<?xml version="1.0"?>
<div class="add">

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

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b><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>2</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>  = <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K27" target="_self">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K3">Re</a> <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"><font color="Maroon">a<sub>1</sub></font>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="sin_cos.html#K30" target="_self">P_dt</a> </span>)</span></span>)</span></span>)</span>;<br/>
<b>consider </b><font color="Maroon">Cr</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="E2:113"/><i><font color="Green">E53</font></i>: 
for <font color="Olive">th</font> being  <a href="real_1.html#NM1">Real</a> holds <br/> ( <font color="Olive">th</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">Cr</font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">th</font>] )
 <b>and </b><br/><a NAME="E3:113"/><i><font color="Green">E55</font></i>: 
for <font color="Olive">th</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">th</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">Cr</font> holds <br/><font color="Maroon">Cr</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">th</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">th</font>)
 <b>from </b><i><a class="ref" href="partfun2.html#S2">PARTFUN2:sch 2</a>();<br/></i>
<a NAME="E4:113"/><i><font color="Green">E56</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">Cr</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K1">REAL</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="113_1"><b>proof </b></a><div class="add">

<a NAME="E1:113_1"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a>  holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">Cr</font>
 
<b>by </b><i/>;<br/>
<a NAME="E2:113_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">Cr</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>hence </b><a NAME="E3:113_1"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">Cr</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="E5:113"/><b>then </b><i><font color="Green">E57</font></i>: 
<font color="Maroon">Cr</font> is <a href="partfun1.html#V1">total</a>
 
<b>by </b><i><a class="ref" href="partfun1.html#D4">PARTFUN1:def 4</a></i>;<br/>
<a NAME="E6:113"/><i><font color="Green">E59</font></i>: 
for <font color="Olive">d</font> being  <a href="real_1.html#NM1">Real</a> holds  <font color="Maroon">Cr</font> <a href="seq_1.html#K2">.</a> <font color="Olive">d</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">d</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K27" target="_self">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K3">Re</a> <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"><font color="Olive">d</font>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="sin_cos.html#K30" target="_self">P_dt</a> </span>)</span></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="113_2"><b>proof </b></a><div class="add">

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

<a NAME="E1:113_2"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">Cr</font>
 
<b>by </b><i/>;<br/>
<a NAME="E2:113_2"/><b>then </b>
<font color="Maroon">Cr</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K27" target="_self">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K3">Re</a> <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"><font color="Maroon">d</font>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="sin_cos.html#K30" target="_self">P_dt</a> </span>)</span></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:113_2"/>
<font color="Maroon">Cr</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K27" target="_self">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K3">Re</a> <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"><font color="Maroon">d</font>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="sin_cos.html#K30" target="_self">P_dt</a> </span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:113"/>
<font color="Maroon">Cr</font> is  <a href="fdiff_1.html#V2">REST-like</a> <a href="partfun1.html#NM1">PartFunc</a> of  <a href="numbers.html#K1">REAL</a> , <a href="numbers.html#K1">REAL</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="113_3"><b>proof </b></a><div class="add">

<a NAME="E1:113_3"/>
for <font color="Olive">hy1</font> being <a href="fdiff_1.html#V1">convergent_to_0</a> <a href="seq_1.html#NM1">Real_Sequence</a> holds <br/> ( <span class="p1">(<span class="default"><font color="Olive">hy1</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">Cr</font> <a href="rfunct_2.html#K1">*</a> <font color="Olive">hy1</font></span>)</span> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive">hy1</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">Cr</font> <a href="rfunct_2.html#K1">*</a> <font color="Olive">hy1</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0 )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="113_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">hy1</font> be  <a href="fdiff_1.html#V1">convergent_to_0</a> <a href="seq_1.html#NM1">Real_Sequence</a>;<br/>

