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<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> ;<br/>

<b>assume </b><a NAME="E1:71"/>
<font color="Maroon" title="c1">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">0,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 ;<br/>

<a NAME="E2:71"/><b>then </b><i><font color="Green" title="E56">A2</font></i>: 
(  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 
<b>by </b><i><a class="ref" href="integra8.html#T37" target="_self" title="INTEGRA8:th.37">Th45</a></i>;<br/>
<a NAME="E3:71"/><i><font color="Green" title="E57">A3</font></i>: 
2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="power.html#K8" title="POWER:func.8">number_e</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1"><b>proof </b></a><div class="add">

<a NAME="E1:71_1"/>
 <a href="sin_cos.html#K29" title="SIN_COS:func.29">exp_R</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="sin_cos.html#T60" title="SIN_COS:th.60">SIN_COS:60</a></i>;<br/>
<b>hence </b><a NAME="E2:71_1"/>
2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="power.html#K8" title="POWER:func.8">number_e</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>by </b><i><a class="ref" href="irrat_1.html#D7" title="IRRAT_1:def.7">IRRAT_1:def 7</a>, <a class="ref" href="xreal_1.html#T131" title="XREAL_1:th.131">XREAL_1:131</a></i>;<br/>


</div><b>end;</b></div>
 <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="c1">A</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="power.html#K8" title="POWER:func.8">number_e</a>  <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="power.html#K8" title="POWER:func.8">number_e</a> </span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
<b>by </b><i><a class="txt" href="integra8.html#E2:71"><i><font color="Green" title="E56">A2</font></i></a>, <a class="ref" href="integra8.html#T55" target="_self" title="INTEGRA8:th.55">The1</a>, <a class="ref" href="integra8.html#T18" target="_self" title="INTEGRA8:th.18">Th20</a>, <a class="ref" href="integra8.html#T19" target="_self" title="INTEGRA8:th.19">Th21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="power.html#K8" title="POWER:func.8">number_e</a>  <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="power.html#K8" title="POWER:func.8">number_e</a> </span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="power.html#K8" title="POWER:func.8">number_e</a> </span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="power.html#K8" title="POWER:func.8">number_e</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T60" title="XCMPLX_1:th.60">XCMPLX_1:60</a>, <a class="txt" href="integra8.html#E3:71"><i><font color="Green" title="E57">A3</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="power.html#K8" title="POWER:func.8">number_e</a>  <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="power.html#K8" title="POWER:func.8">number_e</a> </span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="power.html#K8" title="POWER:func.8">number_e</a> </span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T121" title="XCMPLX_1:th.121">XCMPLX_1:121</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="power.html#K8" title="POWER:func.8">number_e</a>  <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="power.html#K8" title="POWER:func.8">number_e</a> </span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default">1 <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="power.html#K8" title="POWER:func.8">number_e</a> </span>)</span>
<b>by </b><i><a class="ref" href="square_1.html#T59" title="SQUARE_1:th.59">SQUARE_1:59</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="power.html#K8" title="POWER:func.8">number_e</a>  <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="power.html#K8" title="POWER:func.8">number_e</a> </span>)</span>
<b>by </b><i><a class="ref" href="square_1.html#T64" title="SQUARE_1:th.64">SQUARE_1:64</a></i>
;<br/>
<b>hence </b><a NAME="E5:71"/>
 <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="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="power.html#K8" title="POWER:func.8">number_e</a>  <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="power.html#K8" title="POWER:func.8">number_e</a> </span>)</span>
 ;<br/>


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