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<b>let </b><font color="Maroon">s</font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/>

<b>assume </b><a NAME="E1:39"/>
<font color="Maroon">s</font> is <a href="series_1.html#V2" target="_self">absolutely_summable</a>
 ;<br/>

<a NAME="E2:39"/><b>then </b><i><font color="Green">E31</font></i>: 
 <a href="seq_1.html#K22">abs</a> <font color="Maroon">s</font> is <a href="series_1.html#V1" target="_self">summable</a>
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">r</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="E1:39_1"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 ;<br/><b>then consider </b><font color="Maroon">n</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="E3:39_1"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1" target="_self">Partial_Sums</a> <span class="p4">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1" target="_self">Partial_Sums</a> <span class="p4">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i>, </i>;<br/><b>take </b><font color="Maroon">n</font> = <font color="Maroon">n</font>;<br/><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E4:39_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/><a NAME="E5:39_1"/><b>then </b><i><font color="Green">E34</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1" target="_self">Partial_Sums</a> <span class="p4">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1" target="_self">Partial_Sums</a> <span class="p4">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="txt" href="series_1.html#E1"><i><font color="Green">Lemma29</font></i></a></i>;<br/><a NAME="E6:39_1"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">s</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">s</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1" target="_self">Partial_Sums</a> <span class="p4">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1" target="_self">Partial_Sums</a> <span class="p4">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="series_1.html#T1" target="_self">Th1</a>, </i>;<br/><b>hence </b><a NAME="E7:39_1"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">s</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">s</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="txt" href="series_1.html#E2:39"><i><font color="Green">E31</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:39"/>
<font color="Maroon">s</font> is <a href="series_1.html#V1" target="_self">summable</a>
 <b>by </b><i/>;<br/>


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