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

<span class="kw">let </span><font color="Maroon" title="c1">y</font>, <font color="Maroon" title="c2">x</font> be   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K29" title="SIN_COS:func.29">exp_R</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K30" title="SIN_COS:func.30">exp_R</a> <span class="p4">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2</span>)</span> or <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="power.html#K6" title="POWER:func.6">log</a> <a href="power.html#K8" title="POWER:func.8">number_e</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p4">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">y</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <font color="Maroon" title="c1">y</font></span>)</span> or <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <span class="p1">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> <a href="power.html#K8" title="POWER:func.8">number_e</a> ,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p4">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p5">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">y</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <font color="Maroon" title="c1">y</font></span>)</span></span>)</span> )</span><br/>



<span class="kw">set </span><font color="Maroon" title="c3">t</font> =  <a href="sin_cos.html#K29" title="SIN_COS:func.29">exp_R</a> <font color="Maroon" title="c2">x</font>;<br/>
<span class="lab"><font color="Green" title="E47">A1</font></span>:  <a href="quin_1.html#K1" title="QUIN_1:func.1">delta</a> <font color="Maroon" title="c1">y</font>,<span class="p1">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 2</span>)</span>,<font color="Maroon" title="c1">y</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 2</span>)</span> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">4 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c1">y</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c1">y</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="quin_1.html#D1" title="QUIN_1:def.1">QUIN_1:def 1</a></span>
<br/>.= 
4 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p1">(<span class="default">4 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">y</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span>

;<br/>
<span class="kw">assume </span><a NAME="E2:94"/>
<font color="Maroon" title="c1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K29" title="SIN_COS:func.29">exp_R</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K30" title="SIN_COS:func.30">exp_R</a> <span class="p4">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="power.html#K6" title="POWER:func.6">log</a> <a href="power.html#K8" title="POWER:func.8">number_e</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p4">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">y</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <font color="Maroon" title="c1">y</font></span>)</span> or <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <span class="p1">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> <a href="power.html#K8" title="POWER:func.8">number_e</a> ,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p4">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p5">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">y</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <font color="Maroon" title="c1">y</font></span>)</span></span>)</span> )</span><br/>

<a NAME="E3:94"/><span class="kw">then </span>
<font color="Maroon" title="c1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> 2</span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K29" title="SIN_COS:func.29">exp_R</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K30" title="SIN_COS:func.30">exp_R</a> <span class="p5">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T92" title="XCMPLX_1:th.92">XCMPLX_1:92</a></span>;<br/>
<a NAME="E4:94"/><span class="kw">then </span>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="sin_cos.html#K29" title="SIN_COS:func.29">exp_R</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Maroon" title="c1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K29" title="SIN_COS:func.29">exp_R</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p2">(<span class="default">1 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K29" title="SIN_COS:func.29">exp_R</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="sin_cos.html#T60" title="SIN_COS:th.60">SIN_COS:60</a>, <a class="ref" href="taylor_1.html#T4" title="TAYLOR_1:th.4">TAYLOR_1:4</a></span>;<br/>
<a NAME="E5:94"/><span class="kw">then </span>
<font color="Maroon" title="c1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K29" title="SIN_COS:func.29">exp_R</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K29" title="SIN_COS:func.29">exp_R</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K29" title="SIN_COS:func.29">exp_R</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T114" title="XCMPLX_1:th.114">XCMPLX_1:114</a></span>;<br/>
<a NAME="E6:94"/><span class="kw">then </span>
<font color="Maroon" title="c1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K29" title="SIN_COS:func.29">exp_R</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p2">(<span class="default">1 <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K29" title="SIN_COS:func.29">exp_R</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T80" title="XCMPLX_1:th.80">XCMPLX_1:80</a></span>;<br/>
<a NAME="E7:94"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A2</font></span>: 
<font color="Maroon" title="c1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K29" title="SIN_COS:func.29">exp_R</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K29" title="SIN_COS:func.29">exp_R</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span>
 
