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

<a NAME="E1:28"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<b>by </b><i><a class="ref" href="sin_cos.html#D22" title="SIN_COS:def.22">SIN_COS:def 22</a></i>;<br/>
<a NAME="E2:28"/><b>then </b><i><font color="Green" title="E19">A1</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<b>by </b><i><a class="ref" href="seq_1.html#D6" title="SEQ_1:def.6">SEQ_1:def 6</a></i>;<br/>
<a NAME="E3:28"/><b>then </b><i><font color="Green" title="E20">A2</font></i>: 
<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a>  <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<b>by </b><i><a class="ref" href="fdiff_1.html#T28" title="FDIFF_1:th.28">FDIFF_1:28</a>, <a class="ref" href="fcont_3.html#T4" title="FCONT_3:th.4">FCONT_3:4</a>, <a class="ref" href="sin_cos.html#T72" title="SIN_COS:th.72">SIN_COS:72</a></i>;<br/>
<a NAME="E4:28"/><b>then </b><i><font color="Green" title="E21">A3</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a> </span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<b>by </b><i><a class="ref" href="fdiff_1.html#D8" title="FDIFF_1:def.8">FDIFF_1:def 8</a></i>;<br/>
<a NAME="E5:28"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K19" title="SIN_COS:func.19">sin</a>  <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <a href="sin_cos.html#K19" title="SIN_COS:func.19">sin</a> </span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<b>by </b><i><a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>;<br/>
<a NAME="E6:28"/><b>then </b><i><font color="Green" title="E22">A4</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K19" title="SIN_COS:func.19">sin</a>  <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<b>by </b><i><a class="ref" href="sin_cos.html#D20" title="SIN_COS:def.20">SIN_COS:def 20</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c1">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:28_1"/><i><font color="Green" title="E23">A5</font></i>: 
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a> </span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c2">z</font> = <font color="Maroon" title="c1">x</font> as  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  <b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/><a NAME="E3:28_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a> </span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a> ,<font color="Maroon" title="c2">z</font></span>)</span>
 <b>by </b><i><a class="ref" href="sin_cos.html#T72" title="SIN_COS:th.72">SIN_COS:72</a>, <a class="ref" href="fcont_3.html#T4" title="FCONT_3:th.4">FCONT_3:4</a>, <a class="txt" href="integra7.html#E2:28"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="integra7.html#E1:28_1"><i><font color="Green" title="E23">A5</font></i></a>, <a class="ref" href="fdiff_1.html#T28" title="FDIFF_1:th.28">FDIFF_1:28</a></i>;<br/><a NAME="E4:28_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a> </span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K19" title="SIN_COS:func.19">sin</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c2">z</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="sin_cos.html#T68" title="SIN_COS:th.68">SIN_COS:68</a></i>;<br/><b>hence </b><a NAME="E5:28_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a> </span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K19" title="SIN_COS:func.19">sin</a>  <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c1">x</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a>, <a class="txt" href="integra7.html#E1:28_1"><i><font color="Green" title="E23">A5</font></i></a>, <a class="txt" href="integra7.html#E6:28"><i><font color="Green" title="E22">A4</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E8:28"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a> </span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <a href="sin_cos.html#K19" title="SIN_COS:func.19">sin</a>  <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<b>by </b><i><a class="txt" href="integra7.html#E4:28"><i><font color="Green" title="E21">A3</font></i></a>, <a class="txt" href="integra7.html#E6:28"><i><font color="Green" title="E22">A4</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/>
<b>hence </b><a NAME="E9:28"/>
<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a>  <a href="integra7.html#R1" title="INTEGRA7:pred.1">is_integral_of</a>  <a href="sin_cos.html#K19" title="SIN_COS:func.19">sin</a> , <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 <b>by </b><i><a class="txt" href="integra7.html#E13"><i><font color="Green" title="E11">Lm1</font></i></a>, <a class="txt" href="integra7.html#E3:28"><i><font color="Green" title="E20">A2</font></i></a></i>;<br/>


</div>
