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

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">r</font> be   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">x</font> be   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">s</font> be   <a href="real_1.html#NM1">Real</a>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:10"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">x</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"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>and </b><br/><a NAME="E2:10"/><i><font color="Green">E32</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s</font> &amp; <font color="Maroon">s</font> <a href="xxreal_0.html#R1">&lt;</a> 1 )
 ;<br/>

<a NAME="E3:10"/>
<font color="Maroon">x</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">0 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span>,<span class="p2">(<span class="default">0 <a href="real_1.html#K3">+</a> <font color="Maroon">r</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E4:10"/><b>then </b><i><font color="Green">E34</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <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="rcomp_1.html#T8">RCOMP_1:8</a></i>;<br/>
<i><font color="Green">E41</font></i>:  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="real_1.html#K4">*</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K5">-</a> 0</span>)</span> = 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">s</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T151">COMPLEX1:151</a></i>
<br/>.= 
<font color="Maroon">s</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
;<br/>
<a NAME="E6:10"/>
 <a href="complex1.html#K18">abs</a> <font color="Maroon">x</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/>
<a NAME="E7:10"/><b>then </b>
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">s</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> <a href="real_1.html#K4">*</a> 1
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T99">XREAL_1:99</a>, </i>;<br/>
<a NAME="E8:10"/><b>then </b>
<font color="Maroon">s</font> <a href="real_1.html#K4">*</a> <font color="Maroon">x</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">0 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span>,<span class="p2">(<span class="default">0 <a href="real_1.html#K3">+</a> <font color="Maroon">r</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T8">RCOMP_1:8</a>, </i>;<br/>
<a NAME="E9:10"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">s</font> <a href="real_1.html#K4">*</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E10:10"/>
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="sin_cos.html#K5">!</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="newton.html#T23">NEWTON:23</a></i>;<br/>
<a NAME="E11:10"/><b>then </b><i><font color="Green">E48</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="sin_cos.html#K5">!</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="sin_cos.html#K5">!</a> 
 
<b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="taylor_1.html#K5">diff</a> <a href="sin_cos.html#K27">exp_R</a> ,<span class="p0"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">s</font> <a href="real_1.html#K4">*</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="prepower.html#K3">|^</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="sin_cos.html#K5">!</a> </span>)</span></span>)</span> = 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="partfun1.html#K2">|</a> <span class="p5"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p0">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">s</font> <a href="real_1.html#K4">*</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="prepower.html#K3">|^</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="sin_cos.html#K5">!</a> </span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">s</font> <a href="real_1.html#K4">*</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="prepower.html#K3">|^</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="sin_cos.html#K5">!</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T70">FUNCT_1:70</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">s</font> <a href="real_1.html#K4">*</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="prepower.html#K3">|^</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="sin_cos.html#K5">!</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T153">COMPLEX1:153</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">s</font> <a href="real_1.html#K4">*</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="prepower.html#K3">|^</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="sin_cos.html#K5">!</a> </span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T151">COMPLEX1:151</a>, </i>
;<br/>
<b>hence </b><a NAME="E13:10"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="taylor_1.html#K5">diff</a> <a href="sin_cos.html#K27">exp_R</a> ,<span class="p0"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">s</font> <a href="real_1.html#K4">*</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="prepower.html#K3">|^</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="sin_cos.html#K5">!</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">s</font> <a href="real_1.html#K4">*</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">x</font></span>)</span> <a href="prepower.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="sin_cos.html#K5">!</a> </span>)</span>
 <b>by </b><i/>;<br/>


</div>