;<br/>
<a NAME="E8:94"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A3</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="sin_cos7.html#E45"><span class="lab"><font color="Green" title="E37">Lm13</font></span></a>, <a class="ref" href="sin_cos.html#T60" title="SIN_COS:th.60">SIN_COS:60</a></span>;<br/>
<a NAME="E9:94"/>
1 <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K29" title="SIN_COS:func.29">exp_R</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="sin_cos7.html#E36"><span class="lab"><font color="Green" title="E28">Lm6</font></span></a></span>;<br/>
<a NAME="E10:94"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A4</font></span>: 
<font color="Maroon" title="c1">y</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K29" title="SIN_COS:func.29">exp_R</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K29" title="SIN_COS:func.29">exp_R</a> <font color="Maroon" title="c2">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:94"><span class="lab"><font color="Green" title="E48">A2</font></span></a>, <a class="ref" href="xcmplx_1.html#T88" title="XCMPLX_1:th.88">XCMPLX_1:88</a></span>;<br/>
<a NAME="E11:94"/><span class="lab"><font color="Green" title="E51">A5</font></span>: 
<font color="Maroon" title="c1">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:94"><span class="lab"><font color="Green" title="E48">A2</font></span></a>, <a class="txt" href="sin_cos7.html#E46"><span class="lab"><font color="Green" title="E38">Lm14</font></span></a>, <a class="ref" href="sin_cos.html#T60" title="SIN_COS:th.60">SIN_COS:60</a></span>;<br/>
<a NAME="E12:94"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A6</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">y</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:94"><span class="lab"><font color="Green" title="E49">A3</font></span></a>, <a class="txt" href="sin_cos7.html#E48"><span class="lab"><font color="Green" title="E40">Lm16</font></span></a></span>;<br/>
<a NAME="E13:94"/>
 <a href="polyeq_1.html#K3" title="POLYEQ_1:func.3">Polynom</a> <font color="Maroon" title="c1">y</font>,<span class="p1">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 2</span>)</span>,<font color="Maroon" title="c1">y</font>,<span class="p1">(<span class="default"><a href="sin_cos.html#K29" title="SIN_COS:func.29">exp_R</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">y</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K29" title="SIN_COS:func.29">exp_R</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 2</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K29" title="SIN_COS:func.29">exp_R</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c1">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="polyeq_1.html#D2" title="POLYEQ_1:def.2">POLYEQ_1:def 2</a></span>;<br/>
<a NAME="E14:94"/><span class="kw">then </span>
(  <a href="sin_cos.html#K29" title="SIN_COS:func.29">exp_R</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> <span class="p3">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 2</span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p2">(<span class="default"><a href="square_1.html#K6" title="SQUARE_1:func.6">sqrt</a> <span class="p3">(<span class="default"><a href="quin_1.html#K1" title="QUIN_1:func.1">delta</a> <font color="Maroon" title="c1">y</font>,<span class="p4">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 2</span>)</span>,<font color="Maroon" title="c1">y</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c1">y</font></span>)</span> or  <a href="sin_cos.html#K29" title="SIN_COS:func.29">exp_R</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> <span class="p3">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 2</span>)</span></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K6" title="SQUARE_1:func.6">sqrt</a> <span class="p3">(<span class="default"><a href="quin_1.html#K1" title="QUIN_1:func.1">delta</a> <font color="Maroon" title="c1">y</font>,<span class="p4">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 2</span>)</span>,<font color="Maroon" title="c1">y</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c1">y</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:94"><span class="lab"><font color="Green" title="E49">A3</font></span></a>, <a class="txt" href="#E11:94"><span class="lab"><font color="Green" title="E51">A5</font></span></a>, <a class="txt" href="#E10:94"><span class="lab"><font color="Green" title="E50">A4</font></span></a>, <a class="txt" href="#E1:94"><span class="lab"><font color="Green" title="E47">A1</font></span></a>, <a class="txt" href="sin_cos7.html#E49"><span class="lab"><font color="Green" title="E41">Lm17</font></span></a>, <a class="ref" href="quin_1.html#T15" title="QUIN_1:th.15">QUIN_1:15</a></span>;<br/>
<a NAME="E15:94"/><span class="kw">then </span>
(  <a href="sin_cos.html#K29" title="SIN_COS:func.29">exp_R</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p2">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p3">(<span class="default">4 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p4">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">y</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c1">y</font></span>)</span> or  <a href="sin_cos.html#K29" title="SIN_COS:func.29">exp_R</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p3">(<span class="default">4 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p4">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">y</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c1">y</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:94"><span class="lab"><font color="Green" title="E47">A1</font></span></a></span>;<br/>
<a NAME="E16:94"/><span class="kw">then </span>
(  <a href="sin_cos.html#K29" title="SIN_COS:func.29">exp_R</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p2">(<span class="default">2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p3">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p4">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">y</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c1">y</font></span>)</span> or  <a href="sin_cos.html#K29" title="SIN_COS:func.29">exp_R</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p2">(<span class="default">2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p3">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p4">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">y</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c1">y</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:94"><span class="lab"><font color="Green" title="E52">A6</font></span></a>, <a class="ref" href="square_1.html#T85" title="SQUARE_1:th.85">SQUARE_1:85</a>, <a class="ref" href="square_1.html#T97" title="SQUARE_1:th.97">SQUARE_1:97</a></span>;<br/>
<a NAME="E17:94"/><span class="kw">then </span>
(  <a href="sin_cos.html#K29" title="SIN_COS:func.29">exp_R</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default">1 <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p4">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">y</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c1">y</font></span>)</span> or  <a href="sin_cos.html#K29" title="SIN_COS:func.29">exp_R</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p4">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">y</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c1">y</font></span>)</span> )
 
