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<span class="kw">let </span><font color="Maroon" title="c1">z</font>, <font color="Maroon" title="c2">w</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">p</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">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>  holds <br/> ex <font color="Olive" title="b2">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="b3">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="b2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">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"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p3"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K16" title="SIN_COS:func.16">Conj</a> <font color="Olive" title="b3">k</font>,<font color="Maroon" title="c1">z</font>,<font color="Maroon" title="c2">w</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b3">k</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">p</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">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"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p3"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K16" title="SIN_COS:func.16">Conj</a> <font color="Olive" title="b2">k</font>,<font color="Maroon" title="c1">z</font>,<font color="Maroon" title="c2">w</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></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="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">p</font> )</span><br/>





<span class="kw">assume </span><a NAME="E1:43"/><span class="lab"><font color="Green" title="E35">A1</font></span>: 
<font color="Maroon" title="c3">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="Red">::  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"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p3"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K16" title="SIN_COS:func.16">Conj</a> <font color="Olive" title="b2">k</font>,<font color="Maroon" title="c1">z</font>,<font color="Maroon" title="c2">w</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></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="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">p</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c4">pp</font> = <font color="Maroon" title="c3">p</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></span>;<br/>
<a NAME="E3:43"/><span class="lab"><font color="Green" title="E36">A2</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="sin_cos.html#T18" target="_self" title="SIN_COS:th.18">Th18</a></span>;<br/>
<a NAME="E4:43"/><span class="lab"><font color="Green" title="E37">A3</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c2">w</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="sin_cos.html#T18" target="_self" title="SIN_COS:th.18">Th18</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">p1</font> =  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">pp</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default">4 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p4">(<span class="default"><span class="p5"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">pp</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default">4 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p4">(<span class="default"><span class="p5"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c2">w</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span></span>)</span></span>)</span></span>)</span>;<br/>
<a NAME="E5:43"/><span class="lab"><font color="Green" title="E38">A4</font></span>: 
 <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">pp</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default">4 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p4">(<span class="default"><span class="p5"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">pp</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default">4 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p4">(<span class="default"><span class="p5"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c2">w</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="txt" href="#E3:43"><span class="lab"><font color="Green" title="E36">A2</font></span></a>, <a class="txt" href="#E4:43"><span class="lab"><font color="Green" title="E37">A3</font></span></a>, <a class="ref" href="xxreal_0.html#T15" title="XXREAL_0:th.15">XXREAL_0:15</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E39">A5</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">k</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  implies <span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="sin_cos.html#K7" title="SIN_COS:func.7">ExpSeq</a> </span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">k</font> )</span><br/><span class="kw">assume </span><a NAME="E1:43_1"/><span class="lab"><font color="Green" title="E40">A6</font></span>: 
<font color="Maroon" title="c6">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="sin_cos.html#K7" title="SIN_COS:func.7">ExpSeq</a> </span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">k</font></span><br/><span class="kw">thus </span><span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="sin_cos.html#K7" title="SIN_COS:func.7">ExpSeq</a> </span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">k</font> = 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="sin_cos.html#K7" title="SIN_COS:func.7">ExpSeq</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T18" title="VALUED_1:th.18">VALUED_1:18</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_1"><span class="lab"><font color="Green" title="E40">A6</font></span></a>, <a class="ref" href="sin_cos.html#T4" target="_self" title="SIN_COS:th.4">Th4</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E7:43"/><span class="lab"><font color="Green" title="E40">A7</font></span>: 
<span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="sin_cos.html#K7" title="SIN_COS:func.7">ExpSeq</a> </span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:43"><span class="lab"><font color="Green" title="E39">A5</font></span></a>, <a class="ref" href="funct_2.html#T18" title="FUNCT_2:th.18">FUNCT_2:18</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">n1</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="E9:43"/><span class="lab"><font color="Green" title="E41">A8</font></span>: 
