<?xml version="1.0"?>
<div class="add">

<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>assume </b><a NAME="E1:29"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<a NAME="E2:29"/>
 <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="sin_cos.html#K25">P_sin</a> </span>)</span> <a href="hidden.html#R1">=</a> <a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="sin_cos.html#T40">SIN_COS:40</a></i>;<br/>
<a NAME="E3:29"/><b>then </b><i><font color="Green">E32</font></i>: 
 <a href="seq_2.html#K2">lim</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="hidden.html#R1">=</a> <a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="series_1.html#D3">SERIES_1:def 3</a></i>;<br/>
<a NAME="E4:29"/><i><font color="Green">E34</font></i>: 
 <a href="series_1.html#K1">Partial_Sums</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="sin_cos.html#K25">P_sin</a> </span>)</span> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i><a class="ref" href="sin_cos.html#T39">SIN_COS:39</a></i>;<br/>
<b>set </b><font color="Maroon">g</font> = <a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>;<br/>
<a NAME="E5:29"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">p</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 holds <br/> ex <font color="Olive">n1</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>for <font color="Olive">m1</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">n1</font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><a href="taylor_2.html#K1" target="_self">Maclaurin</a> <a href="sin_cos.html#K19">sin</a> ,<span class="p5"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p0">(<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> <font color="Olive">m1</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">p</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1"><b>proof </b></a><div class="add">

<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/>

<b>assume </b><a NAME="E1:29_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<b>consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:29_1"/><i><font color="Green">E48</font></i>: 
for <font color="Olive">m</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4">(<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> <font color="Olive">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="seq_2.html#D7">SEQ_2:def 7</a>, , , </i>;<br/>
<b>set </b><font color="Maroon">n1</font> = <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 2;<br/>
<a NAME="E4:29_1"/>
for <font color="Olive">m1</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 2 holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><a href="taylor_2.html#K1" target="_self">Maclaurin</a> <a href="sin_cos.html#K19">sin</a> ,<span class="p5"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p0">(<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> <font color="Olive">m1</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">m1</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:29_1_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">m1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 2
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:29_1_1_1"/>
( <font color="Maroon">m1</font> is <a href="abian.html#V1">even</a> or  not <font color="Maroon">m1</font> is <a href="abian.html#V1">even</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_1_1"><b>suppose </b></a><a NAME="E1:29_1_1_1_1"/>
<font color="Maroon">m1</font> is <a href="abian.html#V1">even</a>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">j</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:29_1_1_1_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">m1</font> <a href="hidden.html#R1">=</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="abian.html#D2">ABIAN:def 2</a></i>;<br/><a NAME="E4:29_1_1_1_1"/><i><font color="Green">E58</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K2">*</a> 2</span>)</span> <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K2">*</a> 2</span>)</span> <a href="real_1.html#K6">/</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T74">XREAL_1:74</a>, <a class="ref" href="taylor_2.html#T1" target="_self">Th1</a>, </i>;<br/><a NAME="E5:29_1_1_1_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E6:29_1_1_1_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">j</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/><a NAME="E7:29_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E8:29_1_1_1_1"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4">(<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="p3">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font>
 <b>by </b><i>, <a class="txt" href="taylor_2.html#E1:29"><i><font color="Green">E31</font></i></a></i>;<br/><b>hence </b><a NAME="E9:29_1_1_1_1"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><a href="taylor_2.html#K1" target="_self">Maclaurin</a> <a href="sin_cos.html#K19">sin</a> ,<span class="p5"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p0">(<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> <font color="Maroon">m1</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font>
 <b>by </b><i>, , , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_1_2"><b>suppose </b></a><a NAME="E1:29_1_1_1_2"/>
not <font color="Maroon">m1</font> is <a href="abian.html#V1">even</a>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">j</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:29_1_1_1_2"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">m1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">j</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="abian.html#T9">ABIAN:9</a></i>;<br/><a NAME="E4:29_1_1_1_2"/><i><font color="Green">E62</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K2">*</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K2">*</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E5:29_1_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K2">*</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">j</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a>, <a class="ref" href="taylor_2.html#T1" target="_self">Th1</a>, </i>;<br/><a NAME="E6:29_1_1_1_2"/><b>then </b>
<font color="Maroon">n</font> <a href="nat_1.html#K2">*</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> <a href="nat_1.html#K2">*</a> 2
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E7:29_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K2">*</a> 2</span>)</span> <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K2">*</a> 2</span>)</span> <a href="real_1.html#K6">/</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T74">XREAL_1:74</a></i>;<br/><a NAME="E8:29_1_1_1_2"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4">(<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> <font color="Maroon">j</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E9:29_1_1_1_2"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><a href="taylor_2.html#K1" target="_self">Maclaurin</a> <a href="sin_cos.html#K19">sin</a> ,<span class="p5"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p0">(<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> <font color="Maroon">m1</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E5:29_1"/>
 ex <font color="Olive">n1</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>for <font color="Olive">m1</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">n1</font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><a href="taylor_2.html#K1" target="_self">Maclaurin</a> <a href="sin_cos.html#K19">sin</a> ,<span class="p5"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p0">(<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> <font color="Olive">m1</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E6:29"/><b>then </b><i><font color="Green">E64</font></i>: 
 <a href="series_1.html#K1">Partial_Sums</a> <span class="p1">(<span class="default"><a href="taylor_2.html#K1" target="_self">Maclaurin</a> <a href="sin_cos.html#K19">sin</a> ,<span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<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> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i><a class="ref" href="seq_2.html#D6">SEQ_2:def 6</a></i>;<br/>
<a NAME="E7:29"/><i><font color="Green">E65</font></i>: 
 <a href="seq_2.html#K2">lim</a> <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="hidden.html#R1">=</a> <a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
 
