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<b>let </b><font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c2">p</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/>



<a NAME="E1:12"/>
 <a href="sin_cos.html#K28" title="SIN_COS:func.28">exp_R</a> <font color="Maroon" title="c1">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/>
<b>then consider </b><font color="Maroon" title="c3">s</font> being   <a href="prepower.html#NM1" title="PREPOWER:NM.1">Rational_Sequence</a><b> such that </b><br/><a NAME="E3:12"/><i><font color="Green" title="E12">A1</font></i>: 
<font color="Maroon" title="c3">s</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 <b>and </b><br/><a NAME="E4:12"/><i><font color="Green" title="E13">A2</font></i>: 
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c3">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font>
 <b>and </b><br/><a NAME="E5:12"/>
<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#K11" title="PREPOWER:func.11">#Q</a> <font color="Maroon" title="c3">s</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 <b>and </b><br/><a NAME="E6:12"/><i><font color="Green" title="E14">A3</font></i>: 
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</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="c1">x</font></span>)</span> <a href="prepower.html#K11" title="PREPOWER:func.11">#Q</a> <font color="Maroon" title="c3">s</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="c1">x</font></span>)</span> <a href="prepower.html#K12" title="PREPOWER:func.12">#R</a> <font color="Maroon" title="c2">p</font>
 <b>by </b><i><a class="ref" href="prepower.html#D8" title="PREPOWER:def.8">PREPOWER:def 8</a></i>;<br/>
<div><i><font color="Green" title="E15">A4</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">ii</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:12_1"/><i><font color="Green" title="E16">A5</font></i>: 
<font color="Maroon" title="c4">ii</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 ;<br/><b>reconsider </b><font color="Maroon" title="c5">i</font> = <font color="Maroon" title="c4">ii</font> as    <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>by </b><i><a class="txt" href="taylor_1.html#E1:12_1"><i><font color="Green" title="E16">A5</font></i></a></i>;<br/><a NAME="E3:12_1"/><i><font color="Green" title="E17">A6</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="seq_1.html#K14" title="SEQ_1:func.14">(#)</a> <font color="Maroon" title="c3">s</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a> 
 <b>by </b><i><a class="ref" href="sin_cos.html#T51" title="SIN_COS:th.51">SIN_COS:51</a></i>;<br/><b>thus </b><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#K11" title="PREPOWER:func.11">#Q</a> <font color="Maroon" title="c3">s</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">ii</font> = 
<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#K8" title="PREPOWER:func.8">#Q</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="prepower.html#K10" title="PREPOWER:func.10">.</a> <font color="Maroon" title="c5">i</font></span>)</span>
<b>by </b><i><a class="ref" href="prepower.html#D7" title="PREPOWER:def.7">PREPOWER:def 7</a></i>
<br/>.= 
 <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="c3">s</font> <a href="prepower.html#K10" title="PREPOWER:func.10">.</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">x</font></span>)</span>
<b>by </b><i><a class="ref" href="taylor_1.html#T7" target="_self" title="TAYLOR_1:th.7">Th7</a></i>
<br/>.= 
 <a href="sin_cos.html#K29" title="SIN_COS:func.29">exp_R</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="seq_1.html#K14" title="SEQ_1:func.14">(#)</a> <font color="Maroon" title="c3">s</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">i</font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T13" title="SEQ_1:th.13">SEQ_1:13</a></i>
<br/>.= 
<a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="seq_1.html#K14" title="SEQ_1:func.14">(#)</a> <font color="Maroon" title="c3">s</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">i</font></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos.html#D27" title="SIN_COS:def.27">SIN_COS:def 27</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="seq_1.html#K14" title="SEQ_1:func.14">(#)</a> <font color="Maroon" title="c3">s</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">ii</font>
<b>by </b><i><a class="txt" href="taylor_1.html#E3:12_1"><i><font color="Green" title="E17">A6</font></i></a>, <a class="ref" href="rfunct_2.html#T21" title="RFUNCT_2:th.21">RFUNCT_2:21</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E8:12"/><i><font color="Green" title="E16">A7</font></i>: 
<font color="Maroon" title="c1">x</font> <a href="seq_1.html#K14" title="SEQ_1:func.14">(#)</a> <font color="Maroon" title="c3">s</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 
<b>by </b><i><a class="txt" href="taylor_1.html#E3:12"><i><font color="Green" title="E12">A1</font></i></a>, <a class="ref" href="seq_2.html#T21" title="SEQ_2:th.21">SEQ_2:21</a></i>;<br/>
<a NAME="E9:12"/><i><font color="Green" title="E17">A8</font></i>: 
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="seq_1.html#K14" title="SEQ_1:func.14">(#)</a> <font color="Maroon" title="c3">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">p</font>
 
<b>by </b><i><a class="txt" href="taylor_1.html#E3:12"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="taylor_1.html#E4:12"><i><font color="Green" title="E13">A2</font></i></a>, <a class="ref" href="seq_2.html#T22" title="SEQ_2:th.22">SEQ_2:22</a></i>;<br/>
<a NAME="E10:12"/>
 <a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">p</font>
 
<b>by </b><i><a class="ref" href="sin_cos.html#T70" title="SIN_COS:th.70">SIN_COS:70</a></i>;<br/>
<a NAME="E11:12"/><b>then </b><i><font color="Green" title="E18">A9</font></i>: 
 <a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="fcont_1.html#R1" title="FCONT_1:pred.1">is_continuous_in</a> <font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">p</font>
 
<b>by </b><i><a class="ref" href="fdiff_1.html#T32" title="FDIFF_1:th.32">FDIFF_1:32</a></i>;<br/>
<a NAME="E12:12"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="seq_1.html#K14" title="SEQ_1:func.14">(#)</a> <font color="Maroon" title="c3">s</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a> 
 
<b>by </b><i><a class="ref" href="sin_cos.html#T51" title="SIN_COS:th.51">SIN_COS:51</a></i>;<br/>
<b>then </b> <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="seq_1.html#K14" title="SEQ_1:func.14">(#)</a> <font color="Maroon" title="c3">s</font></span>)</span></span>)</span> = 
<a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">p</font></span>)</span>
<b>by </b><i><a class="txt" href="taylor_1.html#E8:12"><i><font color="Green" title="E16">A7</font></i></a>, <a class="txt" href="taylor_1.html#E9:12"><i><font color="Green" title="E17">A8</font></i></a>, <a class="txt" href="taylor_1.html#E11:12"><i><font color="Green" title="E18">A9</font></i></a>, <a class="ref" href="fcont_1.html#D1" title="FCONT_1:def.1">FCONT_1:def 1</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">p</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">x</font></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos.html#D27" title="SIN_COS:def.27">SIN_COS:def 27</a></i>
;<br/>
<b>hence </b><a NAME="E14:12"/>
<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#K12" title="PREPOWER:func.12">#R</a> <font color="Maroon" title="c2">p</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="c2">p</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">x</font></span>)</span>
 <b>by </b><i><a class="txt" href="taylor_1.html#E6:12"><i><font color="Green" title="E14">A3</font></i></a>, <a class="txt" href="taylor_1.html#E7:12"><i><font color="Green" title="E15">A4</font></i></a>, <a class="ref" href="seq_1.html#T8" title="SEQ_1:th.8">SEQ_1:8</a></i>;<br/>


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