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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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <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"><a href="numbers.html#K6" title="NUMBERS:func.6">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_cos.html#K27" title="SIN_COS:func.27">exp_R</a> ,<font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="power.html#K8" title="POWER:func.8">number_e</a>  <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 )</span><br/>

<b>assume </b><a NAME="E1:61"/>
<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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 ; <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> <a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a> ,<font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="power.html#K8" title="POWER:func.8">number_e</a>  <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span><br/>

<a NAME="E2:61"/><b>then </b>
(  <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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<b>by </b><i><a class="ref" href="integra8.html#T37" target="_self" title="INTEGRA8:th.37">Th37</a></i>;<br/>
<b>hence </b><a NAME="E3:61"/>
 <a href="integra5.html#K2" title="INTEGRA5:func.2">integral</a> <a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a> ,<font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="power.html#K8" title="POWER:func.8">number_e</a>  <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <b>by </b><i><a class="ref" href="integra8.html#T53" target="_self" title="INTEGRA8:th.53">Th53</a>, <a class="ref" href="irrat_1.html#D7" title="IRRAT_1:def.7">IRRAT_1:def 7</a>, <a class="ref" href="sin_cos.html#T56" title="SIN_COS:th.56">SIN_COS:56</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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