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

<b>let </b><font color="Maroon" title="c1">A</font> be  <a href="integra1.html#V1" title="INTEGRA1:attr.1">closed-interval</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="integra5.html#K2" title="INTEGRA5:func.2">integral</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> 1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <a href="sin_cos.html#K28" title="SIN_COS:func.28">exp_R</a> </span>)</span>,<font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K28" title="SIN_COS:func.28">exp_R</a>  <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> 1,<span class="p4">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#NK2" title="PSCOMP_1:NK.2">sup</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K28" title="SIN_COS:func.28">exp_R</a>  <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> 1,<span class="p4">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#NK3" title="PSCOMP_1:NK.3">inf</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span></span><br/>

<a NAME="E1:24"/><i><font color="Green" title="E15">A1</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#K28" title="SIN_COS:func.28">exp_R</a>  <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> 1,<span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="subset_1.html#K2" title="SUBSET_1:func.2">[#]</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E2:24"/><i><font color="Green" title="E16">A2</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> 1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="subset_1.html#K2" title="SUBSET_1:func.2">[#]</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E3:24"/><i><font color="Green" title="E17">B1</font></i>: 
 <a href="subset_1.html#K2" title="SUBSET_1:func.2">[#]</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> 1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <a href="sin_cos.html#K28" title="SIN_COS:func.28">exp_R</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E4:24"/><i><font color="Green" title="E18">A3</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  holds <br/><span class="p1">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> 1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></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> <font color="Olive" title="b1">x</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  implies <span class="p1">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> 1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></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> <font color="Maroon" title="c2">x</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 )</span><br/>

<b>assume </b><a NAME="E1:24_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> 1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></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> <font color="Maroon" title="c2">x</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span><br/>

<span class="p1">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> 1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</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="p1">(<span class="default">1 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="jordan16.html#D3" title="JORDAN16:def.3">JORDAN16:def 3</a></i>
<br/>.= 
<font color="Maroon" title="c2">x</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1

;<br/>
<b>hence </b><a NAME="E3:24_1"/>
<span class="p1">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> 1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></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> <font color="Maroon" title="c2">x</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E5:24"/><i><font color="Green" title="E19">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>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  holds <br/><span class="p1">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> 1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </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">1 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="jordan16.html#D3" title="JORDAN16:def.3">JORDAN16:def 3</a></i>;<br/>
<a NAME="E6:24"/><i><font color="Green" title="E20">C1</font></i>: 
 <a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> 1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</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="txt" href="#E2:24"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="#E5:24"><i><font color="Green" title="E19">A4</font></i></a>, <a class="ref" href="fdiff_1.html#T31" title="FDIFF_1:th.31">FDIFF_1:31</a></i>;<br/>
<a NAME="E7:24"/>
<span class="p1">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> 1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</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>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<b>by </b><i><a class="ref" href="taylor_1.html#T16" title="TAYLOR_1:th.16">TAYLOR_1:16</a>, <a class="txt" href="#E3:24"><i><font color="Green" title="E17">B1</font></i></a>, <a class="txt" href="#E6:24"><i><font color="Green" title="E20">C1</font></i></a>, <a class="ref" href="fdiff_1.html#T29" title="FDIFF_1:th.29">FDIFF_1:29</a></i>;<br/>
<a NAME="E8:24"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> 1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <a href="sin_cos.html#K28" title="SIN_COS:func.28">exp_R</a> </span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  is <a href="fcont_1.html#V1" title="FCONT_1:attr.1">continuous</a>
 
<b>by </b><i><a class="ref" href="fdiff_1.html#T33" title="FDIFF_1:th.33">FDIFF_1:33</a></i>;<br/>
<a NAME="E9:24"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> 1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <a href="sin_cos.html#K28" title="SIN_COS:func.28">exp_R</a> </span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">A</font> is <a href="fcont_1.html#V1" title="FCONT_1:attr.1">continuous</a>
 
<b>by </b><i><a class="ref" href="fcont_1.html#T17" title="FCONT_1:th.17">FCONT_1:17</a></i>;<br/>
<a NAME="E10:24"/><b>then </b><i><font color="Green" title="E21">A5</font></i>: 
( <span class="p1">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> 1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <a href="sin_cos.html#K28" title="SIN_COS:func.28">exp_R</a>  <a href="integra5.html#R1" title="INTEGRA5:pred.1">is_integrable_on</a> <font color="Maroon" title="c1">A</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> 1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <a href="sin_cos.html#K28" title="SIN_COS:func.28">exp_R</a> </span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">A</font> is <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> )
 
<b>by </b><i><a class="txt" href="#E3:24"><i><font color="Green" title="E17">B1</font></i></a>, <a class="ref" href="integra5.html#T10" title="INTEGRA5:th.10">INTEGRA5:10</a>, <a class="ref" href="integra5.html#T11" title="INTEGRA5:th.11">INTEGRA5:11</a></i>;<br/>
<a NAME="E11:24"/><i><font color="Green" title="E22">A6</font></i>: 
<a href="sin_cos.html#K28" title="SIN_COS:func.28">exp_R</a>  <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> 1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span>)</span> <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="txt" href="#E1:24"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="#E4:24"><i><font color="Green" title="E18">A3</font></i></a>, <a class="ref" href="fdiff_4.html#T55" title="FDIFF_4:th.55">FDIFF_4:55</a></i>;<br/>
<a NAME="E12:24"/><i><font color="Green" title="E23">B</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>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> 1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <a href="sin_cos.html#K28" title="SIN_COS:func.28">exp_R</a> </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> <font color="Olive" title="b1">x</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</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="24_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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> 1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <a href="sin_cos.html#K28" title="SIN_COS:func.28">exp_R</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> <font color="Maroon" title="c2">x</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</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/>

