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

<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#K2">COMPLEX</a> ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a> ;<br/><b>assume </b><a NAME="E1:48_1"/>
 <a href="complex1.html#K6">1r</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E2:48_1"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span>
 <b>by </b><i><a class="ref" href="complex1.html#T148">COMPLEX1:148</a></i>;<br/><a NAME="E3:48_1"/><b>then </b><i><font color="Green">E30</font></i>: 
<a href="complex1.html#K6">1r</a>  <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="complex1.html#K5">0c</a> 
 <b>by </b><i><a class="ref" href="complex1.html#T133">COMPLEX1:133</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means <span class="p1">(<span class="default"><a href="comseq_3.html#K7" target="_self">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="comseq_3.html#K1" target="_self">GeoSeq</a> </span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K11">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="comseq_3.html#K2" target="_self">#N</a> <span class="p3">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="complex1.html#K13">/</a> <span class="p1">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>;<br/><a NAME="E4:48_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1"><b>proof </b></a><div class="add">

<b>thus </b><span class="p1">(<span class="default"><a href="comseq_3.html#K7" target="_self">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="comseq_3.html#K1" target="_self">GeoSeq</a> </span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> 0 = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="comseq_3.html#K1" target="_self">GeoSeq</a> </span>)</span> <a href="comseq_1.html#K1">.</a> 0
<b>by </b><i><a class="ref" href="comseq_3.html#D7" target="_self">Def7</a></i>
<br/>.= 
 <a href="complex1.html#K6">1r</a> 
<b>by </b><i><a class="ref" href="comseq_3.html#D1" target="_self">Def1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#D7">COMPLEX1:def 7</a>, <a class="txt" href="comseq_3.html#E3:48_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="xcmplx_1.html#T60">XCMPLX_1:60</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K11">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="comseq_3.html#K2" target="_self">#N</a> <span class="p3">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="complex1.html#K13">/</a> <span class="p1">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T10">NEWTON:10</a></i>
;<br/>


</div><b>end;</b></div><a NAME="E5:48_1"/><i><font color="Green">E32</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="nat_1.html#K1">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:48_1_2"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>4</sub></font>]
 ;<br/>

<b>thus </b><span class="p1">(<span class="default"><a href="comseq_3.html#K7" target="_self">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="comseq_3.html#K1" target="_self">GeoSeq</a> </span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K7" target="_self">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="comseq_3.html#K1" target="_self">GeoSeq</a> </span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="complex1.html#K8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="comseq_3.html#K1" target="_self">GeoSeq</a> </span>)</span> <a href="comseq_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="comseq_3.html#D7" target="_self">Def7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K11">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="comseq_3.html#K2" target="_self">#N</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="complex1.html#K13">/</a> <span class="p2">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="complex1.html#K8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="comseq_3.html#K2" target="_self">#N</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="complex1.html#K9">*</a> <a href="complex1.html#K6">1r</a> </span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#D7">COMPLEX1:def 7</a>, <a class="txt" href="comseq_3.html#E1:48_1_2"><i><font color="Green">E33</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K11">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="comseq_3.html#K2" target="_self">#N</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="complex1.html#K13">/</a> <span class="p2">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="complex1.html#K8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="comseq_3.html#K2" target="_self">#N</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="complex1.html#K9">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="complex1.html#K13">/</a> <span class="p3">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="comseq_3.html#E3:48_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="complex1.html#D7">COMPLEX1:def 7</a>, <a class="ref" href="xcmplx_1.html#T60">XCMPLX_1:60</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K11">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="comseq_3.html#K2" target="_self">#N</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="complex1.html#K13">/</a> <span class="p2">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="complex1.html#K8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="comseq_3.html#K2" target="_self">#N</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="complex1.html#K9">*</a> <span class="p3">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="complex1.html#K13">/</a> <span class="p2">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>3</sub></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"><span class="p2">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K11">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="comseq_3.html#K2" target="_self">#N</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="complex1.html#K8">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="comseq_3.html#K2" target="_self">#N</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="complex1.html#K9">*</a> <a href="complex1.html#K6">1r</a> </span>)</span> <a href="complex1.html#K11">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="comseq_3.html#K2" target="_self">#N</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="complex1.html#K9">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="complex1.html#K13">/</a> <span class="p1">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T63">XCMPLX_1:63</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K11">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="comseq_3.html#K2" target="_self">#N</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="complex1.html#K8">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="comseq_3.html#K2" target="_self">#N</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="complex1.html#K11">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="comseq_3.html#K2" target="_self">#N</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="complex1.html#K9">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="complex1.html#K13">/</a> <span class="p1">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#D7">COMPLEX1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K11">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="comseq_3.html#K2" target="_self">#N</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="complex1.html#K9">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="complex1.html#K13">/</a> <span class="p1">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K11">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="comseq_3.html#K2" target="_self">#N</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="complex1.html#K13">/</a> <span class="p1">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T11">NEWTON:11</a></i>
;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E6:48_1"/>
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="comseq_3.html#E4:48_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="comseq_3.html#E5:48_1"><i><font color="Green">E32</font></i></a>);</i><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:48"/>
(  <a href="complex1.html#K6">1r</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>2</sub></font> implies for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <span class="p1">(<span class="default"><a href="comseq_3.html#K7" target="_self">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="comseq_3.html#K1" target="_self">GeoSeq</a> </span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K11">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="comseq_3.html#K2" target="_self">#N</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="complex1.html#K13">/</a> <span class="p1">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> )
 ;<br/>


</div>