<a NAME="E1:113_3_1"/><i><font color="Green">E62</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">hy1</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">Cr</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">hy1</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K27" target="_self">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K3">Re</a> <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"><span class="p5">(<span class="default"><font color="Maroon">hy1</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font></span>)</span>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="sin_cos.html#K30" target="_self">P_dt</a> </span>)</span></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="113_3_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<i><font color="Green">E91</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">hy1</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">Cr</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">hy1</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">hy1</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Cr</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">hy1</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T12">SEQ_1:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">hy1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Cr</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">hy1</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#D8">SEQ_1:def 8</a></i>
;<br/>
<a NAME="E2:113_3_1_1"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">hy1</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">Cr</font>
 
<b>by </b><i/>;<br/>
<b>then </b><i><font color="Green">E92</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">hy1</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">Cr</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">hy1</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">hy1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">Cr</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">hy1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="rfunct_2.html#T21">RFUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">hy1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">hy1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K27" target="_self">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p3">(<span class="default"><a href="comseq_3.html#K8">Sum</a> <span class="p4">(<span class="default"><span class="p5"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p0">(<span class="default"><font color="Maroon">hy1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="sin_cos.html#K30" target="_self">P_dt</a> </span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">hy1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">hy1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K27" target="_self">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p3">(<span class="default"><a href="comseq_3.html#K8">Sum</a> <span class="p4">(<span class="default"><span class="p5"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p0">(<span class="default"><font color="Maroon">hy1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="sin_cos.html#K30" target="_self">P_dt</a> </span>)</span></span>)</span></span>)</span></span>)</span>

;<br/>
<a NAME="E4:113_3_1_1"/>
<font color="Maroon">hy1</font> <a href="seq_1.html#NR1" target="_self">is_not_0</a> 
 
<b>by </b><i><a class="ref" href="fdiff_1.html#D1">FDIFF_1:def 1</a></i>;<br/>
<a NAME="E5:113_3_1_1"/><b>then </b>
<font color="Maroon">hy1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="seq_1.html#T7">SEQ_1:7</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">hy1</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">Cr</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">hy1</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> = 
1 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K27" target="_self">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p3">(<span class="default"><a href="comseq_3.html#K8">Sum</a> <span class="p4">(<span class="default"><span class="p5"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p0">(<span class="default"><font color="Maroon">hy1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="sin_cos.html#K30" target="_self">P_dt</a> </span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="xcmplx_0.html#D7">XCMPLX_0:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="sin_cos.html#K27" target="_self">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K3">Re</a> <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"><span class="p5">(<span class="default"><font color="Maroon">hy1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="sin_cos.html#K30" target="_self">P_dt</a> </span>)</span></span>)</span></span>)</span>

