product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">$1 <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c2">j</font></span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|-&gt;</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K21" title="RVSUM_1:func.21">Product</a> <span class="p2">(<span class="default">$1 <a href="newton.html#K1" title="NEWTON:func.1">|-&gt;</a> <font color="Maroon" title="c1">r</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K21" title="RVSUM_1:func.21">Product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">j</font> <a href="newton.html#K1" title="NEWTON:func.1">|-&gt;</a> <font color="Maroon" title="c1">r</font></span>)</span></span>)</span> );<br/>
<a NAME="E3:13"/><span class="lab"><font color="Green" title="E10">A3</font></span>: 
 <a href="rvsum_1.html#K21" title="RVSUM_1:func.21">Product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">j</font> <a href="newton.html#K1" title="NEWTON:func.1">|-&gt;</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:13"><span class="lab"><font color="Green" title="E8">A1</font></span></a>, <a class="ref" href="topreal6.html#T13" target="_self" title="TOPREAL6:th.13">Th13</a></span>;<br/>
<a NAME="E4:13"/><span class="lab"><font color="Green" title="E11">A4</font></span>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">n</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:13_1"/><span class="lab"><font color="Green" title="E12">A5</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">n</font>]
 <span class="kw">and </span><br/><a NAME="E2:13_1"/><span class="lab"><font color="Green" title="E13">A6</font></span>: 
<font color="Maroon" title="c2">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="rvsum_1.html#K21" title="RVSUM_1:func.21">Product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c2">j</font></span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|-&gt;</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K21" title="RVSUM_1:func.21">Product</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|-&gt;</a> <font color="Maroon" title="c1">r</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K21" title="RVSUM_1:func.21">Product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">j</font> <a href="newton.html#K1" title="NEWTON:func.1">|-&gt;</a> <font color="Maroon" title="c1">r</font></span>)</span></span>)</span></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:13_1_1"/>
( <font color="Maroon" title="c2">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> or <font color="Maroon" title="c2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:13_1"><span class="lab"><font color="Green" title="E13">A6</font></span></a>, <a class="ref" href="nat_1.html#T8" title="NAT_1:th.8">NAT_1:8</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:13_1_1_1"/><span class="lab"><font color="Green" title="E14">A7</font></span>: 
<font color="Maroon" title="c2">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="rvsum_1.html#K21" title="RVSUM_1:func.21">Product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c2">j</font></span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|-&gt;</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K21" title="RVSUM_1:func.21">Product</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|-&gt;</a> <font color="Maroon" title="c1">r</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K21" title="RVSUM_1:func.21">Product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">j</font> <a href="newton.html#K1" title="NEWTON:func.1">|-&gt;</a> <font color="Maroon" title="c1">r</font></span>)</span></span>)</span></span><br/><div class="add"> <a href="rvsum_1.html#K21" title="RVSUM_1:func.21">Product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c2">j</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|-&gt;</a> <font color="Maroon" title="c1">r</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K21" title="RVSUM_1:func.21">Product</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c2">j</font></span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|-&gt;</a> <font color="Maroon" title="c1">r</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K21" title="RVSUM_1:func.21">Product</a> <span class="p2">(<span class="default">1 <a href="newton.html#K1" title="NEWTON:func.1">|-&gt;</a> <font color="Maroon" title="c1">r</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T134" title="RVSUM_1:th.134">RVSUM_1:134</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K21" title="RVSUM_1:func.21">Product</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="newton.html#K1" title="NEWTON:func.1">|-&gt;</a> <font color="Maroon" title="c1">r</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K21" title="RVSUM_1:func.21">Product</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">j</font> <a href="newton.html#K1" title="NEWTON:func.1">|-&gt;</a> <font color="Maroon" title="c1">r</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c1">r</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:13_1"><span class="lab"><font color="Green" title="E12">A5</font></span></a>, <a class="txt" href="#E1:13_1_1_1"><span class="lab"><font color="Green" title="E14">A7</font></span></a>, <a class="ref" href="topreal6.html#T10" target="_self" title="TOPREAL6:th.10">Th10</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K21" title="RVSUM_1:func.21">Product</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="newton.html#K1" title="NEWTON:func.1">|-&gt;</a> <font color="Maroon" title="c1">r</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K21" title="RVSUM_1:func.21">Product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">j</font> <a href="newton.html#K1" title="NEWTON:func.1">|-&gt;</a> <font color="Maroon" title="c1">r</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T75" title="XCMPLX_1:th.75">XCMPLX_1:75</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K21" title="RVSUM_1:func.21">Product</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|-&gt;</a> <font color="Maroon" title="c1">r</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K21" title="RVSUM_1:func.21">Product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">j</font> <a href="newton.html#K1" title="NEWTON:func.1">|-&gt;</a> <font color="Maroon" title="c1">r</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="topreal6.html#T12" target="_self" title="TOPREAL6:th.12">Th12</a></span>
;<br/><span class="kw">hence </span><a NAME="E3:13_1_1_1"/>
