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<b>let </b><font color="Maroon" title="c1">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>

<b>assume </b><a NAME="E1:12"/><i><font color="Green" title="E9">A1</font></i>: 
<font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 ;<br/>

<b>take </b><font color="Maroon" title="c2">M</font> = <a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c1">r</font>;<br/>

<b>take </b><font color="Maroon" title="c3">L</font> = <font color="Maroon" title="c1">r</font>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c6">s</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>assume </b><a NAME="E1:12_1_1"/><i><font color="Green" title="E10">A2</font></i>: 
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c1">r</font></span>)</span>,<font color="Maroon" title="c1">r</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> &amp; 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c6">s</font> &amp; <font color="Maroon" title="c6">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 )
 ;<br/><a NAME="E2:12_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default">0 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c1">r</font></span>)</span>,<span class="p2">(<span class="default">0 <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c1">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 <b>by </b><i><a class="txt" href="taylor_2.html#E1:12_1_1"><i><font color="Green" title="E10">A2</font></i></a></i>;<br/><a NAME="E3:12_1_1"/><b>then </b><i><font color="Green" title="E11">A3</font></i>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 0</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">r</font>
 <b>by </b><i><a class="ref" href="rcomp_1.html#T8" title="RCOMP_1:th.8">RCOMP_1:8</a></i>;<br/><a NAME="E4:12_1_1"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c5">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="complex1.html#T132" title="COMPLEX1:th.132">COMPLEX1:132</a></i>;<br/><a NAME="E5:12_1_1"/><b>then </b>
<font color="Maroon" title="c6">s</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c1">r</font>
 <b>by </b><i><a class="txt" href="taylor_2.html#E1:12_1_1"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="taylor_2.html#E3:12_1_1"><i><font color="Green" title="E11">A3</font></i></a>, <a class="ref" href="xreal_1.html#T99" title="XREAL_1:th.99">XREAL_1:99</a></i>;<br/><a NAME="E6:12_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c6">s</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">r</font>
 <b>by </b><i><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a>, <a class="txt" href="taylor_2.html#E1:12_1_1"><i><font color="Green" title="E10">A2</font></i></a></i>;<br/><a NAME="E7:12_1_1"/><b>then </b>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">s</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 0</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">r</font>
 <b>by </b><i><a class="ref" href="complex1.html#T151" title="COMPLEX1:th.151">COMPLEX1:151</a></i>;<br/><a NAME="E8:12_1_1"/><b>then </b><i><font color="Green" title="E12">A4</font></i>: 
<font color="Maroon" title="c6">s</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default">0 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c1">r</font></span>)</span>,<span class="p2">(<span class="default">0 <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c1">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 <b>by </b><i><a class="ref" href="rcomp_1.html#T8" title="RCOMP_1:th.8">RCOMP_1:8</a></i>;<br/><a NAME="E9:12_1_1"/><i><font color="Green" title="E13">A5</font></i>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">s</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="complex1.html#T132" title="COMPLEX1:th.132">COMPLEX1:132</a></i>;<br/><a NAME="E10:12_1_1"/><i><font color="Green" title="E14">A6</font></i>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="prepower.html#K3" title="PREPOWER:func.3">|^</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!</a> </span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="complex1.html#T132" title="COMPLEX1:th.132">COMPLEX1:132</a></i>;<br/><a NAME="E11:12_1_1"/>
(  <a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a>  <a href="subset_1.html#K2" title="SUBSET_1:func.2">[#]</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp; ( for <font color="Olive" title="b1">x0</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> holds  0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a> ,<font color="Olive" title="b1">x0</font> ) )
 <b>by </b><i><a class="ref" href="taylor_1.html#T16" title="TAYLOR_1:th.16">TAYLOR_1:16</a></i>;<br/><a NAME="E12:12_1_1"/><b>then </b><i><font color="Green" title="E15">A7</font></i>: 
 <a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="rfunct_2.html#R3" title="RFUNCT_2:pred.3">is_non_decreasing_on</a>  <a href="subset_1.html#K2" title="SUBSET_1:func.2">[#]</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 <b>by </b><i><a class="ref" href="fdiff_2.html#T39" title="FDIFF_2:th.39">FDIFF_2:39</a></i>;<br/><a NAME="E13:12_1_1"/>
<font color="Maroon" title="c6">s</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : (  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">p</font> &amp; <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">r</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="taylor_2.html#E8:12_1_1"><i><font color="Green" title="E12">A4</font></i></a>, <a class="ref" href="rcomp_1.html#D2" title="RCOMP_1:def.2">RCOMP_1:def 2</a></i>;<br/><a NAME="E14:12_1_1"/><b>then </b><i><font color="Green" title="E16">A8</font></i>: 
 ex <font color="Olive" title="b1">g</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">s</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c5">x</font> &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">g</font> &amp; <font color="Olive" title="b1">g</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">r</font> )
 ;<br/><a NAME="E15:12_1_1"/><i><font color="Green" title="E17">A9</font></i>: 
<font color="Maroon" title="c1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="subset_1.html#K2" title="SUBSET_1:func.2">[#]</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a> </span>)</span>
 <b>by </b><i><a class="ref" href="taylor_1.html#T16" title="TAYLOR_1:th.16">TAYLOR_1:16</a></i>;<br/><a NAME="E16:12_1_1"/>
<a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">s</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="sin_cos.html#T59" title="SIN_COS:th.59">SIN_COS:59</a></i>;<br/><a NAME="E17:12_1_1"/><b>then </b>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">s</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">s</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c5">x</font></span>)</span>
 <b>by </b><i><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/><a NAME="E18:12_1_1"/><b>then </b><i><font color="Green" title="E18">A10</font></i>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">s</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">M</font>
 <b>by </b><i><a class="txt" href="taylor_2.html#E12:12_1_1"><i><font color="Green" title="E15">A7</font></i></a>, <a class="txt" href="taylor_2.html#E14:12_1_1"><i><font color="Green" title="E16">A8</font></i></a>, <a class="ref" href="taylor_1.html#T16" title="TAYLOR_1:th.16">TAYLOR_1:16</a>, <a class="ref" href="rfunct_2.html#T48" title="RFUNCT_2:th.48">RFUNCT_2:48</a>, <a class="txt" href="taylor_2.html#E15:12_1_1"><i><font color="Green" title="E17">A9</font></i></a></i>;<br/><a NAME="E19:12_1_1"/><i><font color="Green" title="E19">A11</font></i>: 
<font color="Maroon" title="c4">n</font> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>by </b><i><a class="ref" href="newton.html#T23" title="NEWTON:th.23">NEWTON:23</a></i>;<br/><i><font color="Green" title="E20">A12</font></i>:  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="prepower.html#K3" title="PREPOWER:func.3">|^</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!</a> </span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="prepower.html#K3" title="PREPOWER:func.3">|^</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T153" title="COMPLEX1:th.153">COMPLEX1:153</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="prepower.html#K3" title="PREPOWER:func.3">|^</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!</a> </span>)</span>
<b>by </b><i><a class="txt" href="taylor_2.html#E19:12_1_1"><i><font color="Green" title="E19">A11</font></i></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="prepower.html#K3" title="PREPOWER:func.3">|^</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!</a> </span>)</span>
<b>by </b><i><a class="ref" href="taylor_2.html#T1" target="_self" title="TAYLOR_2:th.1">Th1</a></i>
;<br/><a NAME="E21:12_1_1"/><i><font color="Green" title="E21">A13</font></i>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">a</font> &amp; <font color="Olive" title="b1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">b</font> holds <br/>for <font color="Olive" title="b3">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/> ( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">a</font> <a href="prepower.html#K3" title="PREPOWER:func.3">|^</a> <font color="Olive" title="b3">n</font> &amp; <font color="Olive" title="b1">a</font> <a href="prepower.html#K3" title="PREPOWER:func.3">|^</a> <font color="Olive" title="b3">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">b</font> <a href="prepower.html#K3" title="PREPOWER:func.3">|^</a> <font color="Olive" title="b3">n</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">a</font>, <font color="Maroon" title="c8">b</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>

