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<b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>



<b>assume </b><a NAME="E1:35"/><i><font color="Green">E26</font></i>: 
( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 )
 ;<br/>

<a NAME="E2:35"/>
 <a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="topgen_3.html#E1:35"><i><font color="Green">E26</font></i></a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/>
<a NAME="E3:35"/><b>then </b><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="prepower.html#K2">GeoSeq</a>  is <a href="series_1.html#V1">summable</a>
 
<b>by </b><i><a class="txt" href="topgen_3.html#E1:35"><i><font color="Green">E26</font></i></a>, <a class="ref" href="series_1.html#T28">SERIES_1:28</a></i>;<br/>
<div><i><font color="Green">E28</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">c<sub>4</sub></font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><a NAME="E1:35_1"/>
( ( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="topgen_3.html#K4" target="_self">-powers</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="newton.html#K2">|^</a> <font color="Maroon">c<sub>4</sub></font> ) or ( not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="topgen_3.html#K4" target="_self">-powers</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 0 ) )
 <b>by </b><i><a class="ref" href="topgen_3.html#D5" target="_self">Def5</a></i>;<br/><b>hence </b><a NAME="E2:35_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="topgen_3.html#K4" target="_self">-powers</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="topgen_3.html#E1:35"><i><font color="Green">E26</font></i></a>, <a class="ref" href="prepower.html#T13">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">c<sub>4</sub></font> = 1;<br/><b>let </b><font color="Maroon">c<sub>5</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="E1:35_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><a NAME="E2:35_2"/><i><font color="Green">E29</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="prepower.html#K2">GeoSeq</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="newton.html#K2">|^</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="prepower.html#D1">PREPOWER:def 1</a></i>;<br/><a NAME="E3:35_2"/>
( ( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="topgen_3.html#K4" target="_self">-powers</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="newton.html#K2">|^</a> <font color="Maroon">c<sub>5</sub></font> ) or ( not <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="topgen_3.html#K4" target="_self">-powers</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 0 ) )
 <b>by </b><i><a class="ref" href="topgen_3.html#D5" target="_self">Def5</a></i>;<br/><b>hence </b><a NAME="E4:35_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="topgen_3.html#K4" target="_self">-powers</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="prepower.html#K2">GeoSeq</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="topgen_3.html#E1:35"><i><font color="Green">E26</font></i></a>, <a class="txt" href="topgen_3.html#E2:35_2"><i><font color="Green">E29</font></i></a>, <a class="ref" href="prepower.html#T13">PREPOWER:13</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:35"/>
<font color="Maroon">c<sub>2</sub></font> <a href="topgen_3.html#K4" target="_self">-powers</a> <font color="Maroon">c<sub>3</sub></font> is <a href="series_1.html#V1">summable</a>
 <b>by </b><i><a class="txt" href="topgen_3.html#E3:35"><i><font color="Green">E27</font></i></a>, <a class="txt" href="topgen_3.html#E4:35"><i><font color="Green">E28</font></i></a>, <a class="ref" href="series_1.html#T22">SERIES_1:22</a></i>;<br/>


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