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<span class="kw">let </span><font color="Maroon" title="c2">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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">X</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">r</font> &amp; <font color="Maroon" title="c2">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 holds <br/> <a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">X</font> <a href="topgen_3.html#K4" title="TOPGEN_3:func.4">-powers</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">r</font> &amp; <font color="Maroon" title="c2">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 implies  <a href="series_1.html#K3" title="SERIES_1:func.3">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> <font color="Maroon" title="c2">r</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a></span>)</span> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:36"/><span class="lab"><font color="Green" title="E28">A1</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">r</font>
 <span class="kw">and </span><br/><a NAME="E2:36"/><span class="lab"><font color="Green" title="E29">A2</font></span>: 
<font color="Maroon" title="c2">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="series_1.html#K3" title="SERIES_1:func.3">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> <font color="Maroon" title="c2">r</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a></span>)</span></span><br/>

<div><span class="lab"><font color="Green" title="E30">A3</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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> <font color="Maroon" title="c2">r</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">n</font> &amp; <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> <font color="Maroon" title="c2">r</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">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="c2">r</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">n</font> )</span><br/><a NAME="E1:36_1"/><span class="lab"><font color="Green" title="E31">A4</font></span>: 
( ( <font color="Maroon" title="c4">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; <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> <font color="Maroon" title="c2">r</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">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="c3">X</font> &amp; <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> <font color="Maroon" title="c2">r</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="topgen_3.html#D5" target="_self" title="TOPGEN_3:def.5">Def5</a></span>;<br/><span class="kw">hence </span><a NAME="E2:36_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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> <font color="Maroon" title="c2">r</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:36"><span class="lab"><font color="Green" title="E28">A1</font></span></a>, <a class="ref" href="prepower.html#T6" title="PREPOWER:th.6">PREPOWER:6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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> <font color="Maroon" title="c2">r</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">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="c2">r</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">n</font></span><br/><a NAME="E3:36_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">r</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c4">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="prepower.html#D1" title="PREPOWER:def.1">PREPOWER:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E4:36_1"/>
<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> <font color="Maroon" title="c2">r</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">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="c2">r</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:36"><span class="lab"><font color="Green" title="E28">A1</font></span></a>, <a class="txt" href="#E1:36_1"><span class="lab"><font color="Green" title="E31">A4</font></span></a>, <a class="ref" href="prepower.html#T6" title="PREPOWER:th.6">PREPOWER:6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E4:36"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c2">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:36"><span class="lab"><font color="Green" title="E28">A1</font></span></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>;<br/>
<a NAME="E5:36"/><span class="kw">then </span>
<font color="Maroon" title="c2">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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:36"><span class="lab"><font color="Green" title="E29">A2</font></span></a>, <a class="ref" href="series_1.html#T24" title="SERIES_1:th.24">SERIES_1:24</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:36"/>
 <a href="series_1.html#K3" title="SERIES_1:func.3">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> <font color="Maroon" title="c2">r</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36"><span class="lab"><font color="Green" title="E30">A3</font></span></a>, <a class="ref" href="series_1.html#T20" title="SERIES_1:th.20">SERIES_1:20</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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