<b>assume </b><a NAME="E1:12_1_1_2"/><i><font color="Green" title="E22">A14</font></i>: 
( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">a</font> &amp; <font color="Maroon" title="c7">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">b</font> )
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b>( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">a</font> <a href="prepower.html#K3" title="PREPOWER:func.3">|^</a> $1 &amp; <font color="Maroon" title="c7">a</font> <a href="prepower.html#K3" title="PREPOWER:func.3">|^</a> $1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">b</font> <a href="prepower.html#K3" title="PREPOWER:func.3">|^</a> $1 );<br/>
<a NAME="E2:12_1_1_2"/><i><font color="Green" title="E23">A15</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:12_1_1_2_1"/><i><font color="Green" title="E24">A16</font></i>: 
<font color="Maroon" title="c8">b</font> <a href="prepower.html#K3" title="PREPOWER:func.3">|^</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="ref" href="newton.html#T9" title="NEWTON:th.9">NEWTON:9</a></i>;<br/>
<b>thus </b><a NAME="E2:12_1_1_2_1"/>
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">a</font> <a href="prepower.html#K3" title="PREPOWER:func.3">|^</a> 0
 <b>by </b><i><a class="ref" href="newton.html#T9" title="NEWTON:th.9">NEWTON:9</a></i>;<br/>

