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<div class="add">

<b>let </b><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b>$1 <a href="sin_cos.html#K4" title="SIN_COS:func.4">!c</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 <a href="sin_cos.html#K5" title="SIN_COS:func.5">!</a> ;<br/>
<a NAME="E1:69"/><i><font color="Green" title="E55">A1</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="ref" href="sin_cos.html#T1" target="_self" title="SIN_COS:th.1">Th1</a>, <a class="ref" href="newton.html#T18" title="NEWTON:th.18">NEWTON:18</a></i>;<br/>
<a NAME="E2:69"/><i><font color="Green" title="E56">A2</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</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#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:69_1"/>
<font color="Maroon" title="c2">k</font> <a href="sin_cos.html#K4" title="SIN_COS:func.4">!c</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">k</font> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!</a> 
 ;<br/>

<b>hence </b><span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="sin_cos.html#K4" title="SIN_COS:func.4">!c</a>  = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="sin_cos.html#T1" target="_self" title="SIN_COS:th.1">Th1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!</a> 
<b>by </b><i><a class="ref" href="newton.html#T21" title="NEWTON:th.21">NEWTON:21</a></i>
;<br/><br/>


</div><b>end;</b></div>
<a NAME="E3:69"/>
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="sin_cos.html#E1:69"><i><font color="Green" title="E55">A1</font></i></a>, <a class="txt" href="sin_cos.html#E2:69"><i><font color="Green" title="E56">A2</font></i></a>);<br/></i>
<b>hence </b><a NAME="E4:69"/>
<font color="Maroon" title="c1">n</font> <a href="sin_cos.html#K4" title="SIN_COS:func.4">!c</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!</a> 
 ;<br/>


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