<b>assume </b><a NAME="E1:24_2"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> 1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <a href="sin_cos.html#K28" title="SIN_COS:func.28">exp_R</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> <font color="Maroon" title="c2">x</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</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="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> 1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <a href="sin_cos.html#K28" title="SIN_COS:func.28">exp_R</a> </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"><span class="p2">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> 1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</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"><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>
<b>by </b><i><a class="ref" href="valued_1.html#T5" title="VALUED_1:th.5">VALUED_1:5</a></i>
<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> <font color="Maroon" title="c2">x</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</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>
<b>by </b><i><a class="ref" href="jordan16.html#D3" title="JORDAN16:def.3">JORDAN16:def 3</a></i>
<br/>.= 
<font color="Maroon" title="c2">x</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</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/>
<b>hence </b><a NAME="E3:24_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> 1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <a href="sin_cos.html#K28" title="SIN_COS:func.28">exp_R</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> <font color="Maroon" title="c2">x</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E13:24"/><i><font color="Green" title="E24">A7</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>  <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="sin_cos.html#K28" title="SIN_COS:func.28">exp_R</a>  <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> 1,<span class="p4">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span>)</span></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> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K28" title="SIN_COS:func.28">exp_R</a>  <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> 1,<span class="p4">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span>)</span></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#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"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> 1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <a href="sin_cos.html#K28" title="SIN_COS:func.28">exp_R</a> </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="24_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">x</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K28" title="SIN_COS:func.28">exp_R</a>  <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> 1,<span class="p4">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span>)</span></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> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K28" title="SIN_COS:func.28">exp_R</a>  <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> 1,<span class="p4">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span>)</span></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#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"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> 1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <a href="sin_cos.html#K28" title="SIN_COS:func.28">exp_R</a> </span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">x</font> )</span><br/>

<b>assume </b><a NAME="E1:24_3"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K28" title="SIN_COS:func.28">exp_R</a>  <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> 1,<span class="p4">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span>)</span></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 class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K28" title="SIN_COS:func.28">exp_R</a>  <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> 1,<span class="p4">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span>)</span></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#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"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> 1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <a href="sin_cos.html#K28" title="SIN_COS:func.28">exp_R</a> </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="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K28" title="SIN_COS:func.28">exp_R</a>  <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> 1,<span class="p4">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span>)</span></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#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">x</font> = 
<font color="Maroon" title="c2">x</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</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>
<b>by </b><i><a class="txt" href="#E1:24"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="#E4:24"><i><font color="Green" title="E18">A3</font></i></a>, <a class="ref" href="fdiff_4.html#T55" title="FDIFF_4:th.55">FDIFF_4:55</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> 1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <a href="sin_cos.html#K28" title="SIN_COS:func.28">exp_R</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="txt" href="#E12:24"><i><font color="Green" title="E23">B</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E3:24_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K28" title="SIN_COS:func.28">exp_R</a>  <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> 1,<span class="p4">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span>)</span></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#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"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> 1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <a href="sin_cos.html#K28" title="SIN_COS:func.28">exp_R</a> </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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E14:24"/>
 <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="sin_cos.html#K28" title="SIN_COS:func.28">exp_R</a>  <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> 1,<span class="p4">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span>)</span></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="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="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> 1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <a href="sin_cos.html#K28" title="SIN_COS:func.28">exp_R</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="#E3:24"><i><font color="Green" title="E17">B1</font></i></a>, <a class="txt" href="#E11:24"><i><font color="Green" title="E22">A6</font></i></a>, <a class="ref" href="fdiff_1.html#D8" title="FDIFF_1:def.8">FDIFF_1:def 8</a></i>;<br/>
<a NAME="E15:24"/><b>then </b>
<span class="p1">(<span class="default"><a href="sin_cos.html#K28" title="SIN_COS:func.28">exp_R</a>  <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> 1,<span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span>)</span></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> <span class="p1">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> 1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <a href="sin_cos.html#K28" title="SIN_COS:func.28">exp_R</a> 
 
<b>by </b><i><a class="txt" href="#E13:24"><i><font color="Green" title="E24">A7</font></i></a>, <a class="ref" href="partfun1.html#T34" title="PARTFUN1:th.34">PARTFUN1:34</a></i>;<br/>
<b>hence </b><a NAME="E16:24"/>
 <a href="integra5.html#K2" title="INTEGRA5:func.2">integral</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> 1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <a href="sin_cos.html#K28" title="SIN_COS:func.28">exp_R</a> </span>)</span>,<font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K28" title="SIN_COS:func.28">exp_R</a>  <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> 1,<span class="p4">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#NK2" title="PSCOMP_1:NK.2">sup</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K28" title="SIN_COS:func.28">exp_R</a>  <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> 1,<span class="p4">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#NK3" title="PSCOMP_1:NK.3">inf</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:24"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="#E4:24"><i><font color="Green" title="E18">A3</font></i></a>, <a class="txt" href="#E10:24"><i><font color="Green" title="E21">A5</font></i></a>, <a class="ref" href="fdiff_4.html#T55" title="FDIFF_4:th.55">FDIFF_4:55</a>, <a class="ref" href="integra5.html#T13" title="INTEGRA5:th.13">INTEGRA5:13</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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