;<br/>
<b>hence </b><a NAME="E7:113_3_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">hy1</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">Cr</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">hy1</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K27" target="_self">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K3">Re</a> <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"><span class="p5">(<span class="default"><font color="Maroon">hy1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="sin_cos.html#K30" target="_self">P_dt</a> </span>)</span></span>)</span></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>deffunc </b><font color="Maroon">H<sub>3</sub></font>(  <a href="real_1.html#NM1">Real</a>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  = <span class="p1">(<span class="default"><a href="sin_cos.html#K27" target="_self">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K3">Re</a> <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"><span class="p5">(<span class="default"><font color="Maroon">hy1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="sin_cos.html#K30" target="_self">P_dt</a> </span>)</span></span>)</span></span>)</span>;<br/>
<b>consider </b><font color="Maroon">rseq</font> being   <a href="seq_1.html#NM1">Real_Sequence</a><b> such that </b><br/><a NAME="E3:113_3_1"/><i><font color="Green">E93</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">rseq</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">n</font>)
 <b>from </b><i><a class="ref" href="seq_1.html#S1">SEQ_1:sch 1</a>();<br/></i>
<b>deffunc </b><font color="Maroon">H<sub>4</sub></font>(   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</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"><span class="p4">(<span class="default"><font color="Maroon">hy1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="sin_cos.html#K30" target="_self">P_dt</a> </span>)</span></span>)</span> <a href="complex1.html#K9">*</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K27" target="_self">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span>,0</span><a href="arytm_0.html#K5">*]</a></span>;<br/>
<b>consider </b><font color="Maroon">cq1</font> being   <a href="comseq_1.html#NM1">Complex_Sequence</a><b> such that </b><br/><a NAME="E5:113_3_1"/><i><font color="Green">E94</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">cq1</font> <a href="comseq_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">n</font>)
 <b>from </b><i><a class="ref" href="comseq_1.html#S1">COMSEQ_1:sch 1</a>();<br/></i>
<a NAME="E6:113_3_1"/><i><font color="Green">E95</font></i>: 
( <font color="Maroon">cq1</font> is <a href="comseq_2.html#V2">convergent</a> &amp;  <a href="comseq_2.html#K2">lim</a> <font color="Maroon">cq1</font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K5">0c</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="113_3_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:113_3_1_2"/><i><font color="Green">E96</font></i>: 
for <font color="Olive">q</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">q</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 holds <br/> ex <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/><span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">cq1</font> <a href="comseq_1.html#K1">.</a> <font color="Olive">m</font></span>)</span> <a href="complex1.html#K11">-</a> <a href="complex1.html#K5">0c</a> </span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">q</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="113_3_1_2_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:113_3_1_2_1"/><i><font color="Green">E97</font></i>: 
<font color="Maroon">q</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<a NAME="E2:113_3_1_2_1"/>
 ex <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/><span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">cq1</font> <a href="comseq_1.html#K1">.</a> <font color="Olive">m</font></span>)</span> <a href="complex1.html#K11">-</a> <a href="complex1.html#K5">0c</a> </span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">q</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="113_3_1_2_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:113_3_1_2_1_1"/>
<a href="sin_cos.html#K27" target="_self">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="sin_cos.html#D17" target="_self">Def17</a></i>;<br/>
<a NAME="E2:113_3_1_2_1_1"/><b>then </b>
<font color="Maroon">q</font> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K27" target="_self">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i>, <a class="ref" href="real_2.html#T127">REAL_2:127</a></i>;<br/>
<b>then consider </b><font color="Maroon">r</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E4:113_3_1_2_1_1"/><i><font color="Green">E98</font></i>: 
( <font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; ( for <font color="Olive">z</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a>   st <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Olive">z</font></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> holds <br/><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"><font color="Olive">z</font> <a href="sin_cos.html#K30" target="_self">P_dt</a> </span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">q</font> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K27" target="_self">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span> ) )
 <b>by </b><i><a class="ref" href="sin_cos.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E5:113_3_1_2_1_1"/>
( <font color="Maroon">hy1</font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <font color="Maroon">hy1</font> <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i><a class="ref" href="fdiff_1.html#D1">FDIFF_1:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E7:113_3_1_2_1_1"/><i><font color="Green">E99</font></i>: 
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">hy1</font> <a href="seq_1.html#K2">.</a> <font color="Olive">m</font></span>)</span> <a href="real_1.html#K5">-</a> 0</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i>, <a class="ref" href="seq_2.html#D7">SEQ_2:def 7</a></i>;<br/>
<div><i><font color="Green">E100</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="113_3_1_2_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:113_3_1_2_1_1_1"/><i><font color="Green">E101</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/><i><font color="Green">E102</font></i>: <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">cq1</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="complex1.html#K11">-</a> <a href="complex1.html#K5">0c</a> </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"><span class="p3">(<span class="default"><a href="comseq_3.html#K8">Sum</a> <span class="p4">(<span class="default"><span class="p5"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p0">(<span class="default"><font color="Maroon">hy1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="sin_cos.html#K30" target="_self">P_dt</a> </span>)</span></span>)</span> <a href="complex1.html#K9">*</a> <span class="p3"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K27" target="_self">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span>,0</span><a href="arytm_0.html#K5">*]</a></span></span>)</span></span><a href="complex1.html#K17">.|</a></span>
