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<b>set </b><font color="Maroon" title="c2">r</font> = 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2;<br/>
<b>let </b><font color="Maroon" title="c3">X</font>, <font color="Maroon" title="c4">Y</font> be  <a href="finset_1.html#NV2" title="FINSET_1:NV.2">infinite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:41"/><i><font color="Green" title="E31">A1</font></i>: 
 <a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="topgen_3.html#K4" title="TOPGEN_3:func.4">-powers</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">Y</font> <a href="topgen_3.html#K4" title="TOPGEN_3:func.4">-powers</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E2:41"/><i><font color="Green" title="E32">A2</font></i>: 
<font color="Maroon" title="c3">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">Y</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c5">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:41"/><i><font color="Green" title="E33">A3</font></i>: 
( ( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; not <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> ) or ( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> &amp; not <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> ) )
 <b>by </b><i><a class="txt" href="topgen_3.html#E2:41"><i><font color="Green" title="E32">A2</font></i></a>, <a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/>
<b>defpred </b><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> ] <b>means </b>( ( $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; not $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> ) or ( $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> &amp; not $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> ) );<br/>
<a NAME="E5:41"/><i><font color="Green" title="E34">A4</font></i>: 
 ex <font color="Olive" title="b1">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">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>]
 
<b>by </b><i><a class="txt" href="topgen_3.html#E4:41"><i><font color="Green" title="E33">A3</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">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> <b> such that </b><br/><a NAME="E7:41"/><i><font color="Green" title="E35">A5</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">i</font>] &amp; ( 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" title="c6">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S5" title="NAT_1:sch.5">NAT_1:sch 5</a>(<a class="txt" href="topgen_3.html#E5:41"><i><font color="Green" title="E34">A4</font></i></a>);<br/></i>
<b>consider </b><font color="Maroon" title="c7">A</font>, <font color="Maroon" title="c8">B</font> being  <a href="finset_1.html#NV2" title="FINSET_1:NV.2">infinite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E9:41"/><i><font color="Green" title="E36">A6</font></i>: 
( ( ( <font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c8">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">Y</font> ) or ( <font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">Y</font> &amp; <font color="Maroon" title="c8">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X</font> ) ) &amp; <font color="Maroon" title="c6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">A</font> &amp; not <font color="Maroon" title="c6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> )
 <b>by </b><i><a class="txt" href="topgen_3.html#E7:41"><i><font color="Green" title="E35">A5</font></i></a></i>;<br/>
<a NAME="E10:41"/>
( <font color="Maroon" title="c7">A</font> <a href="topgen_3.html#K4" title="TOPGEN_3:func.4">-powers</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> is <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a> &amp; <font color="Maroon" title="c8">B</font> <a href="topgen_3.html#K4" title="TOPGEN_3:func.4">-powers</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> is <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a> )
 
<b>by </b><i><a class="ref" href="topgen_3.html#T21" target="_self" title="TOPGEN_3:th.21">Th21</a></i>;<br/>
<a NAME="E11:41"/><b>then </b><i><font color="Green" title="E37">A7</font></i>: 
(  <a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="topgen_3.html#K4" title="TOPGEN_3:func.4">-powers</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="topgen_3.html#K4" title="TOPGEN_3:func.4">-powers</a> <span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">i</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="topgen_3.html#K4" title="TOPGEN_3:func.4">-powers</a> <span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></span>)</span> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> &amp;  <a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">B</font> <a href="topgen_3.html#K4" title="TOPGEN_3:func.4">-powers</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">B</font> <a href="topgen_3.html#K4" title="TOPGEN_3:func.4">-powers</a> <span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">i</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">B</font> <a href="topgen_3.html#K4" title="TOPGEN_3:func.4">-powers</a> <span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></span>)</span> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="series_1.html#T18" title="SERIES_1:th.18">SERIES_1:18</a></i>;<br/>
<a NAME="E12:41"/><i><font color="Green" title="E38">A8</font></i>: 
<span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="topgen_3.html#K4" title="TOPGEN_3:func.4">-powers</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="topgen_3.html#K4" title="TOPGEN_3:func.4">-powers</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="topgen_3.html#T26" target="_self" title="TOPGEN_3:th.26">Th26</a></i>;<br/>
<a NAME="E13:41"/><i><font color="Green" title="E39">A9</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="topgen_3.html#K4" title="TOPGEN_3:func.4">-powers</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c6">i</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c8">B</font> <a href="topgen_3.html#K4" title="TOPGEN_3:func.4">-powers</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 
<b>by </b><i><a class="txt" href="topgen_3.html#E9:41"><i><font color="Green" title="E36">A6</font></i></a>, <a class="ref" href="topgen_3.html#D5" target="_self" title="TOPGEN_3:def.5">Def5</a></i>;<br/>
<a NAME="E14:41"/><i><font color="Green" title="E40">A10</font></i>: 
 <a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">B</font> <a href="topgen_3.html#K4" title="TOPGEN_3:func.4">-powers</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></span>)</span> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c6">i</font>
 
