<?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="firebrick">::  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> <a href="sin_cos.html#K30" title="SIN_COS:func.30">tan</a> implies ( <a href="sin_cos.html#K20" title="SIN_COS:func.20">cos</a> <a href="rfunct_1.html#K6" title="RFUNCT_1:func.6">^</a>  <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <font color="Maroon" title="c1">Z</font> &amp; <span class="p1">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">cos</a> <a href="rfunct_1.html#K6" title="RFUNCT_1:func.6">^</a></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c1">Z</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">cos</a> <a href="rfunct_1.html#K6" title="RFUNCT_1:func.6">^</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <a href="sin_cos.html#K30" title="SIN_COS:func.30">tan</a></span>)</span> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <font color="Maroon" title="c1">Z</font> ) )</span><br/>

<a NAME="E1:59"/><span class="lab"><font color="Green" title="E33">A1</font></span>: 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <a href="sin_cos.html#K17" title="SIN_COS:func.17">sin</a></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</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">cos</a> <a href="rfunct_1.html#K6" title="RFUNCT_1:func.6">^</a></span>)</span></span>)</span> <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">cos</a> <a href="rfunct_1.html#K6" title="RFUNCT_1:func.6">^</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:59"/><span class="lab"><font color="Green" title="E34">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> <a href="sin_cos.html#K30" title="SIN_COS:func.30">tan</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <a href="sin_cos.html#K20" title="SIN_COS:func.20">cos</a> <a href="rfunct_1.html#K6" title="RFUNCT_1:func.6">^</a>  <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <font color="Maroon" title="c1">Z</font> &amp; <span class="p1">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">cos</a> <a href="rfunct_1.html#K6" title="RFUNCT_1:func.6">^</a></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c1">Z</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">cos</a> <a href="rfunct_1.html#K6" title="RFUNCT_1:func.6">^</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <a href="sin_cos.html#K30" title="SIN_COS:func.30">tan</a></span>)</span> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <font color="Maroon" title="c1">Z</font> )</span><br/>

<a NAME="E3:59"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A3</font></span>: 
 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#K20" title="SIN_COS:func.20">cos</a> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">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="fdiff_8.html#T1" title="FDIFF_8:th.1">FDIFF_8:1</a></span>;<br/>
<a NAME="E4:59"/><span class="kw">then </span>
<a href="sin_cos.html#K20" title="SIN_COS:func.20">cos</a> <a href="rfunct_1.html#K6" title="RFUNCT_1:func.6">^</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="ref" href="fdiff_4.html#T39" title="FDIFF_4:th.39">FDIFF_4:39</a></span>;<br/>
<a NAME="E5:59"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A4</font></span>: 
 <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">cos</a> <a href="rfunct_1.html#K6" title="RFUNCT_1:func.6">^</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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">Z</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fdiff_1.html#D7" title="FDIFF_1:def.7">FDIFF_1:def 7</a></span>;<br/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <a href="sin_cos.html#K30" title="SIN_COS:func.30">tan</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#K17" title="SIN_COS:func.17">sin</a> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">cos</a> <a href="rfunct_1.html#K6" title="RFUNCT_1:func.6">^</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rfunct_1.html#T31" title="RFUNCT_1:th.31">RFUNCT_1:31</a>, <a class="ref" href="sin_cos.html#D26" title="SIN_COS:def.26">SIN_COS:def 26</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <a href="sin_cos.html#K17" title="SIN_COS:func.17">sin</a></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</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">cos</a> <a href="rfunct_1.html#K6" title="RFUNCT_1:func.6">^</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#D4" title="VALUED_1:def.4">VALUED_1:def 4</a></span>
;<br/>
<a NAME="E7:59"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A5</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">cos</a> <a href="rfunct_1.html#K6" title="RFUNCT_1:func.6">^</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:59"><span class="lab"><font color="Green" title="E34">A2</font></span></a>, <a class="txt" href="#E1:59"><span class="lab"><font color="Green" title="E33">A1</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E8:59"/><span class="lab"><font color="Green" title="E38">A6</font></span>: 
 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">cos</a> <a href="rfunct_1.html#K6" title="RFUNCT_1:func.6">^</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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">cos</a> <a href="rfunct_1.html#K6" title="RFUNCT_1:func.6">^</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <a href="sin_cos.html#K30" title="SIN_COS:func.30">tan</a></span>)</span> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_2"><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="firebrick">::  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"><a href="sin_cos.html#K20" title="SIN_COS:func.20">cos</a> <a href="rfunct_1.html#K6" title="RFUNCT_1:func.6">^</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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">cos</a> <a href="rfunct_1.html#K6" title="RFUNCT_1:func.6">^</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <a href="sin_cos.html#K30" title="SIN_COS:func.30">tan</a></span>)</span> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</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><br/>

<a NAME="E1:59_2"/><span class="lab"><font color="Green" title="E39">A7</font></span>: 
 <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">cos</a> <a href="rfunct_1.html#K6" title="RFUNCT_1:func.6">^</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <a href="sin_cos.html#K17" title="SIN_COS:func.17">sin</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <a href="sin_cos.html#K30" title="SIN_COS:func.30">tan</a>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rfunct_1.html#T31" title="RFUNCT_1:th.31">RFUNCT_1:31</a>, <a class="ref" href="sin_cos.html#D26" title="SIN_COS:def.26">SIN_COS:def 26</a></span>;<br/>
