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

<b>thus </b><span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K18" target="_self">exp</a> <span class="p3"><a href="arytm_0.html#K5">[*</a><span class="default">0,<font color="Maroon">th</font></span><a href="arytm_0.html#K5">*]</a></span></span>)</span></span><a href="complex1.html#K17">.|</a></span> = 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K8">Sum</a> <span class="p3">(<span class="default"><span class="p4"><a href="arytm_0.html#K5">[*</a><span class="default">0,<font color="Maroon">th</font></span><a href="arytm_0.html#K5">*]</a></span> <a href="sin_cos.html#K6" target="_self">ExpSeq</a> </span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span>
<b>by </b><i/>
<br/>.= 
1
<b>by </b><i/>
;<br/>

<b>let </b><font color="Maroon">th</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<a NAME="E2:63"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">th</font> is   <a href="real_1.html#NM1">Real</a>
 
<b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E3:63"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K19" target="_self">sin</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">th</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:63"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K20" target="_self">sin</a> <font color="Maroon">th</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1
 ;<br/>

<a NAME="E5:63"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K22" target="_self">cos</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">th</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E6:63"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K23" target="_self">cos</a> <font color="Maroon">th</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1
 ;<br/>


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