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

<b>let </b><font color="Maroon" title="c1">Z</font> be  <a href="rcomp_1.html#V3" title="RCOMP_1:attr.3">open</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/>

<b>assume </b><a NAME="E1:20"/><i><font color="Green" title="E7">A1</font></i>: 
<font color="Maroon" title="c1">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="taylor_1.html#K3" title="TAYLOR_1:func.3">ln</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a> </span>)</span>
 ;<br/>

<a NAME="E2:20"/><i><font color="Green" title="E8">A2</font></i>: 
for <font color="Olive" title="b1">x</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> <font color="Maroon" title="c1">Z</font> holds <br/><a href="sin_cos.html#K19" title="SIN_COS:func.19">sin</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">x</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>

<b>assume </b><a NAME="E1:20_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Z</font>
 ;<br/>

<a NAME="E2:20_1"/><b>then </b>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a> 
 
<b>by </b><i><a class="txt" href="fdiff_9.html#E1:20"><i><font color="Green" title="E7">A1</font></i></a>, <a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></i>;<br/>
<b>hence </b><a NAME="E3:20_1"/>
<a href="sin_cos.html#K19" title="SIN_COS:func.19">sin</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 <b>by </b><i><a class="ref" href="rfunct_1.html#T13" title="RFUNCT_1:th.13">RFUNCT_1:13</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:20"/><i><font color="Green" title="E9">A3</font></i>: 
for <font color="Olive" title="b1">x</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> <font color="Maroon" title="c1">Z</font> holds <br/><a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">x</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>

<b>assume </b><a NAME="E1:20_2"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Z</font>
 ;<br/>

<a NAME="E2:20_2"/><b>then </b>
<a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> 0
 
<b>by </b><i><a class="txt" href="fdiff_9.html#E1:20"><i><font color="Green" title="E7">A1</font></i></a>, <a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a>, <a class="ref" href="taylor_1.html#T18" title="TAYLOR_1:th.18">TAYLOR_1:18</a></i>;<br/>
<a NAME="E3:20_2"/><b>then </b>
 ex <font color="Olive" title="b1">g</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> &amp; 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">g</font> )
 
<b>by </b><i><a class="txt" href="fdiff_9.html#E16"><i><font color="Green" title="E6">Lm1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:20_2"/>
<a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c2">x</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:20"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="taylor_1.html#K3" title="TAYLOR_1:func.3">ln</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a> </span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a> 
 
<b>by </b><i><a class="ref" href="relat_1.html#T44" title="RELAT_1:th.44">RELAT_1:44</a></i>;<br/>
<a NAME="E5:20"/><b>then </b><i><font color="Green" title="E10">A4</font></i>: 
<font color="Maroon" title="c1">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a> 
 
<b>by </b><i><a class="txt" href="fdiff_9.html#E1:20"><i><font color="Green" title="E7">A1</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E6:20"/><i><font color="Green" title="E11">A5</font></i>: 
for <font color="Olive" title="b1">x</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> <font color="Maroon" title="c1">Z</font> holds <br/> <a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a>  <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">x</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>

<b>assume </b><a NAME="E1:20_3"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Z</font>
 ;<br/>

<a NAME="E2:20_3"/><b>then </b>
<a href="sin_cos.html#K19" title="SIN_COS:func.19">sin</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="fdiff_9.html#E2:20"><i><font color="Green" title="E8">A2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:20_3"/>
 <a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a>  <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c2">x</font>
 <b>by </b><i><a class="ref" href="fdiff_9.html#T2" target="_self" title="FDIFF_9:th.2">Th2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:20"/><i><font color="Green" title="E12">A6</font></i>: 
for <font color="Olive" title="b1">x</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> <font color="Maroon" title="c1">Z</font> holds <br/><a href="taylor_1.html#K3" title="TAYLOR_1:func.3">ln</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a>  <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">x</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>

<b>assume </b><a NAME="E1:20_4"/><i><font color="Green" title="E13">A7</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Z</font>
 ;<br/>

<a NAME="E2:20_4"/><b>then </b><i><font color="Green" title="E14">A8</font></i>: 
 <a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a>  <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c2">x</font>
 
<b>by </b><i><a class="txt" href="fdiff_9.html#E6:20"><i><font color="Green" title="E11">A5</font></i></a></i>;<br/>
<a NAME="E3:20_4"/>
( <a href="sin_cos.html#K19" title="SIN_COS:func.19">sin</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 &amp; <a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c2">x</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 )
 
<b>by </b><i><a class="txt" href="fdiff_9.html#E2:20"><i><font color="Green" title="E8">A2</font></i></a>, <a class="txt" href="fdiff_9.html#E3:20"><i><font color="Green" title="E9">A3</font></i></a>, <a class="txt" href="fdiff_9.html#E1:20_4"><i><font color="Green" title="E13">A7</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:20_4"/>
<a href="taylor_1.html#K3" title="TAYLOR_1:func.3">ln</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a>  <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c2">x</font>
 <b>by </b><i><a class="ref" href="taylor_1.html#T20" title="TAYLOR_1:th.20">TAYLOR_1:20</a>, <a class="txt" href="fdiff_9.html#E2:20_4"><i><font color="Green" title="E14">A8</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:20"/><b>then </b><i><font color="Green" title="E13">A9</font></i>: 
<a href="taylor_1.html#K3" title="TAYLOR_1:func.3">ln</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a>  <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <font color="Maroon" title="c1">Z</font>
 
