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

<span class="kw">let </span><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> ; <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="c1">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K18" title="BINOP_2:func.18">/</a> 2</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a> </span>)</span> <a href="rfunct_1.html#K3" title="RFUNCT_1:func.3">/</a> <a href="sin_cos.html#K28" title="SIN_COS:func.28">exp_R</a> </span>)</span></span>)</span> implies ( <span class="p1">(<span class="default">1 <a href="binop_2.html#K18" title="BINOP_2:func.18">/</a> 2</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a> </span>)</span> <a href="rfunct_1.html#K3" title="RFUNCT_1:func.3">/</a> <a href="sin_cos.html#K28" title="SIN_COS:func.28">exp_R</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"><span class="p3">(<span class="default">1 <a href="binop_2.html#K18" title="BINOP_2:func.18">/</a> 2</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a> </span>)</span> <a href="rfunct_1.html#K3" title="RFUNCT_1:func.3">/</a> <a href="sin_cos.html#K28" title="SIN_COS:func.28">exp_R</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#K23" title="SIN_COS:func.23">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="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K28" title="SIN_COS:func.28">exp_R</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font></span>)</span> ) ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:10"/><span class="lab"><font color="Green" title="E6">A1</font></span>: 
<font color="Maroon" title="c1">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K18" title="BINOP_2:func.18">/</a> 2</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a> </span>)</span> <a href="rfunct_1.html#K3" title="RFUNCT_1:func.3">/</a> <a href="sin_cos.html#K28" title="SIN_COS:func.28">exp_R</a> </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"> ( <span class="p1">(<span class="default">1 <a href="binop_2.html#K18" title="BINOP_2:func.18">/</a> 2</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a> </span>)</span> <a href="rfunct_1.html#K3" title="RFUNCT_1:func.3">/</a> <a href="sin_cos.html#K28" title="SIN_COS:func.28">exp_R</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"><span class="p3">(<span class="default">1 <a href="binop_2.html#K18" title="BINOP_2:func.18">/</a> 2</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a> </span>)</span> <a href="rfunct_1.html#K3" title="RFUNCT_1:func.3">/</a> <a href="sin_cos.html#K28" title="SIN_COS:func.28">exp_R</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#K23" title="SIN_COS:func.23">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="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K28" title="SIN_COS:func.28">exp_R</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font></span>)</span> ) )</span><br/>

<a NAME="E2:10"/><span class="lab"><font color="Green" title="E7">A2</font></span>: 
<font color="Maroon" title="c1">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a> </span>)</span> <a href="rfunct_1.html#K3" title="RFUNCT_1:func.3">/</a> <a href="sin_cos.html#K28" title="SIN_COS:func.28">exp_R</a> </span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:10"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="ref" href="valued_1.html#D5" title="VALUED_1:def.5">VALUED_1:def 5</a></span>;<br/>
<a NAME="E3:10"/><span class="kw">then </span><span class="lab"><font color="Green" title="E8">A3</font></span>: 
<font color="Maroon" title="c1">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a> </span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <a href="sin_cos.html#K28" title="SIN_COS:func.28">exp_R</a> </span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K28" title="SIN_COS:func.28">exp_R</a>  <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rfunct_1.html#D4" title="RFUNCT_1:def.4">RFUNCT_1:def 4</a></span>;<br/>
<a NAME="E4:10"/><span class="lab"><font color="Green" title="E9">A4</font></span>: 
<font color="Maroon" title="c1">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a> </span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:10"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="ref" href="xboole_1.html#T18" title="XBOOLE_1:th.18">XBOOLE_1:18</a></span>;<br/>
<a NAME="E5:10"/><span class="kw">then </span><span class="lab"><font color="Green" title="E10">A5</font></span>: 
( <a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</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="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</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="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#K23" title="SIN_COS:func.23">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="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font></span>)</span> ) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fdiff_7.html#T39" title="FDIFF_7:th.39">FDIFF_7:39</a></span>;<br/>
<a NAME="E6:10"/><span class="lab"><font color="Green" title="E11">A6</font></span>: 
<span class="p1">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a> </span>)</span> <a href="rfunct_1.html#K3" title="RFUNCT_1:func.3">/</a> <a href="sin_cos.html#K28" title="SIN_COS:func.28">exp_R</a>  <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <font color="Maroon" title="c1">Z</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:10"><span class="lab"><font color="Green" title="E7">A2</font></span></a>, <a class="ref" href="fdiff_7.html#T43" title="FDIFF_7:th.43">FDIFF_7:43</a></span>;<br/>
<a NAME="E7:10"/><span class="kw">then </span><span class="lab"><font color="Green" title="E12">A7</font></span>: 
<span class="p1">(<span class="default">1 <a href="binop_2.html#K18" title="BINOP_2:func.18">/</a> 2</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a> </span>)</span> <a href="rfunct_1.html#K3" title="RFUNCT_1:func.3">/</a> <a href="sin_cos.html#K28" title="SIN_COS:func.28">exp_R</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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fdiff_2.html#T19" title="FDIFF_2:th.19">FDIFF_2:19</a>, <a class="txt" href="integr13.html#E3"><span class="lab"><font color="Green" title="E3">A</font></span></a></span>;<br/>
<a NAME="E8:10"/>
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"><span class="p3">(<span class="default">1 <a href="binop_2.html#K18" title="BINOP_2:func.18">/</a> 2</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a> </span>)</span> <a href="rfunct_1.html#K3" title="RFUNCT_1:func.3">/</a> <a href="sin_cos.html#K28" title="SIN_COS:func.28">exp_R</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#K23" title="SIN_COS:func.23">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="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K28" title="SIN_COS:func.28">exp_R</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">x</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <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#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Z</font> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K18" title="BINOP_2:func.18">/</a> 2</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a> </span>)</span> <a href="rfunct_1.html#K3" title="RFUNCT_1:func.3">/</a> <a href="sin_cos.html#K28" title="SIN_COS:func.28">exp_R</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#K23" title="SIN_COS:func.23">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="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K28" title="SIN_COS:func.28">exp_R</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">x</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:10_1"/><span class="lab"><font color="Green" title="E13">A8</font></span>: 
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K18" title="BINOP_2:func.18">/</a> 2</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a> </span>)</span> <a href="rfunct_1.html#K3" title="RFUNCT_1:func.3">/</a> <a href="sin_cos.html#K28" title="SIN_COS:func.28">exp_R</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#K23" title="SIN_COS:func.23">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="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K28" title="SIN_COS:func.28">exp_R</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">x</font></span>)</span></span><br/>

