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<b>let </b><font color="Maroon">x</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">p</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>



<a NAME="E1:12"/>
 <a href="sin_cos.html#K28">exp_R</a> <font color="Maroon">x</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="sin_cos.html#T60">SIN_COS:60</a></i>;<br/>
<b>then consider </b><font color="Maroon">s</font> being   <a href="prepower.html#NM1">Rational_Sequence</a><b> such that </b><br/><a NAME="E3:12"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">s</font> is <a href="seq_2.html#V4">convergent</a>
 <b>and </b><br/><a NAME="E4:12"/><i><font color="Green">E35</font></i>: 
 <a href="seq_2.html#K2">lim</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font>
 <b>and </b><br/><a NAME="E5:12"/>
<span class="p1">(<span class="default"><a href="sin_cos.html#K28">exp_R</a> <font color="Maroon">x</font></span>)</span> <a href="prepower.html#K11">#Q</a> <font color="Maroon">s</font> is <a href="seq_2.html#V4">convergent</a>
 <b>and </b><br/><a NAME="E6:12"/><i><font color="Green">E36</font></i>: 
 <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K28">exp_R</a> <font color="Maroon">x</font></span>)</span> <a href="prepower.html#K11">#Q</a> <font color="Maroon">s</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K28">exp_R</a> <font color="Maroon">x</font></span>)</span> <a href="prepower.html#K12">#R</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="prepower.html#D8">PREPOWER:def 8</a></i>;<br/>
<div><i><font color="Green">E39</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">ii</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:12_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">ii</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 ;<br/><b>reconsider </b><font color="Maroon">i</font> = <font color="Maroon">ii</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/><a NAME="E3:12_1"/><i><font color="Green">E41</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">s</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <a href="sin_cos.html#K27">exp_R</a> 
 <b>by </b><i><a class="ref" href="sin_cos.html#T51">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">exp_R</a> <font color="Maroon">x</font></span>)</span> <a href="prepower.html#K11">#Q</a> <font color="Maroon">s</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">ii</font> = 
<span class="p1">(<span class="default"><a href="sin_cos.html#K28">exp_R</a> <font color="Maroon">x</font></span>)</span> <a href="prepower.html#K8">#Q</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="prepower.html#K10">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i><a class="ref" href="prepower.html#D7">PREPOWER:def 7</a></i>
<br/>.= 
 <a href="sin_cos.html#K28">exp_R</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="prepower.html#K10">.</a> <font color="Maroon">i</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="sin_cos.html#K29">exp_R</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">s</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T13">SEQ_1:13</a></i>
<br/>.= 
<a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">s</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos.html#D27">SIN_COS:def 27</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="rfunct_2.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">ii</font>
<b>by </b><i>, <a class="ref" href="rfunct_2.html#T21">RFUNCT_2:21</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E8:12"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">x</font> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">s</font> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i>, <a class="ref" href="seq_2.html#T21">SEQ_2:21</a></i>;<br/>
<a NAME="E9:12"/><i><font color="Green">E46</font></i>: 
 <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">s</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">p</font>
 
<b>by </b><i>, , <a class="ref" href="seq_2.html#T22">SEQ_2:22</a></i>;<br/>
<a NAME="E10:12"/>
 <a href="sin_cos.html#K27">exp_R</a>  <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Maroon">x</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="sin_cos.html#T70">SIN_COS:70</a></i>;<br/>
<a NAME="E11:12"/><b>then </b><i><font color="Green">E48</font></i>: 
 <a href="sin_cos.html#K27">exp_R</a>  <a href="fcont_1.html#R1">is_continuous_in</a> <font color="Maroon">x</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="fdiff_1.html#T32">FDIFF_1:32</a></i>;<br/>
<a NAME="E12:12"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">s</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <a href="sin_cos.html#K27">exp_R</a> 
 
<b>by </b><i><a class="ref" href="sin_cos.html#T51">SIN_COS:51</a></i>;<br/>
<b>then </b> <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="rfunct_2.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">s</font></span>)</span></span>)</span> = 
<a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">p</font></span>)</span>
<b>by </b><i>, , , <a class="ref" href="fcont_1.html#D1">FCONT_1:def 1</a></i>
<br/>.= 
 <a href="sin_cos.html#K28">exp_R</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos.html#D27">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">exp_R</a> <font color="Maroon">x</font></span>)</span> <a href="prepower.html#K12">#R</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="sin_cos.html#K28">exp_R</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="seq_1.html#T8">SEQ_1:8</a></i>;<br/>


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