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<b>let </b><font color="Maroon" title="c1">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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">m</font>, <font color="Olive" title="b2">n</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> holds  <span class="p1">(<span class="default"><a href="sin_cos.html#K28" title="SIN_COS:func.28">exp_R</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="prepower.html#K10" title="PREPOWER:func.10">#R</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">m</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Olive" title="b2">n</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"><span class="p2">(<span class="default"><font color="Olive" title="b1">m</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Olive" title="b2">n</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">x</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c2">m</font>, <font color="Maroon" title="c3">n</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="sin_cos.html#K28" title="SIN_COS:func.28">exp_R</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="prepower.html#K10" title="PREPOWER:func.10">#R</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c3">n</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"><span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">x</font></span>)</span></span><br/>



<b>thus </b> <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">m</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">x</font></span>)</span> = 
 <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="c1">x</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">m</font></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T76" title="XCMPLX_1:th.76">XCMPLX_1:76</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="c1">x</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span> <a href="prepower.html#K10" title="PREPOWER:func.10">#R</a> <font color="Maroon" title="c2">m</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/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K28" title="SIN_COS:func.28">exp_R</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="prepower.html#K10" title="PREPOWER:func.10">#R</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span> <a href="prepower.html#K10" title="PREPOWER:func.10">#R</a> <font color="Maroon" title="c2">m</font>
<b>by </b><i><a class="ref" href="taylor_1.html#T5" target="_self" title="TAYLOR_1:th.5">Th5</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="c1">x</font></span>)</span> <a href="prepower.html#K10" title="PREPOWER:func.10">#R</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c2">m</font></span>)</span>
<b>by </b><i><a class="ref" href="prepower.html#T105" title="PREPOWER:th.105">PREPOWER:105</a>, <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="c1">x</font></span>)</span> <a href="prepower.html#K10" title="PREPOWER:func.10">#R</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T76" title="XCMPLX_1:th.76">XCMPLX_1:76</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="c1">x</font></span>)</span> <a href="prepower.html#K10" title="PREPOWER:func.10">#R</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c3">n</font></span>)</span>

; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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