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<span class="kw">let </span><font color="Maroon" title="c1">th</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">th</font> <a href="sin_cos.html#K23" title="SIN_COS:func.23">P_sin</a></span>)</span> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  &amp;  <a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">th</font> <a href="sin_cos.html#K23" title="SIN_COS:func.23">P_sin</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K10" title="COMSEQ_3:func.10">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">th</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a></span>)</span> <a href="sin_cos.html#K4" title="SIN_COS:func.4">ExpSeq</a></span>)</span></span>)</span> &amp;  <a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">th</font> <a href="sin_cos.html#K24" title="SIN_COS:func.24">P_cos</a></span>)</span> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  &amp;  <a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">th</font> <a href="sin_cos.html#K24" title="SIN_COS:func.24">P_cos</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K10" title="COMSEQ_3:func.10">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">th</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a></span>)</span> <a href="sin_cos.html#K4" title="SIN_COS:func.4">ExpSeq</a></span>)</span></span>)</span> )</span><br/>

<a NAME="E1:66"/><span class="lab"><font color="Green" title="E51">A1</font></span>: 
 <a href="comseq_3.html#K10" title="COMSEQ_3:func.10">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">th</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a></span>)</span> <a href="sin_cos.html#K4" title="SIN_COS:func.4">ExpSeq</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K7" title="COMSEQ_3:func.7">Re</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">th</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a></span>)</span> <a href="sin_cos.html#K4" title="SIN_COS:func.4">ExpSeq</a></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p3">(<span class="default"><a href="comseq_3.html#K8" title="COMSEQ_3:func.8">Im</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">th</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a></span>)</span> <a href="sin_cos.html#K4" title="SIN_COS:func.4">ExpSeq</a></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="comseq_3.html#T53" title="COMSEQ_3:th.53">COMSEQ_3:53</a></span>;<br/>
<a NAME="E2:66"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A2</font></span>: 
 <a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K7" title="COMSEQ_3:func.7">Re</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">th</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a></span>)</span> <a href="sin_cos.html#K4" title="SIN_COS:func.4">ExpSeq</a></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K10" title="COMSEQ_3:func.10">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">th</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a></span>)</span> <a href="sin_cos.html#K4" title="SIN_COS:func.4">ExpSeq</a></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T12" title="COMPLEX1:th.12">COMPLEX1:12</a></span>;<br/>
<a NAME="E3:66"/><span class="lab"><font color="Green" title="E53">A3</font></span>: 
 <a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K8" title="COMSEQ_3:func.8">Im</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">th</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a></span>)</span> <a href="sin_cos.html#K4" title="SIN_COS:func.4">ExpSeq</a></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K10" title="COMSEQ_3:func.10">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">th</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a></span>)</span> <a href="sin_cos.html#K4" title="SIN_COS:func.4">ExpSeq</a></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:66"><span class="lab"><font color="Green" title="E51">A1</font></span></a>, <a class="ref" href="complex1.html#T12" title="COMPLEX1:th.12">COMPLEX1:12</a></span>;<br/>
<a NAME="E4:66"/><span class="lab"><font color="Green" title="E54">A4</font></span>: 
(  <a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K7" title="COMSEQ_3:func.7">Re</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">th</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a></span>)</span> <a href="sin_cos.html#K4" title="SIN_COS:func.4">ExpSeq</a></span>)</span></span>)</span> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K7" title="COMSEQ_3:func.7">Re</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">th</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a></span>)</span> <a href="sin_cos.html#K4" title="SIN_COS:func.4">ExpSeq</a></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K10" title="COMSEQ_3:func.10">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">th</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a></span>)</span> <a href="sin_cos.html#K4" title="SIN_COS:func.4">ExpSeq</a></span>)</span></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:66"><span class="lab"><font color="Green" title="E52">A2</font></span></a>, <a class="ref" href="series_1.html#D2" title="SERIES_1:def.2">SERIES_1:def 2</a>, <a class="ref" href="series_1.html#D3" title="SERIES_1:def.3">SERIES_1:def 3</a></span>;<br/>
<a NAME="E5:66"/><span class="lab"><font color="Green" title="E55">A5</font></span>: 