<b>by </b><i><a class="txt" href="sin_cos.html#E2:113_3_1_2_1_1_1"><i><font color="Green">E102</font></i></a></i>
<br/>.= 
<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"><span class="p5">(<span class="default"><font color="Maroon">hy1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="sin_cos.html#K30" target="_self">P_dt</a> </span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K27" target="_self">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span>,0</span><a href="arytm_0.html#K5">*]</a></span></span><a href="complex1.html#K17">.|</a></span>
<b>by </b><i><a class="ref" href="complex1.html#T151">COMPLEX1:151</a></i>
<br/>.= 
<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"><span class="p5">(<span class="default"><font color="Maroon">hy1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="sin_cos.html#K30" target="_self">P_dt</a> </span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p5"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p0">(<span class="default"><a href="sin_cos.html#K27" target="_self">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span>,0</span><a href="arytm_0.html#K5">*]</a></span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="complex1.html#K4">Im</a> <span class="p5"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p0">(<span class="default"><a href="sin_cos.html#K27" target="_self">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span>,0</span><a href="arytm_0.html#K5">*]</a></span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#D16">COMPLEX1:def 16</a></i>
;<br/><a NAME="E3:113_3_1_2_1_1_1"/>
<span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K27" target="_self">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K27" target="_self">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">0 <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span>
 <b>by </b><i><a class="ref" href="complex1.html#T27">COMPLEX1:27</a></i>;<br/><a NAME="E4:113_3_1_2_1_1_1"/><b>then </b><i><font color="Green">E103</font></i>: 
(  <a href="complex1.html#K3">Re</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K27" target="_self">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="hidden.html#R1">=</a> <a href="sin_cos.html#K27" target="_self">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font> &amp;  <a href="complex1.html#K4">Im</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K27" target="_self">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="ref" href="complex1.html#T28">COMPLEX1:28</a></i>;<br/><a NAME="E5:113_3_1_2_1_1_1"/><i><font color="Green">E104</font></i>: 
<a href="sin_cos.html#K27" target="_self">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="sin_cos.html#D17" target="_self">Def17</a></i>;<br/><a NAME="E6:113_3_1_2_1_1_1"/><b>then </b><i><font color="Green">E105</font></i>: 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">cq1</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="complex1.html#K11">-</a> <a href="complex1.html#K5">0c</a> </span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="hidden.html#R1">=</a> <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"><span class="p5">(<span class="default"><font color="Maroon">hy1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="sin_cos.html#K30" target="_self">P_dt</a> </span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K27" target="_self">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span>
 <b>by </b><i><a class="txt" href="sin_cos.html#E12:113"><i><font color="Green">E110</font></i></a>, <a class="txt" href="sin_cos.html#E13:113"><i><font color="Green">E111</font></i></a>, <a class="ref" href="square_1.html#T89">SQUARE_1:89</a></i>;<br/><a NAME="E7:113_3_1_2_1_1_1"/><i><font color="Green">E106</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">hy1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> 0</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="txt" href="sin_cos.html#E7:113_3_1"><i><font color="Green">E107</font></i></a>, <a class="txt" href="sin_cos.html#E11:113"><i><font color="Green">E109</font></i></a></i>;<br/><a NAME="E8:113_3_1_2_1_1_1"/>
<span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">hy1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">hy1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">0 <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span>
 <b>by </b><i><a class="ref" href="complex1.html#T27">COMPLEX1:27</a></i>;<br/><a NAME="E9:113_3_1_2_1_1_1"/><b>then </b>
(  <a href="complex1.html#K3">Re</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">hy1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">hy1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> &amp;  <a href="complex1.html#K4">Im</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">hy1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="ref" href="complex1.html#T28">COMPLEX1:28</a></i>;<br/><b>then </b><span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">hy1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span>,0</span><a href="arytm_0.html#K5">*]</a></span></span><a href="complex1.html#K17">.|</a></span> = 
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">hy1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default">0 <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#D16">COMPLEX1:def 16</a></i>
<br/>.= 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">hy1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T158">COMPLEX1:158</a></i>
;<br/><a NAME="E11:113_3_1_2_1_1_1"/><b>then </b>
<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"><span class="p5">(<span class="default"><font color="Maroon">hy1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="sin_cos.html#K30" target="_self">P_dt</a> </span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">q</font> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K27" target="_self">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span>
 <b>by </b><i><a class="txt" href="sin_cos.html#E7:113_3_1_2_1_1_1"><i><font color="Green">E106</font></i></a>, <a class="txt" href="sin_cos.html#E19:113"><i><font color="Green">E114</font></i></a></i>;<br/><a NAME="E12:113_3_1_2_1_1_1"/><b>then </b>
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">cq1</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="complex1.html#K11">-</a> <a href="complex1.html#K5">0c</a> </span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K27" target="_self">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K27" target="_self">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="sin_cos.html#E15:113"><i><font color="Green">E113</font></i></a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><b>hence </b><a NAME="E13:113_3_1_2_1_1_1"/>
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">cq1</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="complex1.html#K11">-</a> <a href="complex1.html#K5">0c</a> </span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">q</font>
 <b>by </b><i>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>;<br/></div><b>end;</b></div>
<b>take </b>
<font color="Maroon">k</font>
;<br/>

