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

<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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_cos2.html#K1" title="SIN_COS2:func.1">sinh</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"><span class="p3">(<span class="default"><span class="p4">(<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_cos2.html#K4" title="SIN_COS2:func.4">cosh</a> </span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <a href="sin_cos2.html#K1" title="SIN_COS2:func.1">sinh</a> </span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><a href="measure6.html#NK4" title="MEASURE6:NK.4">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"><span class="p3">(<span class="default"><span class="p4">(<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_cos2.html#K4" title="SIN_COS2:func.4">cosh</a> </span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <a href="sin_cos2.html#K1" title="SIN_COS2:func.1">sinh</a> </span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><a href="measure6.html#NK5" title="MEASURE6:NK.5">inf</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span></span><br/>

<a NAME="E1:30"/><span class="lab"><font color="Green" title="E22">A1</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>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</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="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_cos2.html#K4" title="SIN_COS2:func.4">cosh</a> </span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <a href="sin_cos2.html#K1" title="SIN_COS2:func.1">sinh</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> holds <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="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_cos2.html#K4" title="SIN_COS2:func.4">cosh</a> </span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <a href="sin_cos2.html#K1" title="SIN_COS2:func.1">sinh</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#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_cos2.html#K1" title="SIN_COS2:func.1">sinh</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="30_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>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</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="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_cos2.html#K4" title="SIN_COS2:func.4">cosh</a> </span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <a href="sin_cos2.html#K1" title="SIN_COS2:func.1">sinh</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> implies <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="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_cos2.html#K4" title="SIN_COS2:func.4">cosh</a> </span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <a href="sin_cos2.html#K1" title="SIN_COS2:func.1">sinh</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#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_cos2.html#K1" title="SIN_COS2:func.1">sinh</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="kw">assume </span><a NAME="E1:30_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</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="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_cos2.html#K4" title="SIN_COS2:func.4">cosh</a> </span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <a href="sin_cos2.html#K1" title="SIN_COS2:func.1">sinh</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 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"><span class="p4">(<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_cos2.html#K4" title="SIN_COS2:func.4">cosh</a> </span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <a href="sin_cos2.html#K1" title="SIN_COS2:func.1">sinh</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#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_cos2.html#K1" title="SIN_COS2:func.1">sinh</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"><span class="p3">(<span class="default"><span class="p4">(<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_cos2.html#K4" title="SIN_COS2:func.4">cosh</a> </span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <a href="sin_cos2.html#K1" title="SIN_COS2:func.1">sinh</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#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">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_cos2.html#K1" title="SIN_COS2:func.1">sinh</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="ref" href="integr11.html#T10" target="_self" title="INTEGR11:th.10">Th10</a></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="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_cos2.html#K1" title="SIN_COS2:func.1">sinh</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="ref" href="jordan16.html#D3" title="JORDAN16:def.3">JORDAN16:def 3</a></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_cos2.html#K1" title="SIN_COS2:func.1">sinh</a> </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="valued_1.html#T5" title="VALUED_1:th.5">VALUED_1:5</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E3:30_1"/>
<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="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_cos2.html#K4" title="SIN_COS2:func.4">cosh</a> </span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <a href="sin_cos2.html#K1" title="SIN_COS2:func.1">sinh</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#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_cos2.html#K1" title="SIN_COS2:func.1">sinh</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:30"/><span class="lab"><font color="Green" title="E23">A2</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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_cos2.html#K1" title="SIN_COS2:func.1">sinh</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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E3:30"/><span class="kw">then </span>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</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="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_cos2.html#K4" title="SIN_COS2:func.4">cosh</a> </span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <a href="sin_cos2.html#K1" title="SIN_COS2:func.1">sinh</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="relat_1.html#NK2" title="RELAT_1:NK.2">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_cos2.html#K1" title="SIN_COS2:func.1">sinh</a> </span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="integr11.html#T10" target="_self" title="INTEGR11:th.10">Th10</a>, <a class="ref" href="fdiff_1.html#D8" title="FDIFF_1:def.8">FDIFF_1:def 8</a></span>;<br/>
<a NAME="E4:30"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A3</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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_cos2.html#K4" title="SIN_COS2:func.4">cosh</a> </span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <a href="sin_cos2.html#K1" title="SIN_COS2:func.1">sinh</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="hidden.html#R1" title="HIDDEN:pred.1">=</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_cos2.html#K1" title="SIN_COS2:func.1">sinh</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="ref" href="partfun1.html#T34" title="PARTFUN1:th.34">PARTFUN1:34</a></span>;<br/>
<a NAME="E5:30"/>
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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>  &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>  <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>  ) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="jordan16.html#D3" title="JORDAN16:def.3">JORDAN16:def 3</a></span>;<br/>
<a NAME="E6:30"/><span class="kw">then </span>
 <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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fdiff_1.html#T31" title="FDIFF_1:th.31">FDIFF_1:31</a></span>;<br/>
<a NAME="E7:30"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A4</font></span>: 
<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_cos2.html#K1" title="SIN_COS2:func.1">sinh</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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:30"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="ref" href="fdiff_1.html#T29" title="FDIFF_1:th.29">FDIFF_1:29</a>, <a class="ref" href="fdiff_1.html#T33" title="FDIFF_1:th.33">FDIFF_1:33</a>, <a class="ref" href="sin_cos2.html#T34" title="SIN_COS2:th.34">SIN_COS2:34</a></span>;<br/>
<a NAME="E8:30"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A5</font></span>: 
<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_cos2.html#K1" title="SIN_COS2:func.1">sinh</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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fcont_1.html#T17" title="FCONT_1:th.17">FCONT_1:17</a></span>;<br/>
<a NAME="E9:30"/>
<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_cos2.html#K1" title="SIN_COS2:func.1">sinh</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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:30"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="txt" href="#E7:30"><span class="lab"><font color="Green" title="E25">A4</font></span></a>, <a class="ref" href="fcont_1.html#T17" title="FCONT_1:th.17">FCONT_1:17</a>, <a class="ref" href="integra5.html#T10" title="INTEGRA5:th.10">INTEGRA5:10</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:30"/>
 <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_cos2.html#K1" title="SIN_COS2:func.1">sinh</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"><span class="p3">(<span class="default"><span class="p4">(<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_cos2.html#K4" title="SIN_COS2:func.4">cosh</a> </span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <a href="sin_cos2.html#K1" title="SIN_COS2:func.1">sinh</a> </span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><a href="measure6.html#NK4" title="MEASURE6:NK.4">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"><span class="p3">(<span class="default"><span class="p4">(<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_cos2.html#K4" title="SIN_COS2:func.4">cosh</a> </span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <a href="sin_cos2.html#K1" title="SIN_COS2:func.1">sinh</a> </span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><a href="measure6.html#NK5" title="MEASURE6:NK.5">inf</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:30"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="txt" href="#E8:30"><span class="lab"><font color="Green" title="E26">A5</font></span></a>, <a class="txt" href="#E4:30"><span class="lab"><font color="Green" title="E24">A3</font></span></a>, <a class="ref" href="integr11.html#T10" target="_self" title="INTEGR11:th.10">Th10</a>, <a class="ref" href="integra5.html#T11" title="INTEGRA5:th.11">INTEGRA5:11</a>, <a class="ref" href="integra5.html#T13" title="INTEGRA5:th.13">INTEGRA5:13</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/>


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