(  <a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K8" title="COMSEQ_3:func.8">Im</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">th</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a></span>)</span> <a href="sin_cos.html#K4" title="SIN_COS:func.4">ExpSeq</a></span>)</span></span>)</span> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K8" title="COMSEQ_3:func.8">Im</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">th</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a></span>)</span> <a href="sin_cos.html#K4" title="SIN_COS:func.4">ExpSeq</a></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K10" title="COMSEQ_3:func.10">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">th</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a></span>)</span> <a href="sin_cos.html#K4" title="SIN_COS:func.4">ExpSeq</a></span>)</span></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:66"><span class="lab"><font color="Green" title="E53">A3</font></span></a>, <a class="ref" href="series_1.html#D2" title="SERIES_1:def.2">SERIES_1:def 2</a>, <a class="ref" href="series_1.html#D3" title="SERIES_1:def.3">SERIES_1:def 3</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E56">A6</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c2">p</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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">p</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>  implies  ex <font color="Olive" title="b1">n</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 <br/> for <font color="Olive" title="b2">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="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">k</font> holds <br/> <a href="complex1.html#K18" title="COMPLEX1:func.18">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#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">th</font> <a href="sin_cos.html#K24" title="SIN_COS:func.24">P_cos</a></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">k</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p3">(<span class="default"><a href="comseq_3.html#K10" title="COMSEQ_3:func.10">Sum</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">th</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a></span>)</span> <a href="sin_cos.html#K4" title="SIN_COS:func.4">ExpSeq</a></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">p</font> )</span><br/><span class="kw">assume </span><a NAME="E1:66_1"/>
<font color="Maroon" title="c2">p</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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">n</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 <br/> for <font color="Olive" title="b2">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="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">k</font> holds <br/> <a href="complex1.html#K18" title="COMPLEX1:func.18">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#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">th</font> <a href="sin_cos.html#K24" title="SIN_COS:func.24">P_cos</a></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">k</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p3">(<span class="default"><a href="comseq_3.html#K10" title="COMSEQ_3:func.10">Sum</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">th</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a></span>)</span> <a href="sin_cos.html#K4" title="SIN_COS:func.4">ExpSeq</a></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">p</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">n</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> <span class="kw"> such that </span><br/><a NAME="E3:66_1"/><span class="lab"><font color="Green" title="E57">A7</font></span>: 
 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" title="c3">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> holds <br/> <a href="complex1.html#K18" title="COMPLEX1:func.18">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#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p4">(<span class="default"><a href="comseq_3.html#K7" title="COMSEQ_3:func.7">Re</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c1">th</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a></span>)</span> <a href="sin_cos.html#K4" title="SIN_COS:func.4">ExpSeq</a></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p3">(<span class="default"><a href="comseq_3.html#K10" title="COMSEQ_3:func.10">Sum</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">th</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a></span>)</span> <a href="sin_cos.html#K4" title="SIN_COS:func.4">ExpSeq</a></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:66"><span class="lab"><font color="Green" title="E54">A4</font></span></a>, <a class="ref" href="seq_2.html#D7" title="SEQ_2:def.7">SEQ_2:def 7</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c4">n</font> = <font color="Maroon" title="c3">n</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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" title="c4">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> holds <br/> <a href="complex1.html#K18" title="COMPLEX1:func.18">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#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">th</font> <a href="sin_cos.html#K24" title="SIN_COS:func.24">P_cos</a></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p3">(<span class="default"><a href="comseq_3.html#K10" title="COMSEQ_3:func.10">Sum</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">th</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a></span>)</span> <a href="sin_cos.html#K4" title="SIN_COS:func.4">ExpSeq</a></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">p</font></span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">k</font> implies  <a href="complex1.html#K18" title="COMPLEX1:func.18">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#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">th</font> <a href="sin_cos.html#K24" title="SIN_COS:func.24">P_cos</a></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p3">(<span class="default"><a href="comseq_3.html#K10" title="COMSEQ_3:func.10">Sum</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">th</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a></span>)</span> <a href="sin_cos.html#K4" title="SIN_COS:func.4">ExpSeq</a></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">p</font> )</span><br/><span class="kw">assume </span><a NAME="E1:66_1_1"/><span class="lab"><font color="Green" title="E58">A8</font></span>: 