<b>thus </b><a NAME="E9:113_3_1_2_1_1"/>
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/><span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">cq1</font> <a href="comseq_1.html#K1">.</a> <font color="Olive">m</font></span>)</span> <a href="complex1.html#K11">-</a> <a href="complex1.html#K5">0c</a> </span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">q</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:113_3_1_2_1"/>
 ex <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/><span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">cq1</font> <a href="comseq_1.html#K1">.</a> <font color="Olive">m</font></span>)</span> <a href="complex1.html#K11">-</a> <a href="complex1.html#K5">0c</a> </span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">q</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:113_3_1_2"/><b>then </b>
<font color="Maroon">cq1</font> is <a href="comseq_2.html#V2">convergent</a>
 
<b>by </b><i><a class="ref" href="comseq_2.html#D4">COMSEQ_2:def 4</a></i>;<br/>
<b>hence </b><a NAME="E3:113_3_1_2"/>
( <font color="Maroon">cq1</font> is <a href="comseq_2.html#V2">convergent</a> &amp;  <a href="comseq_2.html#K2">lim</a> <font color="Maroon">cq1</font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K5">0c</a>  )
 <b>by </b><i>, <a class="ref" href="comseq_2.html#D5">COMSEQ_2:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:113_3_1"/><i><font color="Green">E107</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <span class="p1">(<span class="default"><a href="comseq_3.html#K3">Re</a> <font color="Maroon">cq1</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">rseq</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="113_3_1_3"><b>proof </b></a><div class="add">

