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<b>let </b><font color="Maroon">a</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="real_1.html#NM1">Real</a>;<br/>



<b>assume </b><a NAME="E1:6"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<a NAME="E2:6"/><b>then </b><i><font color="Green">E36</font></i>: 
<font color="Maroon">a</font> <a href="prepower.html#K13">#R</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">x</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="prepower.html#T95">PREPOWER:95</a></i>;<br/>
<a NAME="E3:6"/><i><font color="Green">E37</font></i>: 
(  <a href="power.html#K8">number_e</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="power.html#K8">number_e</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 )
 
<b>by </b><i><a class="ref" href="taylor_1.html#T11">TAYLOR_1:11</a></i>;<br/>
<a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="power.html#K6">log</a> <a href="power.html#K8">number_e</a> ,<font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> = 
<a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="power.html#K6">log</a> <a href="power.html#K8">number_e</a> ,<font color="Maroon">a</font></span>)</span></span>)</span>

<br/>.= 
<a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="power.html#K6">log</a> <a href="power.html#K8">number_e</a> ,<span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="power.html#K4">to_power</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="power.html#T63">POWER:63</a>, , </i>
<br/>.= 
<a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="power.html#K6">log</a> <a href="power.html#K8">number_e</a> ,<span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="prepower.html#K13">#R</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="power.html#D2">POWER:def 2</a>, </i>
<br/>.= 
<font color="Maroon">a</font> <a href="prepower.html#K13">#R</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="taylor_1.html#T15">TAYLOR_1:15</a>, </i>
;<br/>
<b>hence </b><a NAME="E5:6"/>
<a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="power.html#K6">log</a> <a href="power.html#K8">number_e</a> ,<font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="prepower.html#K13">#R</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">x</font></span>)</span>
 ;<br/>


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