<a NAME="E2:10_1"/><span class="lab"><font color="Green" title="E14">A9</font></span>: 
 <a href="sin_cos.html#K28" title="SIN_COS:func.28">exp_R</a>  <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c2">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="sin_cos.html#T70" title="SIN_COS:th.70">SIN_COS:70</a></span>;<br/>
<a NAME="E3:10_1"/><span class="lab"><font color="Green" title="E15">A10</font></span>: 
<a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c2">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:10"><span class="lab"><font color="Green" title="E10">A5</font></span></a>, <a class="txt" href="#E1:10_1"><span class="lab"><font color="Green" title="E13">A8</font></span></a>, <a class="ref" href="fdiff_1.html#T16" title="FDIFF_1:th.16">FDIFF_1:16</a></span>;<br/>
<a NAME="E4:10_1"/><span class="lab"><font color="Green" title="E16">A11</font></span>: 
<span class="p1">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a> </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#K20" title="SIN_COS:func.20">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="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:10"><span class="lab"><font color="Green" title="E9">A4</font></span></a>, <a class="txt" href="#E1:10_1"><span class="lab"><font color="Green" title="E13">A8</font></span></a>, <a class="ref" href="valued_1.html#T13" title="VALUED_1:th.13">VALUED_1:13</a></span>;<br/>
<a NAME="E5:10_1"/><span class="lab"><font color="Green" title="E17">A12</font></span>: 
<a href="sin_cos.html#K28" title="SIN_COS:func.28">exp_R</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="sin_cos.html#T59" title="SIN_COS:th.59">SIN_COS:59</a></span>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K18" title="BINOP_2:func.18">/</a> 2</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a> </span>)</span> <a href="rfunct_1.html#K3" title="RFUNCT_1:func.3">/</a> <a href="sin_cos.html#K28" title="SIN_COS:func.28">exp_R</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> = 
<span class="p1">(<span class="default">1 <a href="binop_2.html#K18" title="BINOP_2:func.18">/</a> 2</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</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"><span class="p3">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a> </span>)</span> <a href="rfunct_1.html#K3" title="RFUNCT_1:func.3">/</a> <a href="sin_cos.html#K28" title="SIN_COS:func.28">exp_R</a> </span>)</span>,<font color="Maroon" title="c2">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="integr13.html#E3"><span class="lab"><font color="Green" title="E3">A</font></span></a>, <a class="txt" href="#E1:10"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="txt" href="#E6:10"><span class="lab"><font color="Green" title="E11">A6</font></span></a>, <a class="txt" href="#E1:10_1"><span class="lab"><font color="Green" title="E13">A8</font></span></a>, <a class="ref" href="fdiff_1.html#T28" title="FDIFF_1:th.28">FDIFF_1:28</a></span>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="binop_2.html#K18" title="BINOP_2:func.18">/</a> 2</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <span class="p5">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a> </span>)</span>,<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#K28" title="SIN_COS:func.28">exp_R</a>  <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="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <a href="sin_cos.html#K28" title="SIN_COS:func.28">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"><span class="p5">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</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></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K28" title="SIN_COS:func.28">exp_R</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E2:10_1"><span class="lab"><font color="Green" title="E14">A9</font></span></a>, <a class="txt" href="#E3:10_1"><span class="lab"><font color="Green" title="E15">A10</font></span></a>, <a class="txt" href="#E5:10_1"><span class="lab"><font color="Green" title="E17">A12</font></span></a>, <a class="ref" href="fdiff_2.html#T14" title="FDIFF_2:th.14">FDIFF_2:14</a></span>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="binop_2.html#K18" title="BINOP_2:func.18">/</a> 2</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</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> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K28" title="SIN_COS:func.28">exp_R</a>  <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="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <a href="sin_cos.html#K28" title="SIN_COS:func.28">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"><span class="p5">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</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></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K28" title="SIN_COS:func.28">exp_R</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E5:10"><span class="lab"><font color="Green" title="E10">A5</font></span></a>, <a class="txt" href="#E1:10_1"><span class="lab"><font color="Green" title="E13">A8</font></span></a>, <a class="ref" href="fdiff_1.html#D8" title="FDIFF_1:def.8">FDIFF_1:def 8</a></span>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="binop_2.html#K18" title="BINOP_2:func.18">/</a> 2</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="sin_cos.html#K23" title="SIN_COS:func.23">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="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p5">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <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="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K28" title="SIN_COS:func.28">exp_R</a>  <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="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <a href="sin_cos.html#K28" title="SIN_COS:func.28">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"><span class="p5">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</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></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K28" title="SIN_COS:func.28">exp_R</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E4:10"><span class="lab"><font color="Green" title="E9">A4</font></span></a>, <a class="txt" href="#E1:10_1"><span class="lab"><font color="Green" title="E13">A8</font></span></a>, <a class="ref" href="fdiff_7.html#T39" title="FDIFF_7:th.39">FDIFF_7:39</a></span>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="binop_2.html#K18" title="BINOP_2:func.18">/</a> 2</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="sin_cos.html#K23" title="SIN_COS:func.23">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="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p5">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <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="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K28" title="SIN_COS:func.28">exp_R</a>  <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="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K28" title="SIN_COS:func.28">exp_R</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</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"><span class="p5">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">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="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p5">(<span class="default"><a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</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></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K28" title="SIN_COS:func.28">exp_R</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E4:10_1"><span class="lab"><font color="Green" title="E16">A11</font></span></a>, <a class="ref" href="sin_cos.html#T70" title="SIN_COS:th.70">SIN_COS:70</a></span>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="binop_2.html#K18" title="BINOP_2:func.18">/</a> 2</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><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="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  <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="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K28" title="SIN_COS:func.28">exp_R</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K28" title="SIN_COS:func.28">exp_R</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</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#K28" title="SIN_COS:func.28">exp_R</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></span>)</span>

