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<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">s</font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/>



<b>set </b><font color="Maroon">s1</font> =  <a href="seq_1.html#K22">abs</a> <font color="Maroon">s</font>;<br/>
<a NAME="E1:49"/><i><font color="Green">E41</font></i>: 
<span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">s</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <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"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">s</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="absvalue.html#T11">ABSVALUE:11</a></i>;<br/>
<a NAME="E2:49"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">s</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:49"/>
<span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">s</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>


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