for <font color="Olive" title="b1">k</font>, <font color="Olive" title="b2">l</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="c6">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Maroon" title="c6">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">l</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#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p4">(<span class="default"><span class="p5"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</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"><span class="p3">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p4">(<span class="default"><span class="p5"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">l</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">pp</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default">4 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p4">(<span class="default"><span class="p5"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">pp</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default">4 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p4">(<span class="default"><span class="p5"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c2">w</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:43"><span class="lab"><font color="Green" title="E38">A4</font></span></a>, <a class="txt" href="#E7:43"><span class="lab"><font color="Green" title="E40">A7</font></span></a>, <a class="ref" href="comseq_3.html#T4" title="COMSEQ_3:th.4">COMSEQ_3:4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">n2</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="E11:43"/><span class="lab"><font color="Green" title="E42">A9</font></span>: 
for <font color="Olive" title="b1">k</font>, <font color="Olive" title="b2">l</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="c7">n2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Maroon" title="c7">n2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">l</font> holds <br/><span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">w</font> <a href="sin_cos.html#K7" title="SIN_COS:func.7">ExpSeq</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="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">w</font> <a href="sin_cos.html#K7" title="SIN_COS:func.7">ExpSeq</a> </span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">l</font></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">pp</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default">4 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p4">(<span class="default"><span class="p5"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">pp</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default">4 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p4">(<span class="default"><span class="p5"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c2">w</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:43"><span class="lab"><font color="Green" title="E38">A4</font></span></a>, <a class="ref" href="comseq_3.html#T47" title="COMSEQ_3:th.47">COMSEQ_3:47</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">n3</font> = <font color="Maroon" title="c6">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">n2</font>;<br/>
<span class="kw">take </span><font color="Maroon" title="c9">n4</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c6">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">n2</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">n2</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="c9">n4</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"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p3"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K16" title="SIN_COS:func.16">Conj</a> <font color="Olive" title="b1">k</font>,<font color="Maroon" title="c1">z</font>,<font color="Maroon" title="c2">w</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></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="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">p</font></span><br/>

<div><span class="lab"><font color="Green" title="E43">A10</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="43_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c10">n</font>, <font color="Maroon" title="c11">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="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">l</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">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">k</font> holds <br/><span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K16" title="SIN_COS:func.16">Conj</a> <font color="Maroon" title="c11">k</font>,<font color="Maroon" title="c1">z</font>,<font color="Maroon" title="c2">w</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">l</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">w</font> <a href="sin_cos.html#K7" title="SIN_COS:func.7">ExpSeq</a> </span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">w</font> <a href="sin_cos.html#K7" title="SIN_COS:func.7">ExpSeq</a> </span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Olive" title="b1">l</font></span>)</span></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span></span><br/><div><span class="lab"><font color="Green" title="E44">A11</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="43_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c12">l</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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c12">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">k</font> implies <span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K16" title="SIN_COS:func.16">Conj</a> <font color="Maroon" title="c11">k</font>,<font color="Maroon" title="c1">z</font>,<font color="Maroon" title="c2">w</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">l</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">w</font> <a href="sin_cos.html#K7" title="SIN_COS:func.7">ExpSeq</a> </span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">w</font> <a href="sin_cos.html#K7" title="SIN_COS:func.7">ExpSeq</a> </span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c12">l</font></span>)</span></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> )</span><br/><span class="kw">assume </span><a NAME="E1:43_2_1"/><span class="lab"><font color="Green" title="E45">A12</font></span>: 
<font color="Maroon" title="c12">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K16" title="SIN_COS:func.16">Conj</a> <font color="Maroon" title="c11">k</font>,<font color="Maroon" title="c1">z</font>,<font color="Maroon" title="c2">w</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">l</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">w</font> <a href="sin_cos.html#K7" title="SIN_COS:func.7">ExpSeq</a> </span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">w</font> <a href="sin_cos.html#K7" title="SIN_COS:func.7">ExpSeq</a> </span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c12">l</font></span>)</span></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span></span><br/><span class="kw">thus </span><span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K16" title="SIN_COS:func.16">Conj</a> <font color="Maroon" title="c11">k</font>,<font color="Maroon" title="c1">z</font>,<font color="Maroon" title="c2">w</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">l</font> = 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K16" title="SIN_COS:func.16">Conj</a> <font color="Maroon" title="c11">k</font>,<font color="Maroon" title="c1">z</font>,<font color="Maroon" title="c2">w</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">l</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T18" title="VALUED_1:th.18">VALUED_1:18</a></span>