 <a href="rvsum_1.html#K21" title="RVSUM_1:func.21">Product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c2">j</font></span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|-&gt;</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K21" title="RVSUM_1:func.21">Product</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|-&gt;</a> <font color="Maroon" title="c1">r</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K21" title="RVSUM_1:func.21">Product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">j</font> <a href="newton.html#K1" title="NEWTON:func.1">|-&gt;</a> <font color="Maroon" title="c1">r</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:13_1_1_1"><span class="lab"><font color="Green" title="E14">A7</font></span></a>, <a class="ref" href="nat_d.html#T38" title="NAT_D:th.38">NAT_D:38</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:13_1_1_2"/><span class="lab"><font color="Green" title="E14">A8</font></span>: 
<font color="Maroon" title="c2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="rvsum_1.html#K21" title="RVSUM_1:func.21">Product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c2">j</font></span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|-&gt;</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K21" title="RVSUM_1:func.21">Product</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|-&gt;</a> <font color="Maroon" title="c1">r</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K21" title="RVSUM_1:func.21">Product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">j</font> <a href="newton.html#K1" title="NEWTON:func.1">|-&gt;</a> <font color="Maroon" title="c1">r</font></span>)</span></span>)</span></span><br/><div class="add"><span class="kw">hence </span> <a href="rvsum_1.html#K21" title="RVSUM_1:func.21">Product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c2">j</font></span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|-&gt;</a> <font color="Maroon" title="c1">r</font></span>)</span> = 
 <a href="rvsum_1.html#K21" title="RVSUM_1:func.21">Product</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="newton.html#K1" title="NEWTON:func.1">|-&gt;</a> <font color="Maroon" title="c1">r</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T234" title="XREAL_1:th.234">XREAL_1:234</a></span>
<br/>.= 
1
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T124" title="RVSUM_1:th.124">RVSUM_1:124</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K21" title="RVSUM_1:func.21">Product</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|-&gt;</a> <font color="Maroon" title="c1">r</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K21" title="RVSUM_1:func.21">Product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">j</font> <a href="newton.html#K1" title="NEWTON:func.1">|-&gt;</a> <font color="Maroon" title="c1">r</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:13"><span class="lab"><font color="Green" title="E10">A3</font></span></a>, <a class="txt" href="#E1:13_1_1_2"><span class="lab"><font color="Green" title="E14">A8</font></span></a>, <a class="ref" href="xcmplx_1.html#T60" title="XCMPLX_1:th.60">XCMPLX_1:60</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E5:13"/><span class="lab"><font color="Green" title="E12">A9</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:13_2"/><span class="lab"><font color="Green" title="E13">A10</font></span>: 
<font color="Maroon" title="c2">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="rvsum_1.html#K21" title="RVSUM_1:func.21">Product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c2">j</font></span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|-&gt;</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K21" title="RVSUM_1:func.21">Product</a> <span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="newton.html#K1" title="NEWTON:func.1">|-&gt;</a> <font color="Maroon" title="c1">r</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K21" title="RVSUM_1:func.21">Product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">j</font> <a href="newton.html#K1" title="NEWTON:func.1">|-&gt;</a> <font color="Maroon" title="c1">r</font></span>)</span></span>)</span></span><br/>

<a NAME="E2:13_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T3" title="NAT_1:th.3">NAT_1:3</a></span>;<br/>
<span class="kw">hence </span> <a href="rvsum_1.html#K21" title="RVSUM_1:func.21">Product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c2">j</font></span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|-&gt;</a> <font color="Maroon" title="c1">r</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K21" title="RVSUM_1:func.21">Product</a> <span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="newton.html#K1" title="NEWTON:func.1">|-&gt;</a> <font color="Maroon" title="c1">r</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K21" title="RVSUM_1:func.21">Product</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T40" title="NAT_D:th.40">NAT_D:40</a>, <a class="ref" href="rvsum_1.html#T124" title="RVSUM_1:th.124">RVSUM_1:124</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K21" title="RVSUM_1:func.21">Product</a> <span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="newton.html#K1" title="NEWTON:func.1">|-&gt;</a> <font color="Maroon" title="c1">r</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K21" title="RVSUM_1:func.21">Product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">j</font> <a href="newton.html#K1" title="NEWTON:func.1">|-&gt;</a> <font color="Maroon" title="c1">r</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:13_2"><span class="lab"><font color="Green" title="E13">A10</font></span></a>, <a class="ref" href="nat_1.html#T3" title="NAT_1:th.3">NAT_1:3</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E6:13"/>
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E5:13"><span class="lab"><font color="Green" title="E12">A9</font></span></a>, <a class="txt" href="#E4:13"><span class="lab"><font color="Green" title="E11">A4</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E7:13"/>
 <a href="rvsum_1.html#K21" title="RVSUM_1:func.21">Product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c2">j</font></span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|-&gt;</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K21" title="RVSUM_1:func.21">Product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="newton.html#K1" title="NEWTON:func.1">|-&gt;</a> <font color="Maroon" title="c1">r</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K21" title="RVSUM_1:func.21">Product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">j</font> <a href="newton.html#K1" title="NEWTON:func.1">|-&gt;</a> <font color="Maroon" title="c1">r</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:13"><span class="lab"><font color="Green" title="E9">A2</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