<font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="complex1.html#K18" title="COMPLEX1:func.18">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#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">th</font> <a href="sin_cos.html#K24" title="SIN_COS:func.24">P_cos</a></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p3">(<span class="default"><a href="comseq_3.html#K10" title="COMSEQ_3:func.10">Sum</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">th</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a></span>)</span> <a href="sin_cos.html#K4" title="SIN_COS:func.4">ExpSeq</a></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">p</font></span><br/><a NAME="E2:66_1_1"/>
(  <a href="complex1.html#K18" title="COMPLEX1:func.18">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#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">th</font> <a href="sin_cos.html#K24" title="SIN_COS:func.24">P_cos</a></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p3">(<span class="default"><a href="comseq_3.html#K10" title="COMSEQ_3:func.10">Sum</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">th</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a></span>)</span> <a href="sin_cos.html#K4" title="SIN_COS:func.4">ExpSeq</a></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">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#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p4">(<span class="default"><a href="comseq_3.html#K7" title="COMSEQ_3:func.7">Re</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c1">th</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a></span>)</span> <a href="sin_cos.html#K4" title="SIN_COS:func.4">ExpSeq</a></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">k</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p3">(<span class="default"><a href="comseq_3.html#K10" title="COMSEQ_3:func.10">Sum</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">th</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a></span>)</span> <a href="sin_cos.html#K4" title="SIN_COS:func.4">ExpSeq</a></span>)</span></span>)</span></span>)</span></span>)</span> &amp; 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c5">k</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="sin_cos.html#T35" target="_self" title="SIN_COS:th.35">Th38</a></span>;<br/><span class="kw">hence </span><a NAME="E3:66_1_1"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">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#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">th</font> <a href="sin_cos.html#K24" title="SIN_COS:func.24">P_cos</a></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p3">(<span class="default"><a href="comseq_3.html#K10" title="COMSEQ_3:func.10">Sum</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">th</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a></span>)</span> <a href="sin_cos.html#K4" title="SIN_COS:func.4">ExpSeq</a></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:66_1"><span class="lab"><font color="Green" title="E57">A7</font></span></a>, <a class="txt" href="#E1:66_1_1"><span class="lab"><font color="Green" title="E58">A8</font></span></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E5:66_1"/>
 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" title="c4">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> holds <br/> <a href="complex1.html#K18" title="COMPLEX1:func.18">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#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">th</font> <a href="sin_cos.html#K24" title="SIN_COS:func.24">P_cos</a></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p3">(<span class="default"><a href="comseq_3.html#K10" title="COMSEQ_3:func.10">Sum</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">th</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a></span>)</span> <a href="sin_cos.html#K4" title="SIN_COS:func.4">ExpSeq</a></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E7:66"/><span class="kw">then </span>
 <a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">th</font> <a href="sin_cos.html#K24" title="SIN_COS:func.24">P_cos</a></span>)</span> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_2.html#D6" title="SEQ_2:def.6">SEQ_2:def 6</a></span>;<br/>
<a NAME="E8:66"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A9</font></span>: 
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">th</font> <a href="sin_cos.html#K24" title="SIN_COS:func.24">P_cos</a></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K10" title="COMSEQ_3:func.10">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">th</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a></span>)</span> <a href="sin_cos.html#K4" title="SIN_COS:func.4">ExpSeq</a></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:66"><span class="lab"><font color="Green" title="E56">A6</font></span></a>, <a class="ref" href="seq_2.html#D7" title="SEQ_2:def.7">SEQ_2:def 7</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E58">A10</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="66_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c2">p</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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">p</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>  implies  ex <font color="Olive" title="b1">n</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 <br/> for <font color="Olive" title="b2">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="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">k</font> holds <br/> <a href="complex1.html#K18" title="COMPLEX1:func.18">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#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">th</font> <a href="sin_cos.html#K23" title="SIN_COS:func.23">P_sin</a></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">k</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p3">(<span class="default"><a href="comseq_3.html#K10" title="COMSEQ_3:func.10">Sum</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">th</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a></span>)</span> <a href="sin_cos.html#K4" title="SIN_COS:func.4">ExpSeq</a></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">p</font> )</span><br/><span class="kw">assume </span><a NAME="E1:66_2"/>
