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

<b>set </b><font color="Maroon">c<sub>2</sub></font> = 1 <a href="xcmplx_0.html#K7">/</a> 2;<br/>
<b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be  <a href="finset_1.html#V1">infinite</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:41"/><i><font color="Green">E31</font></i>: 
 <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="topgen_3.html#K4" target="_self">-powers</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="topgen_3.html#K4" target="_self">-powers</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E2:41"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:41"/><i><font color="Green">E33</font></i>: 
( ( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; not <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> ) or ( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; not <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> ) )
 <b>by </b><i><a class="txt" href="topgen_3.html#E2:41"><i><font color="Green">E32</font></i></a>, <a class="ref" href="tarski.html#T2">TARSKI:2</a></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> ] means ( ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; not <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> ) or ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; not <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> ) );<br/>
<a NAME="E5:41"/><i><font color="Green">E34</font></i>: 
ex <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font>]
 
<b>by </b><i><a class="txt" href="topgen_3.html#E4:41"><i><font color="Green">E33</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E7:41"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>6</sub></font>] &amp; ( for <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font>] holds <br/><font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S5">NAT_1:sch 5</a>(<a class="txt" href="topgen_3.html#E5:41"><i><font color="Green">E34</font></i></a>);<br/></i>
<b>consider </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font> being  <a href="finset_1.html#V1">infinite</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E9:41"/><i><font color="Green">E36</font></i>: 
( ( ( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> ) or ( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> ) ) &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> &amp; not <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="txt" href="topgen_3.html#E7:41"><i><font color="Green">E35</font></i></a></i>;<br/>
<a NAME="E10:41"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="topgen_3.html#K4" target="_self">-powers</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> is <a href="series_1.html#V1">summable</a> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="topgen_3.html#K4" target="_self">-powers</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> is <a href="series_1.html#V1">summable</a> )
 
<b>by </b><i><a class="ref" href="topgen_3.html#T21" target="_self">Th21</a></i>;<br/>
<a NAME="E11:41"/><b>then </b><i><font color="Green">E37</font></i>: 
(  <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="topgen_3.html#K4" target="_self">-powers</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="topgen_3.html#K4" target="_self">-powers</a> <span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="topgen_3.html#K4" target="_self">-powers</a> <span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span> <a href="seqm_3.html#K1">^\</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> &amp;  <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="topgen_3.html#K4" target="_self">-powers</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="topgen_3.html#K4" target="_self">-powers</a> <span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="topgen_3.html#K4" target="_self">-powers</a> <span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span> <a href="seqm_3.html#K1">^\</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="series_1.html#T18">SERIES_1:18</a></i>;<br/>
<a NAME="E12:41"/><i><font color="Green">E38</font></i>: 
<span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="topgen_3.html#K4" target="_self">-powers</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="topgen_3.html#K4" target="_self">-powers</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="topgen_3.html#T26" target="_self">Th26</a></i>;<br/>
<a NAME="E13:41"/><i><font color="Green">E39</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="topgen_3.html#K4" target="_self">-powers</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="newton.html#K2">|^</a> <font color="Maroon">c<sub>6</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="topgen_3.html#K4" target="_self">-powers</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i><a class="txt" href="topgen_3.html#E9:41"><i><font color="Green">E36</font></i></a>, <a class="ref" href="topgen_3.html#D5" target="_self">Def5</a></i>;<br/>
<a NAME="E14:41"/><i><font color="Green">E40</font></i>: 
 <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="topgen_3.html#K4" target="_self">-powers</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span> <a href="seqm_3.html#K1">^\</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="newton.html#K2">|^</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="topgen_3.html#T25" target="_self">Th25</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] means ( <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> implies <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="topgen_3.html#K4" target="_self">-powers</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="topgen_3.html#K4" target="_self">-powers</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font> );<br/>
<a NAME="E15:41"/><i><font color="Green">E41</font></i>: 
( <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="topgen_3.html#K4" target="_self">-powers</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="topgen_3.html#K4" target="_self">-powers</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> 0 &amp; <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="topgen_3.html#K4" target="_self">-powers</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="topgen_3.html#K4" target="_self">-powers</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> 0 )
 
<b>by </b><i><a class="ref" href="series_1.html#D1">SERIES_1:def 1</a></i>;<br/>
<a NAME="E16:41"/><b>then </b>
( ( ( 0 <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> &amp; <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="topgen_3.html#K4" target="_self">-powers</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="newton.html#K2">|^</a> 0 ) or ( not 0 <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> &amp; <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="topgen_3.html#K4" target="_self">-powers</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a> 0 ) ) &amp; ( ( 0 <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> &amp; <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="topgen_3.html#K4" target="_self">-powers</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="newton.html#K2">|^</a> 0 ) or ( not 0 <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> &amp; <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="topgen_3.html#K4" target="_self">-powers</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a> 0 ) ) &amp; ( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> implies ( 0 <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> iff 0 <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> ) ) )
 
