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<div class="add">

<b>let </b><font color="Maroon" title="c1">x</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>

<a NAME="E1:17"/>
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds   <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="sin_cos.html#K7" title="SIN_COS:func.7">ExpSeq</a> </span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="sin_cos.html#K7" title="SIN_COS:func.7">ExpSeq</a> </span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<a NAME="E1:17_1"/><i><font color="Green" title="E13">A1</font></i>: 
<font color="Maroon" title="c2">n</font> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="newton.html#T23" title="NEWTON:th.23">NEWTON:23</a></i>;<br/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="sin_cos.html#K7" title="SIN_COS:func.7">ExpSeq</a> </span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c2">n</font></span>)</span> = 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos.html#D9" title="SIN_COS:def.9">SIN_COS:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T153" title="COMPLEX1:th.153">COMPLEX1:153</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!</a> </span>)</span>
<b>by </b><i><a class="txt" href="taylor_2.html#E1:17_1"><i><font color="Green" title="E13">A1</font></i></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!</a> </span>)</span>
<b>by </b><i><a class="ref" href="taylor_2.html#T1" target="_self" title="TAYLOR_2:th.1">Th1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="sin_cos.html#K7" title="SIN_COS:func.7">ExpSeq</a> </span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c2">n</font>
<b>by </b><i><a class="ref" href="sin_cos.html#D9" title="SIN_COS:def.9">SIN_COS:def 9</a></i>
;<br/>
<b>hence </b><a NAME="E3:17_1"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="sin_cos.html#K7" title="SIN_COS:func.7">ExpSeq</a> </span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="sin_cos.html#K7" title="SIN_COS:func.7">ExpSeq</a> </span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c2">n</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:17"/><b>then </b>
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="sin_cos.html#K7" title="SIN_COS:func.7">ExpSeq</a>  <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="sin_cos.html#K7" title="SIN_COS:func.7">ExpSeq</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="seq_1.html#T16" title="SEQ_1:th.16">SEQ_1:16</a></i>;<br/>
<a NAME="E3:17"/><b>then </b>
 <a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="sin_cos.html#K7" title="SIN_COS:func.7">ExpSeq</a> </span>)</span> is <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a>
 
<b>by </b><i><a class="ref" href="sin_cos.html#T49" title="SIN_COS:th.49">SIN_COS:49</a></i>;<br/>
<b>hence </b><a NAME="E4:17"/>
<font color="Maroon" title="c1">x</font> <a href="sin_cos.html#K7" title="SIN_COS:func.7">ExpSeq</a>  is <a href="series_1.html#V2" title="SERIES_1:attr.2">absolutely_summable</a>
 <b>by </b><i><a class="ref" href="series_1.html#D5" title="SERIES_1:def.5">SERIES_1:def 5</a></i>;<br/>


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