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<b>let </b><font color="Maroon" title="c2">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c3">r</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>



<b>assume </b><a NAME="E1:35"/><i><font color="Green" title="E26">A1</font></i>: 
( 0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">r</font> &amp; <font color="Maroon" title="c3">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 )
 ;<br/>

<a NAME="E2:35"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c3">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">r</font>
 
<b>by </b><i><a class="txt" href="topgen_3.html#E1:35"><i><font color="Green" title="E26">A1</font></i></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/>
<a NAME="E3:35"/><b>then </b><i><font color="Green" title="E27">A2</font></i>: 
<font color="Maroon" title="c3">r</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a>  is <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a>
 
<b>by </b><i><a class="txt" href="topgen_3.html#E1:35"><i><font color="Green" title="E26">A1</font></i></a>, <a class="ref" href="series_1.html#T28" title="SERIES_1:th.28">SERIES_1:28</a></i>;<br/>
<div><i><font color="Green" title="E28">A3</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><a NAME="E1:35_1"/>
( ( <font color="Maroon" title="c4">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="topgen_3.html#K4" title="TOPGEN_3:func.4">-powers</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">r</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c4">n</font> ) or ( not <font color="Maroon" title="c4">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="topgen_3.html#K4" title="TOPGEN_3:func.4">-powers</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 ) )
 <b>by </b><i><a class="ref" href="topgen_3.html#D5" target="_self" title="TOPGEN_3:def.5">Def5</a></i>;<br/><b>hence </b><a NAME="E2:35_1"/>
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="topgen_3.html#K4" title="TOPGEN_3:func.4">-powers</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">n</font>
 <b>by </b><i><a class="txt" href="topgen_3.html#E1:35"><i><font color="Green" title="E26">A1</font></i></a>, <a class="ref" href="prepower.html#T13" title="PREPOWER:th.13">PREPOWER:13</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_2"><b>now </b></a><div class="add"><b>take </b><font color="Maroon" title="c4">m</font> = 1;<br/><b>let </b><font color="Maroon" title="c5">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:35_2"/>
<font color="Maroon" title="c4">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">n</font>
 ;<br/><a NAME="E2:35_2"/><i><font color="Green" title="E29">A4</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">r</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c5">n</font>
 <b>by </b><i><a class="ref" href="prepower.html#D1" title="PREPOWER:def.1">PREPOWER:def 1</a></i>;<br/><a NAME="E3:35_2"/>
( ( <font color="Maroon" title="c5">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="topgen_3.html#K4" title="TOPGEN_3:func.4">-powers</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">r</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c5">n</font> ) or ( not <font color="Maroon" title="c5">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="topgen_3.html#K4" title="TOPGEN_3:func.4">-powers</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 ) )
 <b>by </b><i><a class="ref" href="topgen_3.html#D5" target="_self" title="TOPGEN_3:def.5">Def5</a></i>;<br/><b>hence </b><a NAME="E4:35_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="topgen_3.html#K4" title="TOPGEN_3:func.4">-powers</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">n</font>
 <b>by </b><i><a class="txt" href="topgen_3.html#E1:35"><i><font color="Green" title="E26">A1</font></i></a>, <a class="txt" href="topgen_3.html#E2:35_2"><i><font color="Green" title="E29">A4</font></i></a>, <a class="ref" href="prepower.html#T13" title="PREPOWER:th.13">PREPOWER:13</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:35"/>
<font color="Maroon" title="c2">X</font> <a href="topgen_3.html#K4" title="TOPGEN_3:func.4">-powers</a> <font color="Maroon" title="c3">r</font> is <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a>
 <b>by </b><i><a class="txt" href="topgen_3.html#E3:35"><i><font color="Green" title="E27">A2</font></i></a>, <a class="txt" href="topgen_3.html#E4:35"><i><font color="Green" title="E28">A3</font></i></a>, <a class="ref" href="series_1.html#T22" title="SERIES_1:th.22">SERIES_1:22</a></i>;<br/>


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