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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>; <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">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#K28" title="SIN_COS:func.28">exp_R</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="prepower.html#K10" title="PREPOWER:func.10">#R</a> <font color="Maroon" title="c1">i</font> <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="c1">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">x</font></span>)</span></span><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> ; <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="c2">x</font></span>)</span> <a href="prepower.html#K10" title="PREPOWER:func.10">#R</a> <font color="Maroon" title="c1">i</font> <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="c1">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">x</font></span>)</span></span><br/>



<b>consider </b><font color="Maroon" title="c3">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E2:7"/><i><font color="Green" title="E7">A1</font></i>: 
( <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font> or <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c3">n</font> )
 <b>by </b><i><a class="ref" href="int_1.html#T8" title="INT_1:th.8">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" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font> or <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c3">n</font> )
 </span><b>by </b><i><a class="txt" href="taylor_1.html#E2:7"><i><font color="Green" title="E7">A1</font></i></a></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" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font>
 ; <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="c2">x</font></span>)</span> <a href="prepower.html#K10" title="PREPOWER:func.10">#R</a> <font color="Maroon" title="c1">i</font> <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="c1">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">x</font></span>)</span></span><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" title="SIN_COS:func.28">exp_R</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="prepower.html#K10" title="PREPOWER:func.10">#R</a> <font color="Maroon" title="c1">i</font> <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="c1">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">x</font></span>)</span>
 <b>by </b><i><a class="txt" href="taylor_1.html#E5"><i><font color="Green" title="E4">Lm1</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><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" title="E8">A2</font></i>: 
<font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c3">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="sin_cos.html#K28" title="SIN_COS:func.28">exp_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> <font color="Maroon" title="c2">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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#K10" title="PREPOWER:func.10">#R</a> <font color="Maroon" title="c1">i</font></span><br/><div class="add"><b>hence </b> <a href="sin_cos.html#K28" title="SIN_COS:func.28">exp_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> <font color="Maroon" title="c2">x</font></span>)</span> = 
 <a href="sin_cos.html#K28" title="SIN_COS:func.28">exp_R</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span>

<br/>.= 
1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <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="c3">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="taylor_1.html#T4" target="_self" title="TAYLOR_1:th.4">Th4</a></i>
<br/>.= 
1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <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="c2">x</font></span>)</span> <a href="prepower.html#K10" title="PREPOWER:func.10">#R</a> <font color="Maroon" title="c3">n</font></span>)</span>
<b>by </b><i><a class="txt" href="taylor_1.html#E5"><i><font color="Green" title="E4">Lm1</font></i></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#K10" title="PREPOWER:func.10">#R</a> <font color="Maroon" title="c1">i</font>
<b>by </b><i><a class="txt" href="taylor_1.html#E1:7_1_1_2"><i><font color="Green" title="E8">A2</font></i></a>, <a class="ref" href="prepower.html#T90" title="PREPOWER:th.90">PREPOWER:90</a>, <a class="ref" href="sin_cos.html#T60" title="SIN_COS:th.60">SIN_COS:60</a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:7"/>
<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#K10" title="PREPOWER:func.10">#R</a> <font color="Maroon" title="c1">i</font> <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="c1">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">x</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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