<a NAME="E2:59_2"/><span class="kw">then </span>
 <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"><span class="p3">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">cos</a> <a href="rfunct_1.html#K6" title="RFUNCT_1:func.6">^</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <a href="sin_cos.html#K17" title="SIN_COS:func.17">sin</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">cos</a> <a href="rfunct_1.html#K6" title="RFUNCT_1:func.6">^</a></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <a href="sin_cos.html#K30" title="SIN_COS:func.30">tan</a></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</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">cos</a> <a href="rfunct_1.html#K6" title="RFUNCT_1:func.6">^</a></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#D4" title="VALUED_1:def.4">VALUED_1:def 4</a></span>;<br/>
<a NAME="E3:59_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A8</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"><span class="p3">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">cos</a> <a href="rfunct_1.html#K6" title="RFUNCT_1:func.6">^</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <a href="sin_cos.html#K17" title="SIN_COS:func.17">sin</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">cos</a> <a href="rfunct_1.html#K6" title="RFUNCT_1:func.6">^</a></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:59"><span class="lab"><font color="Green" title="E34">A2</font></span></a>, <a class="txt" href="#E7:59"><span class="lab"><font color="Green" title="E37">A5</font></span></a>, <a class="ref" href="xboole_1.html#T19" title="XBOOLE_1:th.19">XBOOLE_1:19</a></span>;<br/>
<span class="kw">assume </span><a NAME="E4:59_2"/><span class="lab"><font color="Green" title="E41">A9</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="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">cos</a> <a href="rfunct_1.html#K6" title="RFUNCT_1:func.6">^</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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">cos</a> <a href="rfunct_1.html#K6" title="RFUNCT_1:func.6">^</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <a href="sin_cos.html#K30" title="SIN_COS:func.30">tan</a></span>)</span> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</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><br/>

<span class="kw">then </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">cos</a> <a href="rfunct_1.html#K6" title="RFUNCT_1:func.6">^</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 class="p1">(<span class="default"><a href="sin_cos.html#K17" title="SIN_COS:func.17">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="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E3:59"><span class="lab"><font color="Green" title="E35">A3</font></span></a>, <a class="ref" href="fdiff_4.html#T39" title="FDIFF_4:th.39">FDIFF_4:39</a></span>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">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="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K17" title="SIN_COS:func.17">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="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">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 class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T103" title="XCMPLX_1:th.103">XCMPLX_1:103</a></span>
<br/>.= 
<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"><a href="sin_cos.html#K20" title="SIN_COS:func.20">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="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K17" title="SIN_COS:func.17">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="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">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 class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T99" title="XCMPLX_1:th.99">XCMPLX_1:99</a></span>
<br/>.= 
<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"><a href="sin_cos.html#K20" title="SIN_COS:func.20">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="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K17" title="SIN_COS:func.17">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="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">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="real_1.html#K2" title="REAL_1:func.2">"</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_0.html#D9" title="XCMPLX_0:def.9">XCMPLX_0:def 9</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">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="real_1.html#K2" title="REAL_1:func.2">"</a></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K17" title="SIN_COS:func.17">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="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">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="real_1.html#K2" title="REAL_1:func.2">"</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_0.html#D9" title="XCMPLX_0:def.9">XCMPLX_0:def 9</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">cos</a> <a href="rfunct_1.html#K6" title="RFUNCT_1:func.6">^</a></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="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K17" title="SIN_COS:func.17">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="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">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="real_1.html#K2" title="REAL_1:func.2">"</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:59"><span class="lab"><font color="Green" title="E37">A5</font></span></a>, <a class="txt" href="#E4:59_2"><span class="lab"><font color="Green" title="E41">A9</font></span></a>, <a class="ref" href="rfunct_1.html#D2" title="RFUNCT_1:def.2">RFUNCT_1:def 2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">cos</a> <a href="rfunct_1.html#K6" title="RFUNCT_1:func.6">^</a></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="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K17" title="SIN_COS:func.17">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="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">cos</a> <a href="rfunct_1.html#K6" title="RFUNCT_1:func.6">^</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E7:59"><span class="lab"><font color="Green" title="E37">A5</font></span></a>, <a class="txt" href="#E4:59_2"><span class="lab"><font