<br/>.= 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="sin_cos.html#K7" title="SIN_COS:func.7">ExpSeq</a> </span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">l</font></span>)</span> <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Partial_Sums</a> <span class="p0">(<span class="default"><font color="Maroon" title="c2">w</font> <a href="sin_cos.html#K7" title="SIN_COS:func.7">ExpSeq</a> </span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Partial_Sums</a> <span class="p0">(<span class="default"><font color="Maroon" title="c2">w</font> <a href="sin_cos.html#K7" title="SIN_COS:func.7">ExpSeq</a> </span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c12">l</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_2_1"><span class="lab"><font color="Green" title="E45">A12</font></span></a>, <a class="ref" href="sin_cos.html#D17" target="_self" title="SIN_COS:def.17">Def17</a></span>
<br/>.= 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="sin_cos.html#K7" title="SIN_COS:func.7">ExpSeq</a> </span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">l</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">w</font> <a href="sin_cos.html#K7" title="SIN_COS:func.7">ExpSeq</a> </span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">w</font> <a href="sin_cos.html#K7" title="SIN_COS:func.7">ExpSeq</a> </span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c12">l</font></span>)</span></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T151" title="COMPLEX1:th.151">COMPLEX1:151</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">l</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">w</font> <a href="sin_cos.html#K7" title="SIN_COS:func.7">ExpSeq</a> </span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">w</font> <a href="sin_cos.html#K7" title="SIN_COS:func.7">ExpSeq</a> </span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c12">l</font></span>)</span></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="sin_cos.html#T4" target="_self" title="SIN_COS:th.4">Th4</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">thus </span><a NAME="E2:43_2"/>
for <font color="Olive" title="b1">l</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">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">k</font> holds <br/><span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K16" title="SIN_COS:func.16">Conj</a> <font color="Maroon" title="c11">k</font>,<font color="Maroon" title="c1">z</font>,<font color="Maroon" title="c2">w</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">l</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">w</font> <a href="sin_cos.html#K7" title="SIN_COS:func.7">ExpSeq</a> </span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">w</font> <a href="sin_cos.html#K7" title="SIN_COS:func.7">ExpSeq</a> </span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Olive" title="b1">l</font></span>)</span></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_2"><span class="lab"><font color="Green" title="E44">A11</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E44">A13</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="43_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c10">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="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">l</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">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">k</font> holds <br/><span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K16" title="SIN_COS:func.16">Conj</a> <font color="Maroon" title="c10">k</font>,<font color="Maroon" title="c1">z</font>,<font color="Maroon" title="c2">w</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">l</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p3">(<span class="default"><span class="p4"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c2">w</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span></span>)</span></span>)</span></span><br/><div><span class="lab"><font color="Green" title="E45">A14</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="43_3_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c11">l</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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">k</font> implies <span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K16" title="SIN_COS:func.16">Conj</a> <font color="Maroon" title="c10">k</font>,<font color="Maroon" title="c1">z</font>,<font color="Maroon" title="c2">w</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c11">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c11">l</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p3">(<span class="default"><span class="p4"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c2">w</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span></span>)</span></span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E1:43_3_1"/><span class="lab"><font color="Green" title="E46">A15</font></span>: 
<font color="Maroon" title="c11">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K16" title="SIN_COS:func.16">Conj</a> <font color="Maroon" title="c10">k</font>,<font color="Maroon" title="c1">z</font>,<font color="Maroon" title="c2">w</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c11">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c11">l</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p3">(<span class="default"><span class="p4"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c2">w</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span></span>)</span></span>)</span></span><br/><a NAME="E2:43_3_1"/><span class="lab"><font color="Green" title="E47">A16</font></span>: 