<a NAME="E1:113_3_1_3"/><i><font color="Green">E108</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <span class="p1">(<span class="default"><a href="comseq_3.html#K3">Re</a> <font color="Maroon">cq1</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K27" target="_self">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K3">Re</a> <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"><span class="p5">(<span class="default"><font color="Maroon">hy1</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font></span>)</span>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="sin_cos.html#K30" target="_self">P_dt</a> </span>)</span></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="113_3_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<span class="p1">(<span class="default"><a href="comseq_3.html#K3">Re</a> <font color="Maroon">cq1</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> = 
 <a href="complex1.html#K3">Re</a> <span class="p1">(<span class="default"><font color="Maroon">cq1</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="comseq_3.html#D3">COMSEQ_3:def 3</a></i>
<br/>.= 
 <a href="complex1.html#K3">Re</a> <span class="p1">(<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"><span class="p5">(<span class="default"><font color="Maroon">hy1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="sin_cos.html#K30" target="_self">P_dt</a> </span>)</span></span>)</span> <a href="complex1.html#K9">*</a> <span class="p2"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K27" target="_self">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span>,0</span><a href="arytm_0.html#K5">*]</a></span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="sin_cos.html#K27" target="_self">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K3">Re</a> <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"><span class="p5">(<span class="default"><font color="Maroon">hy1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="sin_cos.html#K30" target="_self">P_dt</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E2:113_3_1_3_1"/>
<span class="p1">(<span class="default"><a href="comseq_3.html#K3">Re</a> <font color="Maroon">cq1</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K27" target="_self">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K3">Re</a> <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"><span class="p5">(<span class="default"><font color="Maroon">hy1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="sin_cos.html#K30" target="_self">P_dt</a> </span>)</span></span>)</span></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<a NAME="E2:113_3_1_3"/>
<font color="Maroon">rseq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K27" target="_self">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K3">Re</a> <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"><span class="p5">(<span class="default"><font color="Maroon">hy1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="sin_cos.html#K30" target="_self">P_dt</a> </span>)</span></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:113_3_1_3"/>
<span class="p1">(<span class="default"><a href="comseq_3.html#K3">Re</a> <font color="Maroon">cq1</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">rseq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E8:113_3_1"/>
<font color="Maroon">rseq</font> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><font color="Maroon">hy1</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">Cr</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">hy1</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="113_3_1_4"><b>proof </b></a><div class="add">

<a NAME="E1:113_3_1_4"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">rseq</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">hy1</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">Cr</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">hy1</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="113_3_1_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<a NAME="E1:113_3_1_4_1"/>
<font color="Maroon">rseq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K27" target="_self">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K3">Re</a> <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"><span class="p5">(<span class="default"><font color="Maroon">hy1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="sin_cos.html#K30" target="_self">P_dt</a> </span>)</span></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E2:113_3_1_4_1"/>
<font color="Maroon">rseq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">hy1</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">Cr</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">hy1</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:113_3_1_4"/>
<font color="Maroon">rseq</font> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><font color="Maroon">hy1</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">Cr</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">hy1</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:113_3_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">hy1</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">Cr</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">hy1</font></span>)</span> <a href="funct_2.html#R2">=</a>  <a href="comseq_3.html#K3">Re</a> <font color="Maroon">cq1</font>
 
<b>by </b><i>, <a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/>
<b>hence </b><a NAME="E10:113_3_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">hy1</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">Cr</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">hy1</font></span>)</span> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">hy1</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">Cr</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">hy1</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i>, <a class="ref" href="complex1.html#T12">COMPLEX1:12</a>, <a class="ref" href="comseq_3.html#T41">COMSEQ_3:41</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:113_3"/>
<font color="Maroon">Cr</font> is  <a href="fdiff_1.html#V2">REST-like</a> <a href="partfun1.html#NM1">PartFunc</a> of  <a href="numbers.html#K1">REAL</a> , <a href="numbers.html#K1">REAL</a> 
 <b>by </b><i>, <a class="ref" href="fdiff_1.html#D3">FDIFF_1:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">PR</font> = <font color="Maroon">Cr</font> as  <a href="fdiff_1.html#V2">REST-like</a> <a href="partfun1.html#NM1">PartFunc</a> of  <a href="numbers.html#K1">REAL</a> , <a href="numbers.html#K1">REAL</a>  ;<br/>
<b>reconsider </b><font color="Maroon">PR</font> = <font color="Maroon">PR</font> as   <a href="fdiff_1.html#NM1">REST</a> ;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b><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>3</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>  = <font color="Maroon">a<sub>1</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K27" target="_self">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span>;<br/>
<b>consider </b><font color="Maroon">CL</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="E11:113"/><i><font color="Green">E109</font></i>: 
for <font color="Olive">th</font> being  <a href="real_1.html#NM1">Real</a> holds <br/> ( <font color="Olive">th</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">CL</font> iff <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">th</font>] )
 <b>and </b><br/><a NAME="E12:113"/><i><font color="Green">E110</font></i>: 
for <font color="Olive">th</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">th</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">CL</font> holds <br/><font color="Maroon">CL</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">th</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">th</font>)
 <b>from </b><i><a class="ref" href="partfun2.html#S2">PARTFUN2:sch 2</a>();<br/></i>
<a NAME="E13:113"/><i><font color="Green">E111</font></i>: 
for <font color="Olive">d</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  holds  <font color="Maroon">CL</font> <a href="seq_1.html#K2">.</a> <font color="Olive">d</font> <a href="hidden.html#R1">=</a> <font color="Olive">d</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K27" target="_self">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="113_4"><b>proof </b></a><div class="add">