<br/>.= 
<span class="p1">(<span class="default">1 <a href="binop_2.html#K18" title="BINOP_2:func.18">/</a> 2</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><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="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  <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="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#K28" title="SIN_COS:func.28">exp_R</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K28" title="SIN_COS:func.28">exp_R</a>  <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="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K28" title="SIN_COS:func.28">exp_R</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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T79" title="XCMPLX_1:th.79">XCMPLX_1:79</a></span>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="binop_2.html#K18" title="BINOP_2:func.18">/</a> 2</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><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="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  <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="binop_2.html#K11" title="BINOP_2:func.11">*</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#K28" title="SIN_COS:func.28">exp_R</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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:10_1"><span class="lab"><font color="Green" title="E17">A12</font></span></a>, <a class="ref" href="xcmplx_1.html#T60" title="XCMPLX_1:th.60">XCMPLX_1:60</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="sin_cos.html#K23" title="SIN_COS:func.23">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="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K28" title="SIN_COS:func.28">exp_R</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">x</font></span>)</span>

;<br/>
<span class="kw">hence </span><a NAME="E7:10_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K18" title="BINOP_2:func.18">/</a> 2</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a> </span>)</span> <a href="rfunct_1.html#K3" title="RFUNCT_1:func.3">/</a> <a href="sin_cos.html#K28" title="SIN_COS:func.28">exp_R</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#K23" title="SIN_COS:func.23">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="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K28" title="SIN_COS:func.28">exp_R</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">x</font></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><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E9:10"/>
( <span class="p1">(<span class="default">1 <a href="binop_2.html#K18" title="BINOP_2:func.18">/</a> 2</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a> </span>)</span> <a href="rfunct_1.html#K3" title="RFUNCT_1:func.3">/</a> <a href="sin_cos.html#K28" title="SIN_COS:func.28">exp_R</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"><span class="p3">(<span class="default">1 <a href="binop_2.html#K18" title="BINOP_2:func.18">/</a> 2</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a> </span>)</span> <a href="rfunct_1.html#K3" title="RFUNCT_1:func.3">/</a> <a href="sin_cos.html#K28" title="SIN_COS:func.28">exp_R</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#K23" title="SIN_COS:func.23">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="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K28" title="SIN_COS:func.28">exp_R</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font></span>)</span> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:10"><span class="lab"><font color="Green" title="E12">A7</font></span></a></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>