;<br/>
<a NAME="E18:94"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A7</font></span>: 
(  <a href="sin_cos.html#K29" title="SIN_COS:func.29">exp_R</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p2">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p3">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">y</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <font color="Maroon" title="c1">y</font> or  <a href="sin_cos.html#K29" title="SIN_COS:func.29">exp_R</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p3">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">y</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <font color="Maroon" title="c1">y</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T92" title="XCMPLX_1:th.92">XCMPLX_1:92</a></span>;<br/>
<a NAME="E19:94"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p2">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">y</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:94"><span class="lab"><font color="Green" title="E49">A3</font></span></a>, <a class="txt" href="#E11:94"><span class="lab"><font color="Green" title="E51">A5</font></span></a>, <a class="txt" href="sin_cos7.html#E50"><span class="lab"><font color="Green" title="E42">Lm18</font></span></a></span>;<br/>
<a NAME="E20:94"/><span class="kw">then </span>
(  <a href="sin_cos.html#K29" title="SIN_COS:func.29">exp_R</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p2">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p3">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">y</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <font color="Maroon" title="c1">y</font> or  <a href="sin_cos.html#K29" title="SIN_COS:func.29">exp_R</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p4">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">y</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default">1 <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p4">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">y</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">y</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default">1 <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p4">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">y</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:94"><span class="lab"><font color="Green" title="E53">A7</font></span></a>, <a class="ref" href="xcmplx_1.html#T92" title="XCMPLX_1:th.92">XCMPLX_1:92</a></span>;<br/>
<a NAME="E21:94"/><span class="kw">then </span>
(  <a href="sin_cos.html#K29" title="SIN_COS:func.29">exp_R</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p2">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p3">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">y</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <font color="Maroon" title="c1">y</font> or  <a href="sin_cos.html#K29" title="SIN_COS:func.29">exp_R</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p4">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">y</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">y</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default">1 <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p4">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">y</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> )
 
