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<span class="kw">let </span><font color="Maroon" title="c3">A</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="measure5.html#V2" title="MEASURE5:attr.2">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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="integra5.html#K2" title="INTEGRA5:func.2">integral</a> (<span class="p1">(<span class="default"><a href="sin_cos2.html#K1" title="SIN_COS2:func.1">sinh</a> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <a href="sin_cos2.html#K3" title="SIN_COS2:func.3">cosh</a></span>)</span>,<font color="Maroon" title="c3">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"><a href="sin_cos2.html#K3" title="SIN_COS2:func.3">cosh</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <font color="Maroon" title="c3">A</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><a href="sin_cos2.html#K3" title="SIN_COS2:func.3">cosh</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c3">A</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="sin_cos2.html#K1" title="SIN_COS2:func.1">sinh</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <font color="Maroon" title="c3">A</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="sin_cos2.html#K1" title="SIN_COS2:func.1">sinh</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c3">A</font></span>)</span></span>)</span></span><br/>

<a NAME="E1:86"/><span class="lab"><font color="Green" title="E67">A1</font></span>: 
 <a href="subset_1.html#K2" title="SUBSET_1:func.2">[#]</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> is   <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>
 
;<br/>
<a NAME="E2:86"/>
<a href="sin_cos2.html#K3" title="SIN_COS2:func.3">cosh</a> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">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="txt" href="integra8.html#E75"><span class="lab"><font color="Green" title="E55">Lm16</font></span></a></span>;<br/>
<a NAME="E3:86"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A2</font></span>: 
 <a href="sin_cos2.html#K3" title="SIN_COS2:func.3">cosh</a>  <a href="integra5.html#R1" title="INTEGRA5:pred.1">is_integrable_on</a> <font color="Maroon" title="c3">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="integra8.html#E50"><span class="lab"><font color="Green" title="E45">Lm10</font></span></a>, <a class="ref" href="integra5.html#T11" onmouseover="rs('integra5/T11')" onmouseout="rh()">INTEGRA5:11</a></span>;<br/>
<a NAME="E4:86"/>
<a href="sin_cos2.html#K1" title="SIN_COS2:func.1">sinh</a> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">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="txt" href="integra8.html#E71"><span class="lab"><font color="Green" title="E52">Lm14</font></span></a></span>;<br/>
<a NAME="E5:86"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A3</font></span>: 
 <a href="sin_cos2.html#K1" title="SIN_COS2:func.1">sinh</a>  <a href="integra5.html#R1" title="INTEGRA5:pred.1">is_integrable_on</a> <font color="Maroon" title="c3">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="integra8.html#E49"><span class="lab"><font color="Green" title="E44">Lm9</font></span></a>, <a class="ref" href="integra5.html#T11" onmouseover="rs('integra5/T11')" onmouseout="rh()">INTEGRA5:11</a></span>;<br/>
<a NAME="E6:86"/>
( <a href="sin_cos2.html#K3" title="SIN_COS2:func.3">cosh</a> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">A</font> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">bounded</a>  &amp; <a href="sin_cos2.html#K1" title="SIN_COS2:func.1">sinh</a> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">A</font> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">bounded</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="integra8.html#E49"><span class="lab"><font color="Green" title="E44">Lm9</font></span></a>, <a class="txt" href="integra8.html#E50"><span class="lab"><font color="Green" title="E45">Lm10</font></span></a>, <a class="txt" href="integra8.html#E71"><span class="lab"><font color="Green" title="E52">Lm14</font></span></a>, <a class="txt" href="integra8.html#E75"><span class="lab"><font color="Green" title="E55">Lm16</font></span></a>, <a class="ref" href="integra5.html#T10" onmouseover="rs('integra5/T10')" onmouseout="rh()">INTEGRA5:10</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:86"/>
 <a href="integra5.html#K2" title="INTEGRA5:func.2">integral</a> (<span class="p1">(<span class="default"><a href="sin_cos2.html#K1" title="SIN_COS2:func.1">sinh</a> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <a href="sin_cos2.html#K3" title="SIN_COS2:func.3">cosh</a></span>)</span>,<font color="Maroon" title="c3">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"><a href="sin_cos2.html#K3" title="SIN_COS2:func.3">cosh</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <font color="Maroon" title="c3">A</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><a href="sin_cos2.html#K3" title="SIN_COS2:func.3">cosh</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c3">A</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="sin_cos2.html#K1" title="SIN_COS2:func.1">sinh</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <font color="Maroon" title="c3">A</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="sin_cos2.html#K1" title="SIN_COS2:func.1">sinh</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c3">A</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:86"><span class="lab"><font color="Green" title="E68">A2</font></span></a>, <a class="txt" href="#E5:86"><span class="lab"><font color="Green" title="E69">A3</font></span></a>, <a class="txt" href="#E1:86"><span class="lab"><font color="Green" title="E67">A1</font></span></a>, <a class="ref" href="integra8.html#T30" target="_self" onmouseover="rs('integra8/T30')" onmouseout="rh()">Th30</a>, <a class="ref" href="integra8.html#T31" target="_self" onmouseover="rs('integra8/T31')" onmouseout="rh()">Th31</a>, <a class="ref" href="integra8.html#T66" target="_self" onmouseover="rs('integra8/T66')" onmouseout="rh()">Th66</a>, <a class="ref" href="sin_cos2.html#T34" onmouseover="rs('sin_cos2/T34')" onmouseout="rh()">SIN_COS2:34</a>, <a class="ref" href="sin_cos2.html#T35" onmouseover="rs('sin_cos2/T35')" onmouseout="rh()">SIN_COS2:35</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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