color="Green" title="E41">A9</font></span></a>, <a class="ref" href="rfunct_1.html#D2" title="RFUNCT_1:def.2">RFUNCT_1:def 2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">cos</a> <a href="rfunct_1.html#K6" title="RFUNCT_1:func.6">^</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <a href="sin_cos.html#K17" title="SIN_COS:func.17">sin</a></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="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">cos</a> <a href="rfunct_1.html#K6" title="RFUNCT_1:func.6">^</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E2:59"><span class="lab"><font color="Green" title="E34">A2</font></span></a>, <a class="txt" href="#E4:59_2"><span class="lab"><font color="Green" title="E41">A9</font></span></a>, <a class="txt" href="#E1:59_2"><span class="lab"><font color="Green" title="E39">A7</font></span></a>, <a class="ref" href="valued_1.html#D4" title="VALUED_1:def.4">VALUED_1:def 4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">cos</a> <a href="rfunct_1.html#K6" title="RFUNCT_1:func.6">^</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <a href="sin_cos.html#K17" title="SIN_COS:func.17">sin</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">cos</a> <a href="rfunct_1.html#K6" title="RFUNCT_1:func.6">^</a></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:59_2"><span class="lab"><font color="Green" title="E41">A9</font></span></a>, <a class="txt" href="#E3:59_2"><span class="lab"><font color="Green" title="E40">A8</font></span></a>, <a class="ref" href="valued_1.html#D4" title="VALUED_1:def.4">VALUED_1:def 4</a></span>
<br/>.= 
<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="sin_cos.html#K20" title="SIN_COS:func.20">cos</a> <a href="rfunct_1.html#K6" title="RFUNCT_1:func.6">^</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <a href="sin_cos.html#K17" title="SIN_COS:func.17">sin</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">cos</a> <a href="rfunct_1.html#K6" title="RFUNCT_1:func.6">^</a></span>)</span></span>)</span> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</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="kw">by</span> <span class="lab"><a class="txt" href="#E4:59_2"><span class="lab"><font color="Green" title="E41">A9</font></span></a>, <a class="ref" href="funct_1.html#T49" title="FUNCT_1:th.49">FUNCT_1:49</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">cos</a> <a href="rfunct_1.html#K6" title="RFUNCT_1:func.6">^</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <a href="sin_cos.html#K30" title="SIN_COS:func.30">tan</a></span>)</span> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</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="kw">by</span> <span class="lab"><a class="ref" href="rfunct_1.html#T31" title="RFUNCT_1:th.31">RFUNCT_1:31</a>, <a class="ref" href="sin_cos.html#D26" title="SIN_COS:def.26">SIN_COS:def 26</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E6:59_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">cos</a> <a href="rfunct_1.html#K6" title="RFUNCT_1:func.6">^</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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">cos</a> <a href="rfunct_1.html#K6" title="RFUNCT_1:func.6">^</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <a href="sin_cos.html#K30" title="SIN_COS:func.30">tan</a></span>)</span> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</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 class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
 <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"><span class="p3">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">cos</a> <a href="rfunct_1.html#K6" title="RFUNCT_1:func.6">^</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <a href="sin_cos.html#K30" title="SIN_COS:func.30">tan</a></span>)</span> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <font color="Maroon" title="c1">Z</font></span>)</span> = 
<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"><span class="p3">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">cos</a> <a href="rfunct_1.html#K6" title="RFUNCT_1:func.6">^</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <a href="sin_cos.html#K30" title="SIN_COS:func.30">tan</a></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c1">Z</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T61" title="RELAT_1:th.61">RELAT_1:61</a></span>
<br/>.= 
<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> <span class="p3">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">cos</a> <a href="rfunct_1.html#K6" title="RFUNCT_1:func.6">^</a></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <a href="sin_cos.html#K30" title="SIN_COS:func.30">tan</a></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c1">Z</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#D4" title="VALUED_1:def.4">VALUED_1:def 4</a></span>
<br/>.= 
<font color="Maroon" title="c1">Z</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:59"><span class="lab"><font color="Green" title="E34">A2</font></span></a>, <a class="txt" href="#E7:59"><span class="lab"><font color="Green" title="E37">A5</font></span></a>, <a class="ref" href="xboole_1.html#T19" title="XBOOLE_1:th.19">XBOOLE_1:19</a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E10:59"/>
( <a href="sin_cos.html#K20" title="SIN_COS:func.20">cos</a> <a href="rfunct_1.html#K6" title="RFUNCT_1:func.6">^</a>  <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <font color="Maroon" title="c1">Z</font> &amp; <span class="p1">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">cos</a> <a href="rfunct_1.html#K6" title="RFUNCT_1:func.6">^</a></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c1">Z</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">cos</a> <a href="rfunct_1.html#K6" title="RFUNCT_1:func.6">^</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <a href="sin_cos.html#K30" title="SIN_COS:func.30">tan</a></span>)</span> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <font color="Maroon" title="c1">Z</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:59"><span class="lab"><font color="Green" title="E35">A3</font></span></a>, <a class="txt" href="#E5:59"><span class="lab"><font color="Green" title="E36">A4</font></span></a>, <a class="txt" href="#E8:59"><span class="lab"><font color="Green" title="E38">A6</font></span></a>, <a class="ref" href="fdiff_4.html#T39" title="FDIFF_4:th.39">FDIFF_4:39</a>, <a class="ref" href="partfun1.html#T5" title="PARTFUN1:th.5">PARTFUN1:5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