<b>thus </b><a NAME="E3:12_1_1_2_1"/>
<font color="Maroon" title="c7">a</font> <a href="prepower.html#K3" title="PREPOWER:func.3">|^</a> 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">b</font> <a href="prepower.html#K3" title="PREPOWER:func.3">|^</a> 0
 <b>by </b><i><a class="txt" href="taylor_2.html#E1:12_1_1_2_1"><i><font color="Green" title="E24">A16</font></i></a>, <a class="ref" href="newton.html#T9" title="NEWTON:th.9">NEWTON:9</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:12_1_1_2"/><i><font color="Green" title="E24">A17</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:12_1_1_2_2"/><i><font color="Green" title="E25">A18</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">k</font>]
 ;<br/>

<a NAME="E2:12_1_1_2_2"/><i><font color="Green" title="E26">A19</font></i>: 
<font color="Maroon" title="c8">b</font> <a href="prepower.html#K3" title="PREPOWER:func.3">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">b</font> <a href="prepower.html#K3" title="PREPOWER:func.3">|^</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c8">b</font>
 
<b>by </b><i><a class="ref" href="newton.html#T11" title="NEWTON:th.11">NEWTON:11</a></i>;<br/>
<a NAME="E3:12_1_1_2_2"/>
0 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c7">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="prepower.html#K3" title="PREPOWER:func.3">|^</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c7">a</font>
 
<b>by </b><i><a class="txt" href="taylor_2.html#E1:12_1_1_2"><i><font color="Green" title="E22">A14</font></i></a>, <a class="txt" href="taylor_2.html#E1:12_1_1_2_2"><i><font color="Green" title="E25">A18</font></i></a>, <a class="ref" href="xreal_1.html#T68" title="XREAL_1:th.68">XREAL_1:68</a></i>;<br/>
<b>hence </b><a NAME="E4:12_1_1_2_2"/>
<font color="Maroon" title="c7">a</font> <a href="prepower.html#K3" title="PREPOWER:func.3">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="newton.html#T11" title="NEWTON:th.11">NEWTON:11</a></i>;<br/>

<a NAME="E5:12_1_1_2_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="prepower.html#K3" title="PREPOWER:func.3">|^</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c7">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">b</font> <a href="prepower.html#K3" title="PREPOWER:func.3">|^</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c8">b</font>
 
<b>by </b><i><a class="txt" href="taylor_2.html#E1:12_1_1_2"><i><font color="Green" title="E22">A14</font></i></a>, <a class="txt" href="taylor_2.html#E1:12_1_1_2_2"><i><font color="Green" title="E25">A18</font></i></a>, <a class="ref" href="xreal_1.html#T68" title="XREAL_1:th.68">XREAL_1:68</a></i>;<br/>
<b>hence </b><a NAME="E6:12_1_1_2_2"/>
<font color="Maroon" title="c7">a</font> <a href="prepower.html#K3" title="PREPOWER:func.3">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">b</font> <a href="prepower.html#K3" title="PREPOWER:func.3">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="newton.html#T11" title="NEWTON:th.11">NEWTON:11</a>, <a class="txt" href="taylor_2.html#E2:12_1_1_2_2"><i><font color="Green" title="E26">A19</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:12_1_1_2"/>
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="taylor_2.html#E2:12_1_1_2"><i><font color="Green" title="E23">A15</font></i></a>, <a class="txt" href="taylor_2.html#E3:12_1_1_2"><i><font color="Green" title="E24">A17</font></i></a>);<br/></i>
<b>hence </b><a NAME="E5:12_1_1_2"/>
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/> ( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">a</font> <a href="prepower.html#K3" title="PREPOWER:func.3">|^</a> <font color="Olive" title="b1">n</font> &amp; <font color="Maroon" title="c7">a</font> <a href="prepower.html#K3" title="PREPOWER:func.3">|^</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">b</font> <a href="prepower.html#K3" title="PREPOWER:func.3">|^</a> <font color="Olive" title="b1">n</font> )
 ;<br/>


