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<b>let </b><font color="Maroon" title="c1">n</font> be  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<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> ;<br/>



<b>reconsider </b><font color="Maroon" title="c3">x</font> = <font color="Maroon" title="c2">x</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E2:5"/><i><font color="Green" title="E5">A1</font></i>: 
 <a href="sin_cos.html#K29" title="SIN_COS:func.29">exp_R</a> <font color="Maroon" title="c3">x</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="sin_cos.html#T60" title="SIN_COS:th.60">SIN_COS:60</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[ <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ] <b>means </b><span class="p1">(<span class="default"><a href="sin_cos.html#K29" title="SIN_COS:func.29">exp_R</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="prepower.html#K10" title="PREPOWER:func.10">#R</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="sin_cos.html#K29" title="SIN_COS:func.29">exp_R</a> <span class="p1">(<span class="default">$1 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c3">x</font></span>)</span>;<br/>
<a NAME="E3:5"/><i><font color="Green" title="E6">A2</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<b>by </b><i><a class="txt" href="taylor_1.html#E2:5"><i><font color="Green" title="E5">A1</font></i></a>, <a class="ref" href="prepower.html#T85" title="PREPOWER:th.85">PREPOWER:85</a>, <a class="ref" href="sin_cos.html#T56" title="SIN_COS:th.56">SIN_COS:56</a></i>;<br/>
<a NAME="E4:5"/><i><font color="Green" title="E7">A3</font></i>: 
for <font color="Olive" title="b1">k</font> being <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">k</font> be  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>

<b>assume </b><a NAME="E1:5_1"/><i><font color="Green" title="E8">A4</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">k</font>]
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c5">k1</font> = <font color="Maroon" title="c4">k</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="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><a href="sin_cos.html#K29" title="SIN_COS:func.29">exp_R</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="prepower.html#K11" title="PREPOWER:func.11">#R</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K29" title="SIN_COS:func.29">exp_R</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="prepower.html#K10" title="PREPOWER:func.10">#R</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K29" title="SIN_COS:func.29">exp_R</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="prepower.html#K11" title="PREPOWER:func.11">#R</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="taylor_1.html#E2:5"><i><font color="Green" title="E5">A1</font></i></a>, <a class="ref" href="prepower.html#T89" title="PREPOWER:th.89">PREPOWER:89</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K29" title="SIN_COS:func.29">exp_R</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="prepower.html#K10" title="PREPOWER:func.10">#R</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K29" title="SIN_COS:func.29">exp_R</a> <font color="Maroon" title="c3">x</font></span>)</span>
<b>by </b><i><a class="txt" href="taylor_1.html#E2:5"><i><font color="Green" title="E5">A1</font></i></a>, <a class="ref" href="prepower.html#T86" title="PREPOWER:th.86">PREPOWER:86</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="c5">k1</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c3">x</font></span>)</span>
<b>by </b><i><a class="txt" href="taylor_1.html#E1:5_1"><i><font color="Green" title="E8">A4</font></i></a>, <a class="ref" href="sin_cos.html#T55" title="SIN_COS:th.55">SIN_COS:55</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="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c3">x</font></span>)</span>

;<br/>


</div><b>end;</b></div>
<a NAME="E5:5"/>
for <font color="Olive" title="b1">n</font> being <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="taylor_1.html#E3:5"><i><font color="Green" title="E6">A2</font></i></a>, <a class="txt" href="taylor_1.html#E4:5"><i><font color="Green" title="E7">A3</font></i></a>);<br/></i>
<b>hence </b><a NAME="E6:5"/>
<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">n</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">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">x</font></span>)</span>
 ;<br/>


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