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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"> ( <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> implies  <a href="integra5.html#K2" title="INTEGRA5:func.2">integral</a> (<a href="sin_cos2.html#K1" title="SIN_COS2:func.1">sinh</a>,<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"><a href="power.html#K4" title="POWER:func.4">number_e</a> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="power.html#K4" title="POWER:func.4">number_e</a></span>)</span> )</span><br/>

<a NAME="E1:64"/>
 <a href="sin_cos.html#K23" title="SIN_COS:func.23">exp_R</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="sin_cos.html#T55" onmouseover="rs('sin_cos/T55')" onmouseout="rh()">SIN_COS:55</a></span>;<br/>
<a NAME="E2:64"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A1</font></span>: 
2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="power.html#K4" title="POWER:func.4">number_e</a> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="irrat_1.html#D7" onmouseover="rs('irrat_1/D7')" onmouseout="rh()">IRRAT_1:def 7</a>, <a class="ref" href="xreal_1.html#T129" onmouseover="rs('xreal_1/T129')" onmouseout="rh()">XREAL_1:129</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:64"/>
<font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</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">  <a href="integra5.html#K2" title="INTEGRA5:func.2">integral</a> (<a href="sin_cos2.html#K1" title="SIN_COS2:func.1">sinh</a>,<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"><a href="power.html#K4" title="POWER:func.4">number_e</a> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="power.html#K4" title="POWER:func.4">number_e</a></span>)</span></span><br/>

<a NAME="E4:64"/><span class="kw">then </span>
<font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<font color="Maroon" title="c2">jj</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
;<br/>
<a NAME="E5:64"/><span class="kw">then </span>
(  <a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra8.html#T37" target="_self" onmouseover="rs('integra8/T37')" onmouseout="rh()">Th37</a></span>;<br/>
<span class="kw">then </span> <a href="integra5.html#K2" title="INTEGRA5:func.2">integral</a> (<a href="sin_cos2.html#K1" title="SIN_COS2:func.1">sinh</a>,<font color="Maroon" title="c3">A</font>) = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="power.html#K4" title="POWER:func.4">number_e</a> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="power.html#K4" title="POWER:func.4">number_e</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra8.html#T18" target="_self" onmouseover="rs('integra8/T18')" onmouseout="rh()">Th18</a>, <a class="ref" href="integra8.html#T19" target="_self" onmouseover="rs('integra8/T19')" onmouseout="rh()">Th19</a>, <a class="ref" href="integra8.html#T55" target="_self" onmouseover="rs('integra8/T55')" onmouseout="rh()">Th55</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="power.html#K4" title="POWER:func.4">number_e</a> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="power.html#K4" title="POWER:func.4">number_e</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="power.html#K4" title="POWER:func.4">number_e</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="power.html#K4" title="POWER:func.4">number_e</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:64"><span class="lab"><font color="Green" title="E55">A1</font></span></a>, <a class="ref" href="xcmplx_1.html#T60" onmouseover="rs('xcmplx_1/T60')" onmouseout="rh()">XCMPLX_1:60</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="power.html#K4" title="POWER:func.4">number_e</a> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="power.html#K4" title="POWER:func.4">number_e</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="power.html#K4" title="POWER:func.4">number_e</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T120" onmouseover="rs('xcmplx_1/T120')" onmouseout="rh()">XCMPLX_1:120</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="power.html#K4" title="POWER:func.4">number_e</a> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="power.html#K4" title="POWER:func.4">number_e</a></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default">1 <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="power.html#K4" title="POWER:func.4">number_e</a></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="power.html#K4" title="POWER:func.4">number_e</a> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="power.html#K4" title="POWER:func.4">number_e</a></span>)</span>

;<br/>
<span class="kw">hence </span><a NAME="E7:64"/>
 <a href="integra5.html#K2" title="INTEGRA5:func.2">integral</a> (<a href="sin_cos2.html#K1" title="SIN_COS2:func.1">sinh</a>,<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"><a href="power.html#K4" title="POWER:func.4">number_e</a> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="power.html#K4" title="POWER:func.4">number_e</a></span>)</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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