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

<a NAME="E1:113_4"/>
<font color="Maroon">d</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="E2:113_4"/><b>then </b><i><font color="Green">E112</font></i>: 
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">CL</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:113_4"/><b>then </b>
<font color="Maroon">CL</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K27" target="_self">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:113_4"/>
<font color="Maroon">CL</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K27" target="_self">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E14:113"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">CL</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K1">REAL</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="113_5"><b>proof </b></a><div class="add">

<a NAME="E1:113_5"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a>  holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">CL</font>
 
<b>by </b><i/>;<br/>
<a NAME="E2:113_5"/><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">CL</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>hence </b><a NAME="E3:113_5"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">CL</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="E15:113"/><b>then </b><i><font color="Green">E113</font></i>: 
<font color="Maroon">CL</font> is <a href="partfun1.html#V1">total</a>
 
<b>by </b><i><a class="ref" href="partfun1.html#D4">PARTFUN1:def 4</a></i>;<br/>
<a NAME="E16:113"/>
<font color="Maroon">CL</font> is  <a href="fdiff_1.html#V3">linear</a> <a href="partfun1.html#NM1">PartFunc</a> of  <a href="numbers.html#K1">REAL</a> , <a href="numbers.html#K1">REAL</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="113_6"><b>proof </b></a><div class="add">

<a NAME="E1:113_6"/>
 ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st <br/>for <font color="Olive">q</font> being  <a href="real_1.html#NM1">Real</a> holds  <font color="Maroon">CL</font> <a href="seq_1.html#K2">.</a> <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <font color="Olive">r</font> <a href="real_1.html#K4">*</a> <font color="Olive">q</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="113_6_1"><b>proof </b></a><div class="add">

<b>take </b>
<a href="sin_cos.html#K27" target="_self">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font>
;<br/>

<b>thus </b><a NAME="E1:113_6_1"/>
for <font color="Olive">q</font> being  <a href="real_1.html#NM1">Real</a> holds  <font color="Maroon">CL</font> <a href="seq_1.html#K2">.</a> <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K27" target="_self">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Olive">q</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:113_6"/>
<font color="Maroon">CL</font> is  <a href="fdiff_1.html#V3">linear</a> <a href="partfun1.html#NM1">PartFunc</a> of  <a href="numbers.html#K1">REAL</a> , <a href="numbers.html#K1">REAL</a> 
 <b>by </b><i>, <a class="ref" href="fdiff_1.html#D4">FDIFF_1:def 4</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">PL</font> = <font color="Maroon">CL</font> as  <a href="fdiff_1.html#V3">linear</a> <a href="partfun1.html#NM1">PartFunc</a> of  <a href="numbers.html#K1">REAL</a> , <a href="numbers.html#K1">REAL</a>  ;<br/>
<b>reconsider </b><font color="Maroon">PL</font> = <font color="Maroon">PL</font> as   <a href="fdiff_1.html#NM2">LINEAR</a> ;<br/>
<a NAME="E19:113"/><i><font color="Green">E114</font></i>: 
 ex <font color="Olive">N</font> being   <a href="rcomp_1.html#M1">Neighbourhood</a> of <font color="Maroon">p</font> st <br/>( <font color="Olive">N</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <a href="sin_cos.html#K27" target="_self">exp_R</a>  &amp; ( for <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <font color="Olive">N</font> holds <br/><span class="p1">(<span class="default"><a href="sin_cos.html#K27" target="_self">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Olive">r</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K27" target="_self">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">PL</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Olive">r</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">PR</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Olive">r</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">p</font></span>)</span></span>)</span> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="113_7"><b>proof </b></a><div class="add">

