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<div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<a NAME="E1:16"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds   <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">x</font></span>)</span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><b>proof </b></a><div class="add">

<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/>

<a NAME="E1:16_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">n</font> <a href="sin_cos.html#K5">!</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="newton.html#T23">NEWTON:23</a></i>;<br/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> = 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="prepower.html#K3">|^</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="sin_cos.html#K5">!</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos.html#D9">SIN_COS:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="prepower.html#K3">|^</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="sin_cos.html#K5">!</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T153">COMPLEX1:153</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="prepower.html#K3">|^</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="sin_cos.html#K5">!</a> </span>)</span>
<b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">x</font></span>)</span> <a href="prepower.html#K3">|^</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="sin_cos.html#K5">!</a> </span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">x</font></span>)</span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
<b>by </b><i><a class="ref" href="sin_cos.html#D9">SIN_COS:def 9</a></i>
;<br/>
<b>hence </b><a NAME="E3:16_1"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">x</font></span>)</span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:16"/><b>then </b>
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">x</font></span>)</span> <a href="sin_cos.html#K7">ExpSeq</a>  <a href="hidden.html#R1">=</a>  <a href="seq_1.html#K22">abs</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="seq_1.html#T16">SEQ_1:16</a></i>;<br/>
<a NAME="E3:16"/><b>then </b>
 <a href="seq_1.html#K22">abs</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span> is <a href="series_1.html#V1">summable</a>
 
<b>by </b><i><a class="ref" href="sin_cos.html#T49">SIN_COS:49</a></i>;<br/>
<b>hence </b><a NAME="E4:16"/>
<font color="Maroon">x</font> <a href="sin_cos.html#K7">ExpSeq</a>  is <a href="series_1.html#V2">absolutely_summable</a>
 <b>by </b><i><a class="ref" href="series_1.html#D5">SERIES_1:def 5</a></i>;<br/>


</div>