<span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K16" title="SIN_COS:func.16">Conj</a> <font color="Maroon" title="c10">k</font>,<font color="Maroon" title="c1">z</font>,<font color="Maroon" title="c2">w</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c11">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c11">l</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">w</font> <a href="sin_cos.html#K7" title="SIN_COS:func.7">ExpSeq</a> </span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">w</font> <a href="sin_cos.html#K7" title="SIN_COS:func.7">ExpSeq</a> </span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c11">l</font></span>)</span></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:43"><span class="lab"><font color="Green" title="E43">A10</font></span></a>, <a class="txt" href="#E1:43_3_1"><span class="lab"><font color="Green" title="E46">A15</font></span></a></span>;<br/><a NAME="E3:43_3_1"/><span class="lab"><font color="Green" title="E48">A17</font></span>: 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">w</font> <a href="sin_cos.html#K7" title="SIN_COS:func.7">ExpSeq</a> </span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c2">w</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="sin_cos.html#T17" target="_self" title="SIN_COS:th.17">Th17</a></span>;<br/><a NAME="E4:43_3_1"/><span class="lab"><font color="Green" title="E49">A18</font></span>: 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">w</font> <a href="sin_cos.html#K7" title="SIN_COS:func.7">ExpSeq</a> </span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">w</font> <a href="sin_cos.html#K7" title="SIN_COS:func.7">ExpSeq</a> </span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c11">l</font></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c2">w</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">w</font> <a href="sin_cos.html#K7" title="SIN_COS:func.7">ExpSeq</a> </span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c11">l</font></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43_3_1"><span class="lab"><font color="Green" title="E48">A17</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><a NAME="E5:43_3_1"/><span class="lab"><font color="Green" title="E50">A19</font></span>: 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">w</font> <a href="sin_cos.html#K7" title="SIN_COS:func.7">ExpSeq</a> </span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c11">l</font></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c2">w</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="sin_cos.html#T17" target="_self" title="SIN_COS:th.17">Th17</a></span>;<br/><a NAME="E6:43_3_1"/><span class="lab"><font color="Green" title="E51">A20</font></span>: 
<span class="p1">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c2">w</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">w</font> <a href="sin_cos.html#K7" title="SIN_COS:func.7">ExpSeq</a> </span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c11">l</font></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c2">w</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c2">w</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:43_3_1"><span class="lab"><font color="Green" title="E50">A19</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><a NAME="E7:43_3_1"/><span class="lab"><font color="Green" title="E52">A21</font></span>: 
( <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">w</font> <a href="sin_cos.html#K7" title="SIN_COS:func.7">ExpSeq</a> </span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">w</font> <a href="sin_cos.html#K7" title="SIN_COS:func.7">ExpSeq</a> </span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c11">l</font></span>)</span></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">w</font> <a href="sin_cos.html#K7" title="SIN_COS:func.7">ExpSeq</a> </span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">w</font> <a href="sin_cos.html#K7" title="SIN_COS:func.7">ExpSeq</a> </span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c11">l</font></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> &amp; <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">w</font> <a href="sin_cos.html#K7" title="SIN_COS:func.7">ExpSeq</a> </span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">w</font> <a href="sin_cos.html#K7" title="SIN_COS:func.7">ExpSeq</a> </span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c11">l</font></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c2">w</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43_3_1"><span class="lab"><font color="Green" title="E49">A18</font></span></a>, <a class="txt" href="#E6:43_3_1"><span class="lab"><font color="Green" title="E51">A20</font></span></a>, <a class="ref" href="complex1.html#T143" title="COMPLEX1:th.143">COMPLEX1:143</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E8:43_3_1"/><span class="lab"><font color="Green" title="E53">A22</font></span>: 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">w</font> <a href="sin_cos.html#K7" title="SIN_COS:func.7">ExpSeq</a> </span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">w</font> <a href="sin_cos.html#K7" title="SIN_COS:func.7">ExpSeq</a> </span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c11">l</font></span>)</span></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c2">w</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:43_3_1"><span class="lab"><font color="Green" title="E52">A21</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E9:43_3_1"/><span class="lab"><font color="Green" title="E54">A23</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c11">l</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="sin_cos.html#T19" target="_self" title="SIN_COS:th.19">Th19</a></span>;<br/><span class="kw">thus </span><a NAME="E10:43_3_1"/>
<span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K16" title="SIN_COS:func.16">Conj</a> <font color="Maroon" title="c10">k</font>,<font color="Maroon" title="c1">z</font>,<font color="Maroon" title="c2">w</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c11">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c11">l</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p3">(<span class="default"><span class="p4"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c2">w</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_3_1"><span class="lab"><font color="Green" title="E47">A16</font></span></a>, <a class="txt" href="#E8:43_3_1"><span class="lab"><font color="Green" title="E53">A22</font></span></a>, <a class="txt" href="#E9:43_3_1"><span class="lab"><font color="Green" title="E54">A23</font></span></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">thus </span><a NAME="E2:43_3"/>
for <font color="Olive" title="b1">l</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">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">k</font> holds <br/><span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K16" title="SIN_COS:func.16">Conj</a> <font color="Maroon" title="c10">k</font>,<font color="Maroon" title="c1">z</font>,<font color="Maroon" title="c2">w</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">l</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p3">(<span class="default"><span class="p4"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c2">w</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_3"><span class="lab"><font color="Green" title="E45">A14</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E45">A24</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="43_4"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c10">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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">n4</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">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"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p3"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K16" title="SIN_COS:func.16">Conj</a> <font color="Maroon" title="c10">k</font>,<font color="Maroon" title="c1">z</font>,<font color="Maroon" title="c2">w</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">p</font> )</span><br/><span class="kw">assume </span><a NAME="E1:43_4"/><span class="lab"><font color="Green" title="E46">A25</font></span>: 