<b>by </b><i><a class="ref" href="topgen_3.html#T25" target="_self" title="TOPGEN_3:th.25">Th25</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</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> ] <b>means </b>( $1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">i</font> implies <span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="topgen_3.html#K4" title="TOPGEN_3:func.4">-powers</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">B</font> <a href="topgen_3.html#K4" title="TOPGEN_3:func.4">-powers</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> $1 );<br/>
<a NAME="E15:41"/><i><font color="Green" title="E41">A11</font></i>: 
( <span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="topgen_3.html#K4" title="TOPGEN_3:func.4">-powers</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="topgen_3.html#K4" title="TOPGEN_3:func.4">-powers</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 0 &amp; <span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">B</font> <a href="topgen_3.html#K4" title="TOPGEN_3:func.4">-powers</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">B</font> <a href="topgen_3.html#K4" title="TOPGEN_3:func.4">-powers</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 0 )
 
<b>by </b><i><a class="ref" href="series_1.html#D1" title="SERIES_1:def.1">SERIES_1:def 1</a></i>;<br/>
<a NAME="E16:41"/><b>then </b>
( ( ( 0 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">A</font> &amp; <span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="topgen_3.html#K4" title="TOPGEN_3:func.4">-powers</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> 0 ) or ( not 0 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">A</font> &amp; <span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="topgen_3.html#K4" title="TOPGEN_3:func.4">-powers</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 ) ) &amp; ( ( 0 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> &amp; <span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">B</font> <a href="topgen_3.html#K4" title="TOPGEN_3:func.4">-powers</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> 0 ) or ( not 0 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> &amp; <span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">B</font> <a href="topgen_3.html#K4" title="TOPGEN_3:func.4">-powers</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 ) ) &amp; ( 0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">i</font> implies ( 0 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">A</font> iff 0 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> ) ) )
 
<b>by </b><i><a class="txt" href="topgen_3.html#E7:41"><i><font color="Green" title="E35">A5</font></i></a>, <a class="txt" href="topgen_3.html#E9:41"><i><font color="Green" title="E36">A6</font></i></a>, <a class="ref" href="topgen_3.html#D5" target="_self" title="TOPGEN_3:def.5">Def5</a></i>;<br/>
<a NAME="E17:41"/><b>then </b><i><font color="Green" title="E42">A12</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 
;<br/>
<div><i><font color="Green" title="E43">A13</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" title="c9">j</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> ;<br/><b>assume </b><a NAME="E1:41_1"/><i><font color="Green" title="E44">A14</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c9">j</font>]
 ;<br/><b>thus </b><a NAME="E2:41_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c9">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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" title="E45">A15</font></i>: 
<font color="Maroon" title="c9">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">i</font>
 ;<br/>

<a NAME="E2:41_1_1"/><b>then </b>
( ( ( <font color="Maroon" title="c9">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">A</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="topgen_3.html#K4" title="TOPGEN_3:func.4">-powers</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> ) or ( not <font color="Maroon" title="c9">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">A</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="topgen_3.html#K4" title="TOPGEN_3:func.4">-powers</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 ) ) &amp; ( ( <font color="Maroon" title="c9">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c8">B</font> <a href="topgen_3.html#K4" title="TOPGEN_3:func.4">-powers</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> ) or ( not <font color="Maroon" title="c9">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c8">B</font> <a href="topgen_3.html#K4" title="TOPGEN_3:func.4">-powers</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 ) ) &amp; ( <font color="Maroon" title="c9">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">A</font> implies <font color="Maroon" title="c9">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> ) &amp; ( <font color="Maroon" title="c9">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> implies <font color="Maroon" title="c9">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">A</font> ) )
 
<b>by </b><i><a class="txt" href="topgen_3.html#E7:41"><i><font color="Green" title="E35">A5</font></i></a>, <a class="txt" href="topgen_3.html#E9:41"><i><font color="Green" title="E36">A6</font></i></a>, <a class="ref" href="topgen_3.html#D5" target="_self" title="TOPGEN_3:def.5">Def5</a></i>;<br/>
<a NAME="E3:41_1_1"/><b>then </b><i><font color="Green" title="E46">A16</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="topgen_3.html#K4" title="TOPGEN_3:func.4">-powers</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">B</font> <a href="topgen_3.html#K4" title="TOPGEN_3:func.4">-powers</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
;<br/>
<a NAME="E4:41_1_1"/>
<font color="Maroon" title="c9">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">i</font>
 