<b>by </b><i>, <a class="ref" href="seq_2.html#D7">SEQ_2:def 7</a>, </i>;<br/>
<a NAME="E8:29"/>
 <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="sin_cos.html#K26">P_cos</a> </span>)</span> <a href="hidden.html#R1">=</a> <a href="sin_cos.html#K22">cos</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="sin_cos.html#T40">SIN_COS:40</a></i>;<br/>
<a NAME="E9:29"/><b>then </b><i><font color="Green">E66</font></i>: 
 <a href="seq_2.html#K2">lim</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="hidden.html#R1">=</a> <a href="sin_cos.html#K22">cos</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="series_1.html#D3">SERIES_1:def 3</a></i>;<br/>
<a NAME="E10:29"/><i><font color="Green">E67</font></i>: 
 <a href="series_1.html#K1">Partial_Sums</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="sin_cos.html#K26">P_cos</a> </span>)</span> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i><a class="ref" href="sin_cos.html#T39">SIN_COS:39</a></i>;<br/>
<b>set </b><font color="Maroon">g</font> = <a href="sin_cos.html#K22">cos</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>;<br/>
<a NAME="E11:29"/><i><font color="Green">E68</font></i>: 
for <font color="Olive">p</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 holds <br/> ex <font color="Olive">n1</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>for <font color="Olive">m1</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">n1</font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><a href="taylor_2.html#K1" target="_self">Maclaurin</a> <a href="sin_cos.html#K22">cos</a> ,<span class="p5"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p0">(<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> <font color="Olive">m1</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">p</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2"><b>proof </b></a><div class="add">