<b>by </b><i><a class="txt" href="topgen_3.html#E7:41"><i><font color="Green">E35</font></i></a>, <a class="txt" href="topgen_3.html#E9:41"><i><font color="Green">E36</font></i></a>, <a class="ref" href="topgen_3.html#D5" target="_self">Def5</a></i>;<br/>
<a NAME="E17:41"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 
;<br/>
<div><i><font color="Green">E43</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>9</sub></font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:41_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>9</sub></font>]
 ;<br/><b>thus </b><a NAME="E2:41_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:41_1_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<a NAME="E2:41_1_1"/><b>then </b>
( ( ( <font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="topgen_3.html#K4" target="_self">-powers</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="newton.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> ) or ( not <font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="topgen_3.html#K4" target="_self">-powers</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 0 ) ) &amp; ( ( <font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="topgen_3.html#K4" target="_self">-powers</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="newton.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> ) or ( not <font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="topgen_3.html#K4" target="_self">-powers</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 0 ) ) &amp; ( <font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> implies <font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> ) &amp; ( <font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> implies <font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> ) )
 
<b>by </b><i><a class="txt" href="topgen_3.html#E7:41"><i><font color="Green">E35</font></i></a>, <a class="txt" href="topgen_3.html#E9:41"><i><font color="Green">E36</font></i></a>, <a class="ref" href="topgen_3.html#D5" target="_self">Def5</a></i>;<br/>
<a NAME="E3:41_1_1"/><b>then </b><i><font color="Green">E46</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="topgen_3.html#K4" target="_self">-powers</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="topgen_3.html#K4" target="_self">-powers</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
;<br/>
<a NAME="E4:41_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="topgen_3.html#E1:41_1_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<b>hence </b><span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="topgen_3.html#K4" target="_self">-powers</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</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="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="topgen_3.html#K4" target="_self">-powers</a> <span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="topgen_3.html#K4" target="_self">-powers</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="topgen_3.html#E1:41_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="series_1.html#D1">SERIES_1:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="topgen_3.html#K4" target="_self">-powers</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="topgen_3.html#E3:41_1_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="series_1.html#D1">SERIES_1:def 1</a></i>
;<br/><br/>


</div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E19:41"/><i><font color="Green">E44</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds <font color="Maroon">S<sub>2</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="topgen_3.html#E17:41"><i><font color="Green">E42</font></i></a>, <a class="txt" href="topgen_3.html#E18:41"><i><font color="Green">E43</font></i></a>);<br/></i>
<a NAME="E20:41"/>
<span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="topgen_3.html#K4" target="_self">-powers</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="topgen_3.html#K4" target="_self">-powers</a> <span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="newton.html#K2">|^</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:41_2_1"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 0 or ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 </span><b>by </b><i><a class="ref" href="nat_1.html#T22">NAT_1:22</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_2_1_1"><b>suppose </b></a><a NAME="E1:41_2_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:41_2_1_1"/>
<span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="topgen_3.html#K4" target="_self">-powers</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="topgen_3.html#K4" target="_self">-powers</a> <span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="newton.html#K2">|^</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="topgen_3.html#E13:41"><i><font color="Green">E39</font></i></a>, <a class="txt" href="topgen_3.html#E15:41"><i><font color="Green">E41</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_2_1_2"><b>suppose </b></a><a NAME="E1:41_2_1_2"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>9</sub></font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:41_2_1_2"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><a NAME="E4:41_2_1_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="topgen_3.html#E3:41_2_1_2"><i><font color="Green">E45</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E5:41_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="topgen_3.html#K4" target="_self">-powers</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="topgen_3.html#K4" target="_self">-powers</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="topgen_3.html#E19:41"><i><font color="Green">E44</font></i></a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="topgen_3.html#K4" target="_self">-powers</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="topgen_3.html#K4" target="_self">-powers</a> <span class="p5">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 0</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="newton.html#K2">|^</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="topgen_3.html#E13:41"><i><font color="Green">E39</font></i></a>, <a class="txt" href="topgen_3.html#E3:41_2_1_2"><i><font color="Green">E45</font></i></a>, <a class="ref" href="series_1.html#D1">SERIES_1:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="topgen_3.html#K4" target="_self">-powers</a> <span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="newton.html#K2">|^</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="topgen_3.html#E13:41"><i><font color="Green">E39</font></i></a>, <a class="txt" href="topgen_3.html#E3:41_2_1_2"><i><font color="Green">E45</font></i></a>, <a class="ref" href="series_1.html#D1">SERIES_1:def 1</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E21:41"/><b>then </b>
 <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="topgen_3.html#K4" target="_self">-powers</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="topgen_3.html#K4" target="_self">-powers</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="topgen_3.html#E11:41"><i><font color="Green">E37</font></i></a>, <a class="txt" href="topgen_3.html#E14:41"><i><font color="Green">E40</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<b>hence </b><a NAME="E22:41"/>
contradiction
 <b>by </b><i><a class="txt" href="topgen_3.html#E1:41"><i><font color="Green">E31</font></i></a>, <a class="txt" href="topgen_3.html#E9:41"><i><font color="Green">E36</font></i></a>, <a class="txt" href="topgen_3.html#E12:41"><i><font color="Green">E38</font></i></a></i>;<br/>


</div>