<b>by </b><i><a class="txt" href="fdiff_9.html#E1:20"><i><font color="Green" title="E7">A1</font></i></a>, <a class="ref" href="fdiff_1.html#T16" title="FDIFF_1:th.16">FDIFF_1:16</a></i>;<br/>
<a NAME="E9:20"/>
for <font color="Olive" title="b1">x</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> <font color="Maroon" title="c1">Z</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="taylor_1.html#K3" title="TAYLOR_1:func.3">ln</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a> </span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c1">Z</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K19" title="SIN_COS:func.19">sin</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">x</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">x</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>

<b>assume </b><a NAME="E1:20_5"/><i><font color="Green" title="E14">A10</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Z</font>
 ;<br/>

<a NAME="E2:20_5"/><b>then </b><i><font color="Green" title="E15">A11</font></i>: 
 <a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a>  <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c2">x</font>
 
<b>by </b><i><a class="txt" href="fdiff_9.html#E6:20"><i><font color="Green" title="E11">A5</font></i></a></i>;<br/>
<a NAME="E3:20_5"/><i><font color="Green" title="E16">A12</font></i>: 
( <a href="sin_cos.html#K19" title="SIN_COS:func.19">sin</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 &amp; <a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c2">x</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 )
 
<b>by </b><i><a class="txt" href="fdiff_9.html#E2:20"><i><font color="Green" title="E8">A2</font></i></a>, <a class="txt" href="fdiff_9.html#E3:20"><i><font color="Green" title="E9">A3</font></i></a>, <a class="txt" href="fdiff_9.html#E1:20_5"><i><font color="Green" title="E14">A10</font></i></a></i>;<br/>
<b>then </b> <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <span class="p1">(<span class="default"><a href="taylor_1.html#K3" title="TAYLOR_1:func.3">ln</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a> </span>)</span>,<font color="Maroon" title="c2">x</font> = 
<span class="p1">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a> ,<font color="Maroon" title="c2">x</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c2">x</font></span>)</span>
<b>by </b><i><a class="ref" href="taylor_1.html#T20" title="TAYLOR_1:th.20">TAYLOR_1:20</a>, <a class="txt" href="fdiff_9.html#E2:20_5"><i><font color="Green" title="E15">A11</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K19" title="SIN_COS:func.19">sin</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c2">x</font></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_9.html#E3:20_5"><i><font color="Green" title="E16">A12</font></i></a>, <a class="ref" href="fdiff_9.html#T2" target="_self" title="FDIFF_9:th.2">Th2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K19" title="SIN_COS:func.19">sin</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K19" title="SIN_COS:func.19">sin</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span>
<b>by </b><i><a class="txt" href="fdiff_9.html#E1:20_5"><i><font color="Green" title="E14">A10</font></i></a>, <a class="txt" href="fdiff_9.html#E5:20"><i><font color="Green" title="E10">A4</font></i></a>, <a class="ref" href="rfunct_1.html#D8" title="RFUNCT_1:def.8">RFUNCT_1:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K19" title="SIN_COS:func.19">sin</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K19" title="SIN_COS:func.19">sin</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K19" title="SIN_COS:func.19">sin</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c2">x</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="sin_cos.html#K19" title="SIN_COS:func.19">sin</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c2">x</font></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_9.html#E3:20_5"><i><font color="Green" title="E16">A12</font></i></a>, <a class="ref" href="xcmplx_1.html#T92" title="XCMPLX_1:th.92">XCMPLX_1:92</a></i>
;<br/>
<b>hence </b><a NAME="E5:20_5"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="taylor_1.html#K3" title="TAYLOR_1:func.3">ln</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a> </span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c1">Z</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K19" title="SIN_COS:func.19">sin</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="fdiff_9.html#E1:20_5"><i><font color="Green" title="E14">A10</font></i></a>, <a class="txt" href="fdiff_9.html#E8:20"><i><font color="Green" title="E13">A9</font></i></a>, <a class="ref" href="fdiff_1.html#D8" title="FDIFF_1:def.8">FDIFF_1:def 8</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E10:20"/>
( <a href="taylor_1.html#K3" title="TAYLOR_1:func.3">ln</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a>  <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <font color="Maroon" title="c1">Z</font> &amp; ( for <font color="Olive" title="b1">x</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> <font color="Maroon" title="c1">Z</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="taylor_1.html#K3" title="TAYLOR_1:func.3">ln</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a> </span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c1">Z</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K19" title="SIN_COS:func.19">sin</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">x</font></span>)</span></span>)</span> ) )
 <b>by </b><i><a class="txt" href="fdiff_9.html#E7:20"><i><font color="Green" title="E12">A6</font></i></a>, <a class="txt" href="fdiff_9.html#E1:20"><i><font color="Green" title="E7">A1</font></i></a>, <a class="ref" href="fdiff_1.html#T16" title="FDIFF_1:th.16">FDIFF_1:16</a></i>;<br/>


</div>
