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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/>



<b>set </b><font color="Maroon">n</font> = <font color="Maroon">i</font>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:8_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 or <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1_1"><b>suppose </b></a><a NAME="E1:8_1_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:8_1_1"/><i><font color="Green">E35</font></i>: 
 <a href="sin_cos.html#K28">exp_R</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">i</font></span>)</span> <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/> <a href="sin_cos.html#K28">exp_R</a> <font color="Maroon">x</font> = 
 <a href="sin_cos.html#K28">exp_R</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">i</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="sin_cos.html#K28">exp_R</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="prepower.html#K12">#R</a> <font color="Maroon">i</font>
<b>by </b><i/>
;<br/><b>hence </b><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> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">i</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="sin_cos.html#K28">exp_R</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="prepower.html#K12">#R</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">i</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="prepower.html#T105">PREPOWER:105</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="sin_cos.html#K28">exp_R</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="prepower.html#K12">#R</a> 1
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T107">XCMPLX_1:107</a></i>
<br/>.= 
 <a href="sin_cos.html#K28">exp_R</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i>, <a class="ref" href="prepower.html#T86">PREPOWER:86</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1_2"><b>suppose </b></a><a NAME="E1:8_1_2"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:8_1_2"/><i><font color="Green">E39</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/><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> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 0</span>)</span> = 
 <a href="sin_cos.html#K29">exp_R</a> 0
<b>by </b><i>, <a class="ref" href="prepower.html#T85">PREPOWER:85</a>, <a class="ref" href="sin_cos.html#T56">SIN_COS:56</a></i>
<br/>.= 
 <a href="sin_cos.html#K28">exp_R</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K7">/</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T49">XCMPLX_1:49</a></i>
;<br/><b>hence </b><a NAME="E4:8_1_2"/>
<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> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="sin_cos.html#K28">exp_R</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">i</font></span>)</span>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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