<b>by </b><i><a class="txt" href="topgen_3.html#E1:41_1_1"><i><font color="Green" title="E45">A15</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<b>hence </b><span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="topgen_3.html#K4" title="TOPGEN_3:func.4">-powers</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">B</font> <a href="topgen_3.html#K4" title="TOPGEN_3:func.4">-powers</a> <span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">j</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="topgen_3.html#K4" title="TOPGEN_3:func.4">-powers</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="topgen_3.html#E1:41_1"><i><font color="Green" title="E44">A14</font></i></a>, <a class="ref" href="series_1.html#D1" title="SERIES_1:def.1">SERIES_1:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">B</font> <a href="topgen_3.html#K4" title="TOPGEN_3:func.4">-powers</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="topgen_3.html#E3:41_1_1"><i><font color="Green" title="E46">A16</font></i></a>, <a class="ref" href="series_1.html#D1" title="SERIES_1:def.1">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" title="E44">A17</font></i>: 
for <font color="Olive" title="b1">j</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>2</sub></font>[<font color="Olive" title="b1">j</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="topgen_3.html#E17:41"><i><font color="Green" title="E42">A12</font></i></a>, <a class="txt" href="topgen_3.html#E18:41"><i><font color="Green" title="E43">A13</font></i></a>);<br/></i>
<a NAME="E20:41"/>
<span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="topgen_3.html#K4" title="TOPGEN_3:func.4">-powers</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">B</font> <a href="topgen_3.html#K4" title="TOPGEN_3:func.4">-powers</a> <span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">i</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c6">i</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" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or  ex <font color="Olive" title="b1">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 )
 </span><b>by </b><i><a class="ref" href="nat_1.html#T6" title="NAT_1:th.6">NAT_1:6</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" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="topgen_3.html#K4" title="TOPGEN_3:func.4">-powers</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">B</font> <a href="topgen_3.html#K4" title="TOPGEN_3:func.4">-powers</a> <span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">i</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c6">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="topgen_3.html#E13:41"><i><font color="Green" title="E39">A9</font></i></a>, <a class="txt" href="topgen_3.html#E15:41"><i><font color="Green" title="E41">A11</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" title="b1">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c9">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E3:41_2_1_2"/><i><font color="Green" title="E45">A18</font></i>: 
<font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 ;<br/><b>reconsider </b><font color="Maroon" title="c10">j</font> = <font color="Maroon" title="c9">j</font> as    <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>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E5:41_2_1_2"/>
<font color="Maroon" title="c10">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">i</font>
 <b>by </b><i><a class="txt" href="topgen_3.html#E3:41_2_1_2"><i><font color="Green" title="E45">A18</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E6:41_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="topgen_3.html#K4" title="TOPGEN_3:func.4">-powers</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">B</font> <a href="topgen_3.html#K4" title="TOPGEN_3:func.4">-powers</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c10">j</font>
 <b>by </b><i><a class="txt" href="topgen_3.html#E19:41"><i><font color="Green" title="E44">A17</font></i></a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="topgen_3.html#K4" title="TOPGEN_3:func.4">-powers</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">i</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" title="SERIES_1:func.1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">B</font> <a href="topgen_3.html#K4" title="TOPGEN_3:func.4">-powers</a> <span class="p5">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c10">j</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 0</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c6">i</font></span>)</span>
<b>by </b><i><a class="txt" href="topgen_3.html#E13:41"><i><font color="Green" title="E39">A9</font></i></a>, <a class="txt" href="topgen_3.html#E3:41_2_1_2"><i><font color="Green" title="E45">A18</font></i></a>, <a class="ref" href="series_1.html#D1" title="SERIES_1:def.1">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" title="SERIES_1:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">B</font> <a href="topgen_3.html#K4" title="TOPGEN_3:func.4">-powers</a> <span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">i</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c6">i</font></span>)</span>
<b>by </b><i><a class="txt" href="topgen_3.html#E13:41"><i><font color="Green" title="E39">A9</font></i></a>, <a class="txt" href="topgen_3.html#E3:41_2_1_2"><i><font color="Green" title="E45">A18</font></i></a>, <a class="ref" href="series_1.html#D1" title="SERIES_1:def.1">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" title="SERIES_1:func.2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">B</font> <a href="topgen_3.html#K4" title="TOPGEN_3:func.4">-powers</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="topgen_3.html#K4" title="TOPGEN_3:func.4">-powers</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">i</font>
 
<b>by </b><i><a class="txt" href="topgen_3.html#E11:41"><i><font color="Green" title="E37">A7</font></i></a>, <a class="txt" href="topgen_3.html#E14:41"><i><font color="Green" title="E40">A10</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">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" title="E31">A1</font></i></a>, <a class="txt" href="topgen_3.html#E9:41"><i><font color="Green" title="E36">A6</font></i></a>, <a class="txt" href="topgen_3.html#E12:41"><i><font color="Green" title="E38">A8</font></i></a></i>;<br/>


</div>