<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/>

<b>assume </b><a NAME="E1:29_2"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<b>consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:29_2"/><i><font color="Green">E70</font></i>: 
for <font color="Olive">m</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4">(<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="Olive">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="seq_2.html#D7">SEQ_2:def 7</a>, <a class="ref" href="taylor_2.html#T1" target="_self">Th1</a>, , </i>;<br/>
<b>set </b><font color="Maroon">n1</font> = <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1;<br/>
<a NAME="E4:29_2"/>
for <font color="Olive">m1</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><a href="taylor_2.html#K1" target="_self">Maclaurin</a> <a href="sin_cos.html#K22">cos</a> ,<span class="p5"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p0">(<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> <font color="Olive">m1</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">m1</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:29_2_1"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">m1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:29_2_1_1"/>
( <font color="Maroon">m1</font> is <a href="abian.html#V1">even</a> or  not <font color="Maroon">m1</font> is <a href="abian.html#V1">even</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_2_1_1_1"><b>suppose </b></a><a NAME="E1:29_2_1_1_1"/>
<font color="Maroon">m1</font> is <a href="abian.html#V1">even</a>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">j</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:29_2_1_1_1"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">m1</font> <a href="hidden.html#R1">=</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="abian.html#D2">ABIAN:def 2</a></i>;<br/><a NAME="E4:29_2_1_1_1"/><i><font color="Green">E87</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K2">*</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K2">*</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/><a NAME="E5:29_2_1_1_1"/>
<font color="Maroon">n</font> <a href="nat_1.html#K2">*</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> <a href="nat_1.html#K2">*</a> 2
 <b>by </b><i><a class="txt" href="taylor_2.html#E4:29"><i><font color="Green">E34</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="txt" href="taylor_2.html#E3:29"><i><font color="Green">E32</font></i></a>, </i>;<br/><a NAME="E6:29_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K2">*</a> 2</span>)</span> <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K2">*</a> 2</span>)</span> <a href="real_1.html#K6">/</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T74">XREAL_1:74</a></i>;<br/><a NAME="E7:29_2_1_1_1"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4">(<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">j</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="txt" href="taylor_2.html#E1:29"><i><font color="Green">E31</font></i></a></i>;<br/><b>hence </b><a NAME="E8:29_2_1_1_1"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><a href="taylor_2.html#K1" target="_self">Maclaurin</a> <a href="sin_cos.html#K22">cos</a> ,<span class="p5"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p0">(<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> <font color="Maroon">m1</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_2_1_1_2"><b>suppose </b></a><a NAME="E1:29_2_1_1_2"/>
not <font color="Maroon">m1</font> is <a href="abian.html#V1">even</a>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">j</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:29_2_1_1_2"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">m1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">j</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="abian.html#T9">ABIAN:9</a></i>;<br/><a NAME="E4:29_2_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K2">*</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K2">*</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a>, <a class="txt" href="taylor_2.html#E3:29"><i><font color="Green">E32</font></i></a>, </i>;<br/><a NAME="E5:29_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K2">*</a> 2</span>)</span> <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K2">*</a> 2</span>)</span> <a href="real_1.html#K6">/</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T74">XREAL_1:74</a></i>;<br/><a NAME="E6:29_2_1_1_2"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4">(<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">j</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="txt" href="taylor_2.html#E1:29"><i><font color="Green">E31</font></i></a></i>;<br/><b>hence </b><a NAME="E7:29_2_1_1_2"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><a href="taylor_2.html#K1" target="_self">Maclaurin</a> <a href="sin_cos.html#K22">cos</a> ,<span class="p5"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p0">(<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> <font color="Maroon">m1</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E5:29_2"/>
 ex <font color="Olive">n1</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>for <font color="Olive">m1</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">n1</font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><a href="taylor_2.html#K1" target="_self">Maclaurin</a> <a href="sin_cos.html#K22">cos</a> ,<span class="p5"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p0">(<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> <font color="Olive">m1</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E12:29"/><b>then </b>
 <a href="series_1.html#K1">Partial_Sums</a> <span class="p1">(<span class="default"><a href="taylor_2.html#K1" target="_self">Maclaurin</a> <a href="sin_cos.html#K22">cos</a> ,<span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<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> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i><a class="ref" href="seq_2.html#D6">SEQ_2:def 6</a></i>;<br/>
<a NAME="E13:29"/><b>then </b>
 <a href="seq_2.html#K2">lim</a> <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="hidden.html#R1">=</a> <a href="sin_cos.html#K22">cos</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="seq_2.html#D7">SEQ_2:def 7</a>, </i>;<br/>
<b>hence </b><a NAME="E14:29"/>
(  <a href="series_1.html#K1">Partial_Sums</a> <span class="p1">(<span class="default"><a href="taylor_2.html#K1" target="_self">Maclaurin</a> <a href="sin_cos.html#K19">sin</a> ,<span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<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> is <a href="seq_2.html#V4">convergent</a> &amp; <a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><a href="taylor_2.html#K1" target="_self">Maclaurin</a> <a href="sin_cos.html#K19">sin</a> ,<span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<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> &amp;  <a href="series_1.html#K1">Partial_Sums</a> <span class="p1">(<span class="default"><a href="taylor_2.html#K1" target="_self">Maclaurin</a> <a href="sin_cos.html#K22">cos</a> ,<span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<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> is <a href="seq_2.html#V4">convergent</a> &amp; <a href="sin_cos.html#K22">cos</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><a href="taylor_2.html#K1" target="_self">Maclaurin</a> <a href="sin_cos.html#K22">cos</a> ,<span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<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> )
 <b>by </b><i>, , , <a class="ref" href="series_1.html#D3">SERIES_1:def 3</a>, <a class="ref" href="seq_2.html#D6">SEQ_2:def 6</a></i>;<br/>


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