<?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="E1:40"/><i><font color="Green" title="E3">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="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K32" title="SIN_COS:func.32">tan</a>  <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">cot</a> </span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:40"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K32" title="SIN_COS:func.32">tan</a>  <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">cot</a> </span>)</span></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> <span class="p1">(<span class="default"><a href="sin_cos.html#K32" title="SIN_COS:func.32">tan</a>  <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">cot</a> </span>)</span>
 
<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="E3:40"/><b>then </b><i><font color="Green" title="E4">A2</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="sin_cos.html#K32" title="SIN_COS:func.32">tan</a>  <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">cot</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="fdiff_10.html#E1:40"><i><font color="Green" title="E3">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="E4:40"/><b>then </b><i><font color="Green" title="E5">A3</font></i>: 
<a href="sin_cos.html#K32" title="SIN_COS:func.32">tan</a>  <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">cot</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="ref" href="fdiff_10.html#T5" target="_self" title="FDIFF_10:th.5">Th5</a></i>;<br/>
<a NAME="E5:40"/><i><font color="Green" title="E6">A4</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#K27" title="SIN_COS:func.27">exp_R</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K32" title="SIN_COS:func.32">tan</a>  <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">cot</a> </span>)</span> <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="40_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="E1:40_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="E2:40_1"/><b>then </b><i><font color="Green" title="E7">A5</font></i>: 
<a href="sin_cos.html#K32" title="SIN_COS:func.32">tan</a>  <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">cot</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_10.html#E4:40"><i><font color="Green" title="E5">A3</font></i></a>, <a class="ref" href="fdiff_1.html#T16" title="FDIFF_1:th.16">FDIFF_1:16</a></i>;<br/>
<a NAME="E3:40_1"/>
 <a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K32" title="SIN_COS:func.32">tan</a>  <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">cot</a> </span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">x</font>
 
<b>by </b><i><a class="ref" href="sin_cos.html#T70" title="SIN_COS:th.70">SIN_COS:70</a></i>;<br/>
<b>hence </b><a NAME="E4:40_1"/>
<a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K32" title="SIN_COS:func.32">tan</a>  <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">cot</a> </span>)</span> <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_10.html#E2:40_1"><i><font color="Green" title="E7">A5</font></i></a>, <a class="ref" href="fdiff_2.html#T13" title="FDIFF_2:th.13">FDIFF_2:13</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:40"/><b>then </b><i><font color="Green" title="E7">A6</font></i>: 
<a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K32" title="SIN_COS:func.32">tan</a>  <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">cot</a> </span>)</span> <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_10.html#E1:40"><i><font color="Green" title="E3">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="E7:40"/>
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="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K32" title="SIN_COS:func.32">tan</a>  <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">cot</a> </span>)</span></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#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K32" title="SIN_COS:func.32">tan</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K33" title="SIN_COS:func.33">cot</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</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#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_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="E1:40_2"/><i><font color="Green" title="E8">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="E2:40_2"/><b>then </b><i><font color="Green" title="E9">A8</font></i>: 
<a href="sin_cos.html#K32" title="SIN_COS:func.32">tan</a>  <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">cot</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_10.html#E4:40"><i><font color="Green" title="E5">A3</font></i></a>, <a class="ref" href="fdiff_1.html#T16" title="FDIFF_1:th.16">FDIFF_1:16</a></i>;<br/>
<a NAME="E3:40_2"/>
 <a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K32" title="SIN_COS:func.32">tan</a>  <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">cot</a> </span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">x</font>
 
<b>by </b><i><a class="ref" href="sin_cos.html#T70" title="SIN_COS:th.70">SIN_COS:70</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="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K32" title="SIN_COS:func.32">tan</a>  <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">cot</a> </span>)</span></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="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a> ,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K32" title="SIN_COS:func.32">tan</a>  <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">cot</a> </span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K32" title="SIN_COS:func.32">tan</a>  <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">cot</a> </span>)</span>,<font color="Maroon" title="c2">x</font></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_10.html#E2:40_2"><i><font color="Green" title="E9">A8</font></i></a>, <a class="ref" href="fdiff_2.html#T13" title="FDIFF_2:th.13">FDIFF_2:13</a></i>
<br/>.= 
<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#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K32" title="SIN_COS:func.32">tan</a>  <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">cot</a> </span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K32" title="SIN_COS:func.32">tan</a>  <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">cot</a> </span>)</span>,<font color="Maroon" title="c2">x</font></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos.html#T70" title="SIN_COS:th.70">SIN_COS:70</a></i>
<br/>.= 
<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#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K32" title="SIN_COS:func.32">tan</a>  <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">cot</a> </span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K32" title="SIN_COS:func.32">tan</a>  <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">cot</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#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">x</font></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_10.html#E4:40"><i><font color="Green" title="E5">A3</font></i></a>, <a class="txt" href="fdiff_10.html#E1:40_2"><i><font color="Green" title="E8">A7</font></i></a>, <a class="ref" href="fdiff_1.html#D8" title="FDIFF_1:def.8">FDIFF_1:def 8</a></i>
<br/>.= 
<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#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K32" title="SIN_COS:func.32">tan</a>  <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">cot</a> </span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</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> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</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#K1" title="SEQ_1:func.1">.</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>
<b>by </b><i><a class="txt" href="fdiff_10.html#E3:40"><i><font color="Green" title="E4">A2</font></i></a>, <a class="txt" href="fdiff_10.html#E1:40_2"><i><font color="Green" title="E8">A7</font></i></a>, <a class="ref" href="fdiff_10.html#T5" target="_self" title="FDIFF_10:th.5">Th5</a></i>
<br/>.= 
<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#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K32" title="SIN_COS:func.32">tan</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K33" title="SIN_COS:func.33">cot</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</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> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</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#K1" title="SEQ_1:func.1">.</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>
<b>by </b><i><a class="txt" href="fdiff_10.html#E3:40"><i><font color="Green" title="E4">A2</font></i></a>, <a class="txt" href="fdiff_10.html#E1:40_2"><i><font color="Green" title="E8">A7</font></i></a>, <a class="ref" href="valued_1.html#T13" title="VALUED_1:th.13">VALUED_1:13</a></i>
;<br/>
<b>hence </b><a NAME="E5:40_2"/>
<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="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K32" title="SIN_COS:func.32">tan</a>  <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">cot</a> </span>)</span></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#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K32" title="SIN_COS:func.32">tan</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K33" title="SIN_COS:func.33">cot</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</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> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</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#K1" title="SEQ_1:func.1">.</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>
 <b>by </b><i><a class="txt" href="fdiff_10.html#E6:40"><i><font color="Green" title="E7">A6</font></i></a>, <a class="txt" href="fdiff_10.html#E1:40_2"><i><font color="Green" title="E8">A7</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="E8:40"/>
( <a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K32" title="SIN_COS:func.32">tan</a>  <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">cot</a> </span>)</span> <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="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K32" title="SIN_COS:func.32">tan</a>  <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">cot</a> </span>)</span></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#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K32" title="SIN_COS:func.32">tan</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K33" title="SIN_COS:func.33">cot</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</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#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span></span>)</span> ) )
 <b>by </b><i><a class="txt" href="fdiff_10.html#E1:40"><i><font color="Green" title="E3">A1</font></i></a>, <a class="txt" href="fdiff_10.html#E5:40"><i><font color="Green" title="E6">A4</font></i></a>, <a class="ref" href="fdiff_1.html#T16" title="FDIFF_1:th.16">FDIFF_1:16</a></i>;<br/>


</div>
