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<b>let </b><font color="Maroon">r</font> be   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">x</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:27"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>and </b><br/><a NAME="E2:27"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<a NAME="E3:27"/>
2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 2 <a href="nat_1.html#K2">*</a> 0
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/>
<a NAME="E4:27"/><b>then </b><i><font color="Green">E34</font></i>: 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> 1 is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="nat_1.html#T60">NAT_1:60</a></i>;<br/>
<a NAME="E5:27"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">m</font> <a href="xcmplx_0.html#K6">-</a> 1 is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="nat_1.html#T60">NAT_1:60</a>, </i>;<br/>
<i><font color="Green">E42</font></i>: <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><a href="taylor_2.html#K1" target="_self">Maclaurin</a> <a href="sin_cos.html#K19">sin</a> ,<span class="p3"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span>,<font color="Maroon">x</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><a href="taylor_2.html#K1" target="_self">Maclaurin</a> <a href="sin_cos.html#K19">sin</a> ,<span class="p4"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span>,<font color="Maroon">x</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="taylor_2.html#K1" target="_self">Maclaurin</a> <a href="sin_cos.html#K19">sin</a> ,<span class="p3"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span>,<font color="Maroon">x</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="series_1.html#D1">SERIES_1:def 1</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><a href="taylor_2.html#K1" target="_self">Maclaurin</a> <a href="sin_cos.html#K19">sin</a> ,<span class="p4"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span>,<font color="Maroon">x</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 0
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><a href="taylor_2.html#K1" target="_self">Maclaurin</a> <a href="sin_cos.html#K19">sin</a> ,<span class="p3"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span>,<font color="Maroon">x</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="sin_cos.html#K25">P_sin</a> </span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span>
<b>by </b><i>, , </i>
;<br/>
<span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><a href="taylor_2.html#K1" target="_self">Maclaurin</a> <a href="sin_cos.html#K22">cos</a> ,<span class="p3"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span>,<font color="Maroon">x</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><a href="taylor_2.html#K1" target="_self">Maclaurin</a> <a href="sin_cos.html#K22">cos</a> ,<span class="p4"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span>,<font color="Maroon">x</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="taylor_2.html#K1" target="_self">Maclaurin</a> <a href="sin_cos.html#K22">cos</a> ,<span class="p3"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span>,<font color="Maroon">x</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="series_1.html#D1">SERIES_1:def 1</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><a href="taylor_2.html#K1" target="_self">Maclaurin</a> <a href="sin_cos.html#K22">cos</a> ,<span class="p4"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span>,<font color="Maroon">x</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="prepower.html#K3">|^</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="prepower.html#K3">|^</a> <span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span> <a href="sin_cos.html#K5">!</a> </span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><a href="taylor_2.html#K1" target="_self">Maclaurin</a> <a href="sin_cos.html#K22">cos</a> ,<span class="p4"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span>,<font color="Maroon">x</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="sin_cos.html#K26">P_cos</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos.html#D25">SIN_COS:def 25</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="sin_cos.html#K26">P_cos</a> </span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="sin_cos.html#K26">P_cos</a> </span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="sin_cos.html#K26">P_cos</a> </span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font>
<b>by </b><i><a class="ref" href="series_1.html#D1">SERIES_1:def 1</a>, </i>
;<br/>
<b>hence </b><a NAME="E8:27"/>
( <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><a href="taylor_2.html#K1" target="_self">Maclaurin</a> <a href="sin_cos.html#K19">sin</a> ,<span class="p3"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span>,<font color="Maroon">x</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="sin_cos.html#K25">P_sin</a> </span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> &amp; <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><a href="taylor_2.html#K1" target="_self">Maclaurin</a> <a href="sin_cos.html#K22">cos</a> ,<span class="p3"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span>,<font color="Maroon">x</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="sin_cos.html#K26">P_cos</a> </span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> )
 <b>by </b><i/>;<br/>


</div>