;<br/>
<a NAME="E22:94"/><span class="kw">then </span>
(  <a href="sin_cos.html#K29" title="SIN_COS:func.29">exp_R</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p2">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p3">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">y</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <font color="Maroon" title="c1">y</font> or  <a href="sin_cos.html#K29" title="SIN_COS:func.29">exp_R</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p2">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">y</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">y</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default">1 <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p4">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">y</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:94"><span class="lab"><font color="Green" title="E52">A6</font></span></a>, <a class="ref" href="square_1.html#D4" title="SQUARE_1:def.4">SQUARE_1:def 4</a></span>;<br/>
<a NAME="E23:94"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A8</font></span>: 
(  <a href="sin_cos.html#K29" title="SIN_COS:func.29">exp_R</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p2">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p3">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">y</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <font color="Maroon" title="c1">y</font> or  <a href="sin_cos.html#K29" title="SIN_COS:func.29">exp_R</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">y</font> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p2">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p3">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">y</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:94"><span class="lab"><font color="Green" title="E49">A3</font></span></a>, <a class="ref" href="xcmplx_1.html#T92" title="XCMPLX_1:th.92">XCMPLX_1:92</a></span>;<br/>
<a NAME="E24:94"/>
(  <a href="power.html#K5" title="POWER:func.5">log</a> <a href="power.html#K8" title="POWER:func.8">number_e</a> ,<span class="p1">(<span class="default"><a href="sin_cos.html#K29" title="SIN_COS:func.29">exp_R</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> &amp; 1 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p4">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">y</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <font color="Maroon" title="c1">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p4">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">y</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <font color="Maroon" title="c1">y</font></span>)</span> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p1">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1</span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:94"><span class="lab"><font color="Green" title="E49">A3</font></span></a>, <a class="txt" href="#E11:94"><span class="lab"><font color="Green" title="E51">A5</font></span></a>, <a class="txt" href="sin_cos7.html#E51"><span class="lab"><font color="Green" title="E43">Lm19</font></span></a>, <a class="ref" href="sin_cos7.html#T1" target="_self" title="SIN_COS7:th.1">Th1</a>, <a class="ref" href="taylor_1.html#T12" title="TAYLOR_1:th.12">TAYLOR_1:12</a></span>;<br/>
<a NAME="E25:94"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A9</font></span>: 
(  <a href="power.html#K6" title="POWER:func.6">log</a> <a href="power.html#K8" title="POWER:func.8">number_e</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p4">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">y</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <font color="Maroon" title="c1">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> or  <a href="power.html#K5" title="POWER:func.5">log</a> <a href="power.html#K8" title="POWER:func.8">number_e</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p4">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p5">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">y</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <font color="Maroon" title="c1">y</font></span>)</span> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p2">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:94"><span class="lab"><font color="Green" title="E54">A8</font></span></a>, <a class="ref" href="xcmplx_1.html#T57" title="XCMPLX_1:th.57">XCMPLX_1:57</a></span>;<br/>
<a NAME="E26:94"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p2">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p3">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">y</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <font color="Maroon" title="c1">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:94"><span class="lab"><font color="Green" title="E49">A3</font></span></a>, <a class="txt" href="#E11:94"><span class="lab"><font color="Green" title="E51">A5</font></span></a>, <a class="txt" href="sin_cos7.html#E51"><span class="lab"><font color="Green" title="E43">Lm19</font></span></a></span>;<br/>
<a NAME="E27:94"/><span class="kw">then </span>
(  <a href="power.html#K6" title="POWER:func.6">log</a> <a href="power.html#K8" title="POWER:func.8">number_e</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p4">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">y</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <font color="Maroon" title="c1">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> or <span class="p1">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> <a href="power.html#K8" title="POWER:func.8">number_e</a> ,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p4">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p5">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">y</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <font color="Maroon" title="c1">y</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:94"><span class="lab"><font color="Green" title="E55">A9</font></span></a>, <a class="txt" href="sin_cos7.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a>, <a class="ref" href="power.html#T63" title="POWER:th.63">POWER:63</a>, <a class="ref" href="taylor_1.html#T11" title="TAYLOR_1:th.11">TAYLOR_1:11</a></span>;<br/>
<span class="kw">hence </span><a NAME="E28:94"/>
( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="power.html#K6" title="POWER:func.6">log</a> <a href="power.html#K8" title="POWER:func.8">number_e</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p4">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">y</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <font color="Maroon" title="c1">y</font></span>)</span> or <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <span class="p1">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> <a href="power.html#K8" title="POWER:func.8">number_e</a> ,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p4">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p5">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">y</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <font color="Maroon" title="c1">y</font></span>)</span></span>)</span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
