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<span class="kw">let </span><font color="Maroon" title="c1">i</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font> being   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  holds  <span class="p1">(<span class="default"><a href="sin_cos.html#K26" title="SIN_COS:func.26">exp_R</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="prepower.html#K9" title="PREPOWER:func.9">#R</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</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#K26" title="SIN_COS:func.26">exp_R</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <font color="Maroon" title="c1">i</font></span>)</span></span><br/><span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="sin_cos.html#K26" title="SIN_COS:func.26">exp_R</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="prepower.html#K9" title="PREPOWER:func.9">#R</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</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#K26" title="SIN_COS:func.26">exp_R</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <font color="Maroon" title="c1">i</font></span>)</span></span><br/>



<span class="kw">set </span><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"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1_1"><span class="kw">suppose </span></a><a NAME="E1:8_1_1"/><span class="lab"><font color="Green" title="E8">A1</font></span>: 
<font color="Maroon" title="c1">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="sin_cos.html#K26" title="SIN_COS:func.26">exp_R</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="prepower.html#K9" title="PREPOWER:func.9">#R</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</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#K26" title="SIN_COS:func.26">exp_R</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <font color="Maroon" title="c1">i</font></span>)</span></span><br/><div class="add"><span class="kw">then </span> <a href="sin_cos.html#K26" title="SIN_COS:func.26">exp_R</a> <font color="Maroon" title="c2">x</font> = 
 <a href="sin_cos.html#K26" title="SIN_COS:func.26">exp_R</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <font color="Maroon" title="c1">i</font></span>)</span> <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <font color="Maroon" title="c1">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T87" title="XCMPLX_1:th.87">XCMPLX_1:87</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="sin_cos.html#K26" title="SIN_COS:func.26">exp_R</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <font color="Maroon" title="c1">i</font></span>)</span></span>)</span> <a href="prepower.html#K9" title="PREPOWER:func.9">#R</a> <font color="Maroon" title="c1">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="taylor_1.html#E7"><span class="lab"><font color="Green" title="E6">Lm2</font></span></a></span>
;<br/><span class="kw">hence </span><span class="p1">(<span class="default"><a href="sin_cos.html#K26" title="SIN_COS:func.26">exp_R</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="prepower.html#K9" title="PREPOWER:func.9">#R</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">i</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="sin_cos.html#K26" title="SIN_COS:func.26">exp_R</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <font color="Maroon" title="c1">i</font></span>)</span></span>)</span> <a href="prepower.html#K9" title="PREPOWER:func.9">#R</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">i</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="prepower.html#T91" title="PREPOWER:th.91">PREPOWER:91</a>, <a class="ref" href="sin_cos.html#T55" title="SIN_COS:th.55">SIN_COS:55</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="sin_cos.html#K26" title="SIN_COS:func.26">exp_R</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <font color="Maroon" title="c1">i</font></span>)</span></span>)</span> <a href="prepower.html#K9" title="PREPOWER:func.9">#R</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8_1_1"><span class="lab"><font color="Green" title="E8">A1</font></span></a>, <a class="ref" href="xcmplx_1.html#T106" title="XCMPLX_1:th.106">XCMPLX_1:106</a></span>
<br/>.= 
 <a href="sin_cos.html#K26" title="SIN_COS:func.26">exp_R</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <font color="Maroon" title="c1">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="prepower.html#T72" title="PREPOWER:th.72">PREPOWER:72</a>, <a class="ref" href="sin_cos.html#T55" title="SIN_COS:th.55">SIN_COS:55</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1_2"><span class="kw">suppose </span></a><a NAME="E1:8_1_2"/><span class="lab"><font color="Green" title="E8">A2</font></span>: 
<font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="sin_cos.html#K26" title="SIN_COS:func.26">exp_R</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="prepower.html#K9" title="PREPOWER:func.9">#R</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</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#K26" title="SIN_COS:func.26">exp_R</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <font color="Maroon" title="c1">i</font></span>)</span></span><br/><div class="add"><span class="p1">(<span class="default"><a href="sin_cos.html#K26" title="SIN_COS:func.26">exp_R</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="prepower.html#K9" title="PREPOWER:func.9">#R</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> = 
 <a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
<span class="kw">by</span> <span class="lab"><a class="ref" href="prepower.html#T71" title="PREPOWER:th.71">PREPOWER:71</a>, <a class="ref" href="sin_cos.html#T51" title="SIN_COS:th.51">SIN_COS:51</a>, <a class="ref" href="sin_cos.html#T55" title="SIN_COS:th.55">SIN_COS:55</a></span>
<br/>.= 
 <a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T49" title="XCMPLX_1:th.49">XCMPLX_1:49</a></span>
;<br/><span class="kw">hence </span><a NAME="E3:8_1_2"/>
<span class="p1">(<span class="default"><a href="sin_cos.html#K26" title="SIN_COS:func.26">exp_R</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="prepower.html#K9" title="PREPOWER:func.9">#R</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</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#K26" title="SIN_COS:func.26">exp_R</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <font color="Maroon" title="c1">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8_1_2"><span class="lab"><font color="Green" title="E8">A2</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

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