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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] means <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a> ;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  =  <a href="complex1.html#K4">Im</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K8">Sum</a> <span class="p2">(<span class="default"><span class="p3"><a href="arytm_0.html#K5">[*</a><span class="default">0,<font color="Maroon">a<sub>1</sub></font></span><a href="arytm_0.html#K5">*]</a></span> <a href="sin_cos.html#K6" target="_self">ExpSeq</a> </span>)</span></span>)</span>;<br/>
<b>consider </b><font color="Maroon">c<sub>1</sub></font> being   <a href="partfun1.html#NM1">PartFunc</a> of  <a href="numbers.html#K1">REAL</a> , <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E2:48_1_1"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] )
 <b>and </b><br/><a NAME="E3:48_1_1"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>)
 <b>from </b><i><a class="ref" href="partfun2.html#S2">PARTFUN2:sch 2</a>();<br/></i>
<b>take </b>
<font color="Maroon">c<sub>1</sub></font>
;<br/>

<a NAME="E4:48_1_1"/>
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K1">REAL</a>  &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  holds <font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K4">Im</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K8">Sum</a> <span class="p2">(<span class="default"><span class="p3"><a href="arytm_0.html#K5">[*</a><span class="default">0,<font color="Olive">b<sub>1</sub></font></span><a href="arytm_0.html#K5">*]</a></span> <a href="sin_cos.html#K6" target="_self">ExpSeq</a> </span>)</span></span>)</span> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:48_1_1_1"/><i><font color="Green">E41</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K1">REAL</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:48_1_1_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a>  holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="sin_cos.html#E2:48_1_1"><i><font color="Green">E39</font></i></a></i>;<br/>
<a NAME="E2:48_1_1_1_1"/><b>then </b>
 <a href="numbers.html#K1">REAL</a>  <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>hence </b><a NAME="E3:48_1_1_1_1"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K1">REAL</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:48_1_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  holds <font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K4">Im</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K8">Sum</a> <span class="p2">(<span class="default"><span class="p3"><a href="arytm_0.html#K5">[*</a><span class="default">0,<font color="Olive">b<sub>1</sub></font></span><a href="arytm_0.html#K5">*]</a></span> <a href="sin_cos.html#K6" target="_self">ExpSeq</a> </span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> ;<br/>

<a NAME="E1:48_1_1_1_2"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="sin_cos.html#E2:48_1_1"><i><font color="Green">E39</font></i></a></i>;<br/>
<a NAME="E2:48_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K4">Im</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K8">Sum</a> <span class="p2">(<span class="default"><span class="p3"><a href="arytm_0.html#K5">[*</a><span class="default">0,<font color="Maroon">c<sub>2</sub></font></span><a href="arytm_0.html#K5">*]</a></span> <a href="sin_cos.html#K6" target="_self">ExpSeq</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="sin_cos.html#E3:48_1_1"><i><font color="Green">E40</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:48_1_1_1_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K4">Im</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K8">Sum</a> <span class="p2">(<span class="default"><span class="p3"><a href="arytm_0.html#K5">[*</a><span class="default">0,<font color="Maroon">c<sub>2</sub></font></span><a href="arytm_0.html#K5">*]</a></span> <a href="sin_cos.html#K6" target="_self">ExpSeq</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="sin_cos.html#E1:48_1_1_1_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:48_1_1_1"/>
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K1">REAL</a>  &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  holds <font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K4">Im</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K8">Sum</a> <span class="p2">(<span class="default"><span class="p3"><a href="arytm_0.html#K5">[*</a><span class="default">0,<font color="Olive">b<sub>1</sub></font></span><a href="arytm_0.html#K5">*]</a></span> <a href="sin_cos.html#K6" target="_self">ExpSeq</a> </span>)</span></span>)</span> ) )
 <b>by </b><i><a class="txt" href="sin_cos.html#E1:48_1_1_1"><i><font color="Green">E41</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:48_1_1"/>
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K1">REAL</a>  &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  holds <font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K4">Im</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K8">Sum</a> <span class="p2">(<span class="default"><span class="p3"><a href="arytm_0.html#K5">[*</a><span class="default">0,<font color="Olive">b<sub>1</sub></font></span><a href="arytm_0.html#K5">*]</a></span> <a href="sin_cos.html#K6" target="_self">ExpSeq</a> </span>)</span></span>)</span> ) )
 ;<br/>


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