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<b>let </b><font color="Maroon" title="c1">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.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">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies (  <a href="power.html#K6" title="POWER:func.6">log</a> <a href="power.html#K8" title="POWER:func.8">number_e</a> ,<font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">r</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 &amp; ( <font color="Maroon" title="c1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies  <a href="power.html#K6" title="POWER:func.6">log</a> <a href="power.html#K8" title="POWER:func.8">number_e</a> ,<font color="Maroon" title="c1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">r</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 ) &amp; (  <a href="power.html#K6" title="POWER:func.6">log</a> <a href="power.html#K8" title="POWER:func.8">number_e</a> ,<font color="Maroon" title="c1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">r</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 implies <font color="Maroon" title="c1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) &amp; ( <font color="Maroon" title="c1">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 implies  <a href="power.html#K6" title="POWER:func.6">log</a> <a href="power.html#K8" title="POWER:func.8">number_e</a> ,<font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">r</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 ) &amp; (  <a href="power.html#K6" title="POWER:func.6">log</a> <a href="power.html#K8" title="POWER:func.8">number_e</a> ,<font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">r</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 implies <font color="Maroon" title="c1">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 ) ) )</span><br/>

<b>assume </b><a NAME="E1:3"/><i><font color="Green" title="E4">A1</font></i>: 
<font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="power.html#K6" title="POWER:func.6">log</a> <a href="power.html#K8" title="POWER:func.8">number_e</a> ,<font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">r</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 &amp; ( <font color="Maroon" title="c1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies  <a href="power.html#K6" title="POWER:func.6">log</a> <a href="power.html#K8" title="POWER:func.8">number_e</a> ,<font color="Maroon" title="c1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">r</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 ) &amp; (  <a href="power.html#K6" title="POWER:func.6">log</a> <a href="power.html#K8" title="POWER:func.8">number_e</a> ,<font color="Maroon" title="c1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">r</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 implies <font color="Maroon" title="c1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) &amp; ( <font color="Maroon" title="c1">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 implies  <a href="power.html#K6" title="POWER:func.6">log</a> <a href="power.html#K8" title="POWER:func.8">number_e</a> ,<font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">r</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 ) &amp; (  <a href="power.html#K6" title="POWER:func.6">log</a> <a href="power.html#K8" title="POWER:func.8">number_e</a> ,<font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">r</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 implies <font color="Maroon" title="c1">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 ) )</span><br/>

<a NAME="E2:3"/><b>then </b>
<font color="Maroon" title="c1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">g</font> where <font color="Olive" title="b1">g</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> :  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">g</font> </span>}</span> 
 
;<br/>
<a NAME="E3:3"/><b>then </b>
<font color="Maroon" title="c1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="xxreal_1.html#T230" title="XXREAL_1:th.230">XXREAL_1:230</a></i>;<br/>
<a NAME="E4:3"/><b>then </b>
 <a href="power.html#K6" title="POWER:func.6">log</a> <a href="power.html#K8" title="POWER:func.8">number_e</a> ,<font color="Maroon" title="c1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="taylor_1.html#K3" title="TAYLOR_1:func.3">ln</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">r</font>
 
<b>by </b><i><a class="ref" href="taylor_1.html#D2" title="TAYLOR_1:def.2">TAYLOR_1:def 2</a></i>;<br/>
<b>hence </b><a NAME="E5:3"/>
(  <a href="power.html#K6" title="POWER:func.6">log</a> <a href="power.html#K8" title="POWER:func.8">number_e</a> ,<font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">r</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 &amp; ( <font color="Maroon" title="c1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies  <a href="power.html#K6" title="POWER:func.6">log</a> <a href="power.html#K8" title="POWER:func.8">number_e</a> ,<font color="Maroon" title="c1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">r</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 ) &amp; (  <a href="power.html#K6" title="POWER:func.6">log</a> <a href="power.html#K8" title="POWER:func.8">number_e</a> ,<font color="Maroon" title="c1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">r</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 implies <font color="Maroon" title="c1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) &amp; ( <font color="Maroon" title="c1">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 implies  <a href="power.html#K6" title="POWER:func.6">log</a> <a href="power.html#K8" title="POWER:func.8">number_e</a> ,<font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">r</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 ) &amp; (  <a href="power.html#K6" title="POWER:func.6">log</a> <a href="power.html#K8" title="POWER:func.8">number_e</a> ,<font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">r</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 implies <font color="Maroon" title="c1">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 ) )
 <b>by </b><i><a class="txt" href="#E1:3"><i><font color="Green" title="E4">A1</font></i></a>, <a class="ref" href="entropy1.html#T2" target="_self" title="ENTROPY1:th.2">Th2</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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