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<b>let </b><font color="Maroon">i</font> be   <a href="int_1.html#NM1">Integer</a>;<br/><b>let </b><font color="Maroon">x</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>



<a NAME="E1:7"/><i><font color="Green">E31</font></i>: 
 <a href="sin_cos.html#K28">exp_R</a> <font color="Maroon">x</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="sin_cos.html#T60">SIN_COS:60</a></i>;<br/>
<b>consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:7"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> or <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">n</font> )
 <b>by </b><i><a class="ref" href="int_1.html#T8">INT_1:8</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:7_1_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> or <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">n</font> )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_1_1"><b>case </b></a><a NAME="E1:7_1_1_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:7_1_1_1"/>
<span class="p1">(<span class="default"><a href="sin_cos.html#K28">exp_R</a> <font color="Maroon">x</font></span>)</span> <a href="prepower.html#K12">#R</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="sin_cos.html#K28">exp_R</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x</font></span>)</span>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_1_2"><b>case </b></a><a NAME="E1:7_1_1_2"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><b>hence </b> <a href="sin_cos.html#K28">exp_R</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x</font></span>)</span> = 
 <a href="sin_cos.html#K28">exp_R</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x</font></span>)</span></span>)</span>

<br/>.= 
1 <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K28">exp_R</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
1 <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K28">exp_R</a> <font color="Maroon">x</font></span>)</span> <a href="prepower.html#K12">#R</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="sin_cos.html#K28">exp_R</a> <font color="Maroon">x</font></span>)</span> <a href="prepower.html#K12">#R</a> <font color="Maroon">i</font>
<b>by </b><i>, , <a class="ref" href="prepower.html#T90">PREPOWER:90</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E5:7"/>
<span class="p1">(<span class="default"><a href="sin_cos.html#K28">exp_R</a> <font color="Maroon">x</font></span>)</span> <a href="prepower.html#K12">#R</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="sin_cos.html#K28">exp_R</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x</font></span>)</span>
 ;<br/>


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