<a NAME="E1:113_7"/><i><font color="Green">E115</font></i>: 
for <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> holds <br/><span class="p1">(<span class="default"><a href="sin_cos.html#K27" target="_self">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Olive">r</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K27" target="_self">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">PL</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Olive">r</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">PR</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Olive">r</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">p</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="113_7_1"><b>proof </b></a><div class="add">

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

<b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">p</font> as   <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="E2:113_7_1"/>
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K5">-</a> <font color="Maroon">p</font></span>)</span>
 
;<br/>
<b>then </b><span class="p1">(<span class="default"><a href="sin_cos.html#K27" target="_self">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">r</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K27" target="_self">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K5">-</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K27" target="_self">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K5">-</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K27" target="_self">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p3">(<span class="default"><a href="comseq_3.html#K8">Sum</a> <span class="p4">(<span class="default"><span class="p5"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p0">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K5">-</a> <font color="Maroon">p</font></span>)</span>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="sin_cos.html#K30" target="_self">P_dt</a> </span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K5">-</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K27" target="_self">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">Cr</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K5">-</a> <font color="Maroon">p</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">PL</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K5">-</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">PR</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K5">-</a> <font color="Maroon">p</font></span>)</span></span>)</span>
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E4:113_7_1"/>
( <font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> implies <span class="p1">(<span class="default"><a href="sin_cos.html#K27" target="_self">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">r</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K27" target="_self">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">PL</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">PR</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">p</font></span>)</span></span>)</span> )
 ;<br/>


</div><b>end;</b></div>
<b>take </b>
<span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
;<br/>

<b>thus </b><a NAME="E2:113_7"/>
( <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> is    <a href="rcomp_1.html#M1">Neighbourhood</a> of <font color="Maroon">p</font> &amp; <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <a href="sin_cos.html#K27" target="_self">exp_R</a>  &amp; ( for <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> holds <br/><span class="p1">(<span class="default"><a href="sin_cos.html#K27" target="_self">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Olive">r</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K27" target="_self">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">PL</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Olive">r</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">PR</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Olive">r</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">p</font></span>)</span></span>)</span> ) )
 <b>by </b><i>, <a class="ref" href="sin_cos.html#D10" target="_self">Def10</a>, <a class="ref" href="rcomp_1.html#D7">RCOMP_1:def 7</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E20:113"/><b>then </b><i><font color="Green">E116</font></i>: 
 <a href="sin_cos.html#K27" target="_self">exp_R</a>  <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="fdiff_1.html#D5">FDIFF_1:def 5</a></i>;<br/>
<a NAME="E21:113"/>
<font color="Maroon">PL</font> <a href="seq_1.html#K2">.</a> 1 <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K27" target="_self">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E22:113"/>
(  <a href="sin_cos.html#K27" target="_self">exp_R</a>  <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Maroon">p</font> &amp;  <a href="fdiff_1.html#K1">diff</a> <a href="sin_cos.html#K27" target="_self">exp_R</a> ,<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <a href="sin_cos.html#K27" target="_self">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font> )
 <b>by </b><i>, , <a class="ref" href="fdiff_1.html#D6">FDIFF_1:def 6</a></i>;<br/>


</div>