<font color="Maroon" title="c2">p</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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">n</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 <br/> for <font color="Olive" title="b2">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="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">k</font> holds <br/> <a href="complex1.html#K18" title="COMPLEX1:func.18">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#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">th</font> <a href="sin_cos.html#K23" title="SIN_COS:func.23">P_sin</a></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">k</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p3">(<span class="default"><a href="comseq_3.html#K10" title="COMSEQ_3:func.10">Sum</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">th</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a></span>)</span> <a href="sin_cos.html#K4" title="SIN_COS:func.4">ExpSeq</a></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">p</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">n</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> <span class="kw"> such that </span><br/><a NAME="E3:66_2"/><span class="lab"><font color="Green" title="E59">A11</font></span>: 
 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" title="c3">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> holds <br/> <a href="complex1.html#K18" title="COMPLEX1:func.18">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#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p4">(<span class="default"><a href="comseq_3.html#K8" title="COMSEQ_3:func.8">Im</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c1">th</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a></span>)</span> <a href="sin_cos.html#K4" title="SIN_COS:func.4">ExpSeq</a></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p3">(<span class="default"><a href="comseq_3.html#K10" title="COMSEQ_3:func.10">Sum</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">th</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a></span>)</span> <a href="sin_cos.html#K4" title="SIN_COS:func.4">ExpSeq</a></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:66"><span class="lab"><font color="Green" title="E55">A5</font></span></a>, <a class="ref" href="seq_2.html#D7" title="SEQ_2:def.7">SEQ_2:def 7</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c4">n</font> = <font color="Maroon" title="c3">n</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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" title="c4">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> holds <br/> <a href="complex1.html#K18" title="COMPLEX1:func.18">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#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">th</font> <a href="sin_cos.html#K23" title="SIN_COS:func.23">P_sin</a></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p3">(<span class="default"><a href="comseq_3.html#K10" title="COMSEQ_3:func.10">Sum</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">th</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a></span>)</span> <a href="sin_cos.html#K4" title="SIN_COS:func.4">ExpSeq</a></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">p</font></span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">k</font> implies  <a href="complex1.html#K18" title="COMPLEX1:func.18">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#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">th</font> <a href="sin_cos.html#K23" title="SIN_COS:func.23">P_sin</a></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p3">(<span class="default"><a href="comseq_3.html#K10" title="COMSEQ_3:func.10">Sum</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">th</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a></span>)</span> <a href="sin_cos.html#K4" title="SIN_COS:func.4">ExpSeq</a></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">p</font> )</span><br/><span class="kw">assume </span><a NAME="E1:66_2_1"/><span class="lab"><font color="Green" title="E60">A12</font></span>: 
<font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="complex1.html#K18" title="COMPLEX1:func.18">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#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">th</font> <a href="sin_cos.html#K23" title="SIN_COS:func.23">P_sin</a></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p3">(<span class="default"><a href="comseq_3.html#K10" title="COMSEQ_3:func.10">Sum</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">th</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a></span>)</span> <a href="sin_cos.html#K4" title="SIN_COS:func.4">ExpSeq</a></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">p</font></span><br/><a NAME="E2:66_2_1"/><span class="lab"><font color="Green" title="E61">A13</font></span>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">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#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">th</font> <a href="sin_cos.html#K23" title="SIN_COS:func.23">P_sin</a></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p3">(<span class="default"><a