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<b>let </b><font color="Maroon" title="c1">i</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>;<br/><b>let </b><font color="Maroon" title="c2">x</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>



<b>set </b><font color="Maroon" title="c3">n</font> = <font color="Maroon" title="c1">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" title="c1">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 or <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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" title="E8">A1</font></i>: 
<font color="Maroon" title="c1">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:8_1_1"/><i><font color="Green" title="E9">A2</font></i>: 
 <a href="sin_cos.html#K28" title="SIN_COS:func.28">exp_R</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c1">i</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>by </b><i><a class="ref" href="sin_cos.html#T60" title="SIN_COS:th.60">SIN_COS:60</a></i>;<br/> <a href="sin_cos.html#K28" title="SIN_COS:func.28">exp_R</a> <font color="Maroon" title="c2">x</font> = 
 <a href="sin_cos.html#K28" title="SIN_COS:func.28">exp_R</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c1">i</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">i</font></span>)</span>
<b>by </b><i><a class="txt" href="taylor_1.html#E1:8_1_1"><i><font color="Green" title="E8">A1</font></i></a>, <a class="ref" href="xcmplx_1.html#T88" title="XCMPLX_1:th.88">XCMPLX_1:88</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="sin_cos.html#K28" title="SIN_COS:func.28">exp_R</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c1">i</font></span>)</span></span>)</span> <a href="prepower.html#K12" title="PREPOWER:func.12">#R</a> <font color="Maroon" title="c1">i</font>
<b>by </b><i><a class="txt" href="taylor_1.html#E7"><i><font color="Green" title="E6">Lm2</font></i></a></i>
;<br/><b>hence </b><span class="p1">(<span class="default"><a href="sin_cos.html#K28" title="SIN_COS:func.28">exp_R</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="prepower.html#K12" title="PREPOWER:func.12">#R</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c1">i</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="sin_cos.html#K28" title="SIN_COS:func.28">exp_R</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c1">i</font></span>)</span></span>)</span> <a href="prepower.html#K12" title="PREPOWER:func.12">#R</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c1">i</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="taylor_1.html#E2:8_1_1"><i><font color="Green" title="E9">A2</font></i></a>, <a class="ref" href="prepower.html#T105" title="PREPOWER:th.105">PREPOWER:105</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="sin_cos.html#K28" title="SIN_COS:func.28">exp_R</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c1">i</font></span>)</span></span>)</span> <a href="prepower.html#K12" title="PREPOWER:func.12">#R</a> 1
<b>by </b><i><a class="txt" href="taylor_1.html#E1:8_1_1"><i><font color="Green" title="E8">A1</font></i></a>, <a class="ref" href="xcmplx_1.html#T107" title="XCMPLX_1:th.107">XCMPLX_1:107</a></i>
<br/>.= 
 <a href="sin_cos.html#K28" title="SIN_COS:func.28">exp_R</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c1">i</font></span>)</span>
<b>by </b><i><a class="txt" href="taylor_1.html#E2:8_1_1"><i><font color="Green" title="E9">A2</font></i></a>, <a class="ref" href="prepower.html#T86" title="PREPOWER:th.86">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" title="E8">A3</font></i>: 
<font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:8_1_2"/><i><font color="Green" title="E9">A4</font></i>: 
 <a href="sin_cos.html#K28" title="SIN_COS:func.28">exp_R</a> <font color="Maroon" title="c2">x</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>by </b><i><a class="ref" href="sin_cos.html#T60" title="SIN_COS:th.60">SIN_COS:60</a></i>;<br/><span class="p1">(<span class="default"><a href="sin_cos.html#K28" title="SIN_COS:func.28">exp_R</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="prepower.html#K12" title="PREPOWER:func.12">#R</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 0</span>)</span> = 
 <a href="sin_cos.html#K29" title="SIN_COS:func.29">exp_R</a> 0
<b>by </b><i><a class="txt" href="taylor_1.html#E2:8_1_2"><i><font color="Green" title="E9">A4</font></i></a>, <a class="ref" href="prepower.html#T85" title="PREPOWER:th.85">PREPOWER:85</a>, <a class="ref" href="sin_cos.html#T56" title="SIN_COS:th.56">SIN_COS:56</a></i>
<br/>.= 
 <a href="sin_cos.html#K28" title="SIN_COS:func.28">exp_R</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T49" title="XCMPLX_1:th.49">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" title="SIN_COS:func.28">exp_R</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="prepower.html#K12" title="PREPOWER:func.12">#R</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c1">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="sin_cos.html#K28" title="SIN_COS:func.28">exp_R</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c1">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="taylor_1.html#E1:8_1_2"><i><font color="Green" title="E8">A3</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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