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<div><i><font color="Green">E47</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:18_1"/>
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k</font>
 ;<br/><a NAME="E2:18_1"/><b>then </b>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="int_1.html#T20">INT_1:20</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K3">+</a> 1
<b>by </b><i><a class="ref" href="binarith.html#T50">BINARITH:50</a></i>
<br/>.= 
<font color="Maroon">k</font>

;<br/><b>hence </b><a NAME="E4:18_1"/>
<font color="Maroon">k</font> <a href="sin_cos.html#K5">!</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="sin_cos.html#K5">!</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="newton.html#T21">NEWTON:21</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:18_2"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/><a NAME="E2:18_2"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/><a NAME="E3:18_2"/><b>then </b>
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font> = 
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">k</font>
<b>by </b><i><a class="ref" href="binarith.html#T50">BINARITH:50</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="real_1.html#K5">-</a> <font color="Maroon">k</font></span>)</span> <a href="real_1.html#K3">+</a> 1

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>
;<br/><b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font></span>)</span> <a href="sin_cos.html#K5">!</a>  = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font></span>)</span> <a href="sin_cos.html#K5">!</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T21">NEWTON:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font></span>)</span> <a href="sin_cos.html#K5">!</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="real_1.html#K5">-</a> <font color="Maroon">k</font></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span>
<b>by </b><i>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font></span>)</span> <a href="sin_cos.html#K5">!</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">k</font></span>)</span>

;<br/><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:18"/>
( ( for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">k</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="sin_cos.html#K5">!</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Olive">k</font> <a href="sin_cos.html#K5">!</a>  ) &amp; ( for <font color="Olive">m</font>, <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive">m</font> <a href="binarith.html#K3">-'</a> <font color="Olive">k</font></span>)</span> <a href="sin_cos.html#K5">!</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Olive">k</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Olive">k</font></span>)</span> <a href="sin_cos.html#K5">!</a>  ) )
 <b>by </b><i/>;<br/>


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