href="comseq_3.html#K10" title="COMSEQ_3:func.10">Sum</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">th</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a></span>)</span> <a href="sin_cos.html#K4" title="SIN_COS:func.4">ExpSeq</a></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">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#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p4">(<span class="default"><a href="comseq_3.html#K8" title="COMSEQ_3:func.8">Im</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c1">th</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a></span>)</span> <a href="sin_cos.html#K4" title="SIN_COS:func.4">ExpSeq</a></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p3">(<span class="default"><a href="comseq_3.html#K10" title="COMSEQ_3:func.10">Sum</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">th</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a></span>)</span> <a href="sin_cos.html#K4" title="SIN_COS:func.4">ExpSeq</a></span>)</span></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="sin_cos.html#T35" target="_self" title="SIN_COS:th.35">Th38</a></span>;<br/><a NAME="E3:66_2_1"/>
2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c5">k</font>
 ;<br/><a NAME="E4:66_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:66_2_1"><span class="lab"><font color="Green" title="E60">A12</font></span></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/><a NAME="E5:66_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><span class="kw">hence </span><a NAME="E6:66_2_1"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">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#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">th</font> <a href="sin_cos.html#K23" title="SIN_COS:func.23">P_sin</a></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p3">(<span class="default"><a href="comseq_3.html#K10" title="COMSEQ_3:func.10">Sum</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">th</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a></span>)</span> <a href="sin_cos.html#K4" title="SIN_COS:func.4">ExpSeq</a></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:66_2"><span class="lab"><font color="Green" title="E59">A11</font></span></a>, <a class="txt" href="#E2:66_2_1"><span class="lab"><font color="Green" title="E61">A13</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E5:66_2"/>
 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" title="c4">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> holds <br/> <a href="complex1.html#K18" title="COMPLEX1:func.18">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#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">th</font> <a href="sin_cos.html#K23" title="SIN_COS:func.23">P_sin</a></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p3">(<span class="default"><a href="comseq_3.html#K10" title="COMSEQ_3:func.10">Sum</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">th</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a></span>)</span> <a href="sin_cos.html#K4" title="SIN_COS:func.4">ExpSeq</a></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E10:66"/><span class="kw">then </span>
 <a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">th</font> <a href="sin_cos.html#K23" title="SIN_COS:func.23">P_sin</a></span>)</span> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_2.html#D6" title="SEQ_2:def.6">SEQ_2:def 6</a></span>;<br/>
<a NAME="E11:66"/><span class="kw">then </span>
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">th</font> <a href="sin_cos.html#K23" title="SIN_COS:func.23">P_sin</a></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K10" title="COMSEQ_3:func.10">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">th</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a></span>)</span> <a href="sin_cos.html#K4" title="SIN_COS:func.4">ExpSeq</a></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:66"><span class="lab"><font color="Green" title="E58">A10</font></span></a>, <a class="ref" href="seq_2.html#D7" title="SEQ_2:def.7">SEQ_2:def 7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:66"/>
(  <a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">th</font> <a href="sin_cos.html#K23" title="SIN_COS:func.23">P_sin</a></span>)</span> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  &amp;  <a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">th</font> <a href="sin_cos.html#K23" title="SIN_COS:func.23">P_sin</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K10" title="COMSEQ_3:func.10">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">th</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a></span>)</span> <a href="sin_cos.html#K4" title="SIN_COS:func.4">ExpSeq</a></span>)</span></span>)</span> &amp;  <a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">th</font> <a href="sin_cos.html#K24" title="SIN_COS:func.24">P_cos</a></span>)</span> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  &amp;  <a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">th</font> <a href="sin_cos.html#K24" title="SIN_COS:func.24">P_cos</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K10" title="COMSEQ_3:func.10">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">th</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a></span>)</span> <a href="sin_cos.html#K4" title="SIN_COS:func.4">ExpSeq</a></span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:66"><span class="lab"><font color="Green" title="E56">A6</font></span></a>, <a class="txt" href="#E8:66"><span class="lab"><font color="Green" title="E57">A9</font></span></a>, <a class="txt" href="#E9:66"><span class="lab"><font color="Green" title="E58">A10</font></span></a>, <a class="ref" href="seq_2.html#D6" title="SEQ_2:def.6">SEQ_2:def 6</a>, <a class="ref" href="series_1.html#D3" title="SERIES_1:def.3">SERIES_1:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
