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<a NAME="E1:30_1_1"/><i><font color="Green">E51</font></i>: 
 ex <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> holds <br/><span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">z</font></span><a href="clvect_1.html#K5">.||</a></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="30_1_1_1"><b>proof </b></a><div class="add">

<b>take </b>
0
;<br/>

<b>thus </b><a NAME="E1:30_1_1_1"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> holds <br/><span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">z</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E2:30_1_1"/><i><font color="Green">E52</font></i>: 
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">z</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a>  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>thus </b><a NAME="E3:30_1_1"/>
<font color="Maroon">z</font> <a href="clopban4.html#K1" target="_self">ExpSeq</a>  is <a href="clopban3.html#V2">norm_summable</a>
 <b>by </b><i>, , <a class="ref" href="clopban3.html#T31">CLOPBAN3:31</a></i>;<br/>


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