</div><b>end;</b></div><a NAME="E22:12_1_1"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c5">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="complex1.html#T132" title="COMPLEX1:th.132">COMPLEX1:132</a></i>;<br/><a NAME="E23:12_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="prepower.html#K3" title="PREPOWER:func.3">|^</a> <font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">L</font> <a href="prepower.html#K3" title="PREPOWER:func.3">|^</a> <font color="Maroon" title="c4">n</font>
 <b>by </b><i><a class="txt" href="taylor_2.html#E21:12_1_1"><i><font color="Green" title="E21">A13</font></i></a>, <a class="txt" href="taylor_2.html#E3:12_1_1"><i><font color="Green" title="E11">A3</font></i></a></i>;<br/><a NAME="E24:12_1_1"/><b>then </b><i><font color="Green" title="E22">A20</font></i>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="prepower.html#K3" title="PREPOWER:func.3">|^</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="prepower.html#K3" title="PREPOWER:func.3">|^</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!</a> </span>)</span>
 <b>by </b><i><a class="txt" href="taylor_2.html#E20:12_1_1"><i><font color="Green" title="E20">A12</font></i></a>, <a class="ref" href="xreal_1.html#T74" title="XREAL_1:th.74">XREAL_1:74</a>, <a class="txt" href="taylor_2.html#E19:12_1_1"><i><font color="Green" title="E19">A11</font></i></a></i>;<br/> <a href="complex1.html#K18" title="COMPLEX1:func.18">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" title="TAYLOR_1:func.5">diff</a> <a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a> ,<span class="p0"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c1">r</font></span>)</span>,<font color="Maroon" title="c1">r</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">s</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="prepower.html#K3" title="PREPOWER:func.3">|^</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!</a> </span>)</span></span>)</span> = 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">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" title="SIN_COS:func.27">exp_R</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">s</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="prepower.html#K3" title="PREPOWER:func.3">|^</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="taylor_2.html#T7" target="_self" title="TAYLOR_2:th.7">Th7</a>, <a class="txt" href="taylor_2.html#E8:12_1_1"><i><font color="Green" title="E12">A4</font></i></a></i>
<br/>.= 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">s</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="prepower.html#K3" title="PREPOWER:func.3">|^</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T75" title="XCMPLX_1:th.75">XCMPLX_1:75</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">s</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="prepower.html#K3" title="PREPOWER:func.3">|^</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T151" title="COMPLEX1:th.151">COMPLEX1:151</a></i>
;<br/><a NAME="E26:12_1_1"/><b>then </b>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">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" title="TAYLOR_1:func.5">diff</a> <a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a> ,<span class="p0"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c1">r</font></span>)</span>,<font color="Maroon" title="c1">r</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">s</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="prepower.html#K3" title="PREPOWER:func.3">|^</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">M</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="prepower.html#K3" title="PREPOWER:func.3">|^</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T68" title="XREAL_1:th.68">XREAL_1:68</a>, <a class="txt" href="taylor_2.html#E9:12_1_1"><i><font color="Green" title="E13">A5</font></i></a>, <a class="txt" href="taylor_2.html#E10:12_1_1"><i><font color="Green" title="E14">A6</font></i></a>, <a class="txt" href="taylor_2.html#E18:12_1_1"><i><font color="Green" title="E18">A10</font></i></a>, <a class="txt" href="taylor_2.html#E24:12_1_1"><i><font color="Green" title="E22">A20</font></i></a></i>;<br/><b>hence </b><a NAME="E27:12_1_1"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">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" title="TAYLOR_1:func.5">diff</a> <a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a> ,<span class="p0"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c1">r</font></span>)</span>,<font color="Maroon" title="c1">r</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">s</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="prepower.html#K3" title="PREPOWER:func.3">|^</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="prepower.html#K3" title="PREPOWER:func.3">|^</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!</a> </span>)</span>
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T75" title="XCMPLX_1:th.75">XCMPLX_1:75</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E2:12_1"/>
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">s</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c1">r</font></span>)</span>,<font color="Maroon" title="c1">r</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> &amp; 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b2">s</font> &amp; <font color="Olive" title="b2">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 holds <br/> <a href="complex1.html#K18" title="COMPLEX1:func.18">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" title="TAYLOR_1:func.5">diff</a> <a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a> ,<span class="p0"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c1">r</font></span>)</span>,<font color="Maroon" title="c1">r</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p4">(<span class="default"><font color="Olive" title="b2">s</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Olive" title="b1">x</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">x</font> <a href="prepower.html#K3" title="PREPOWER:func.3">|^</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="prepower.html#K3" title="PREPOWER:func.3">|^</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!</a> </span>)</span>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:12"/>
( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">M</font> &amp; 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">L</font> &amp; ( for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b2">x</font>, <font color="Olive" title="b3">s</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c1">r</font></span>)</span>,<font color="Maroon" title="c1">r</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> &amp; 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b3">s</font> &amp; <font color="Olive" title="b3">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 holds <br/> <a href="complex1.html#K18" title="COMPLEX1:func.18">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" title="TAYLOR_1:func.5">diff</a> <a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a> ,<span class="p0"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c1">r</font></span>)</span>,<font color="Maroon" title="c1">r</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p4">(<span class="default"><font color="Olive" title="b3">s</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Olive" title="b2">x</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p3">(<span class="default"><font color="Olive" title="b2">x</font> <a href="prepower.html#K3" title="PREPOWER:func.3">|^</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="prepower.html#K3" title="PREPOWER:func.3">|^</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!</a> </span>)</span> ) )
 <b>by </b><i><a class="ref" href="sin_cos.html#T59" title="SIN_COS:th.59">SIN_COS:59</a>, <a class="txt" href="taylor_2.html#E1:12"><i><font color="Green" title="E9">A1</font></i></a></i>;<br/>


</div>