<font color="Maroon" title="c9">n4</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p3"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K16" title="SIN_COS:func.16">Conj</a> <font color="Maroon" title="c10">k</font>,<font color="Maroon" title="c1">z</font>,<font color="Maroon" title="c2">w</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">p</font></span><br/><a NAME="E2:43_4"/><span class="lab"><font color="Green" title="E47">A26</font></span>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">n2</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">n2</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">n2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><a NAME="E3:43_4"/><span class="lab"><font color="Green" title="E48">A27</font></span>: 
<font color="Maroon" title="c6">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">n2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_4"><span class="lab"><font color="Green" title="E46">A25</font></span></a>, <a class="txt" href="#E2:43_4"><span class="lab"><font color="Green" title="E47">A26</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E4:43_4"/><span class="lab"><font color="Green" title="E49">A28</font></span>: 
<font color="Maroon" title="c6">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">n2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><a NAME="E5:43_4"/><span class="lab"><font color="Green" title="E50">A29</font></span>: 
<font color="Maroon" title="c6">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43_4"><span class="lab"><font color="Green" title="E48">A27</font></span></a>, <a class="txt" href="#E4:43_4"><span class="lab"><font color="Green" title="E49">A28</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><div><span class="lab"><font color="Green" title="E51">A30</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="43_4_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c11">l</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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">n2</font> implies <span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K16" title="SIN_COS:func.16">Conj</a> <font color="Maroon" title="c10">k</font>,<font color="Maroon" title="c1">z</font>,<font color="Maroon" title="c2">w</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c11">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">pp</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default">4 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p5">(<span class="default"><span class="p0"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">pp</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default">4 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p5">(<span class="default"><span class="p0"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c2">w</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c11">l</font></span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E1:43_4_1"/><span class="lab"><font color="Green" title="E52">A31</font></span>: 
<font color="Maroon" title="c11">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">n2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K16" title="SIN_COS:func.16">Conj</a> <font color="Maroon" title="c10">k</font>,<font color="Maroon" title="c1">z</font>,<font color="Maroon" title="c2">w</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c11">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">pp</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default">4 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p5">(<span class="default"><span class="p0"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">pp</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default">4 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p5">(<span class="default"><span class="p0"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c2">w</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c11">l</font></span>)</span></span><br/><a NAME="E2:43_4_1"/><span class="lab"><font color="Green" title="E53">A32</font></span>: 
<font color="Maroon" title="c10">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c11">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">k</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c11">l</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43_4"><span class="lab"><font color="Green" title="E48">A27</font></span></a>, <a class="txt" href="#E1:43_4_1"><span class="lab"><font color="Green" title="E52">A31</font></span></a>, <a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E3:43_4_1"/><span class="lab"><font color="Green" title="E54">A33</font></span>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">n2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">n2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><a NAME="E4:43_4_1"/><span class="lab"><font color="Green" title="E55">A34</font></span>: 
<font color="Maroon" title="c9">n4</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c11">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">k</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c11">l</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_4"><span class="lab"><font color="Green" title="E46">A25</font></span></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></span>;<br/><a NAME="E5:43_4_1"/><span class="lab"><font color="Green" title="E56">A35</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">n2</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">n2</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">n2</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">n2</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">n2</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c11">l</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_4_1"><span class="lab"><font color="Green" title="E52">A31</font></span></a>, <a class="ref" href="xreal_1.html#T12" title="XREAL_1:th.12">XREAL_1:12</a></span>;<br/><a NAME="E6:43_4_1"/><span class="lab"><font color="Green" title="E57">A36</font></span>: 
<font color="Maroon" title="c6">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">n2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">k</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c11">l</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43_4_1"><span class="lab"><font color="Green" title="E55">A34</font></span></a>, <a class="txt" href="#E5:43_4_1"><span class="lab"><font color="Green" title="E56">A35</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E7:43_4_1"/><span class="lab"><font color="Green" title="E58">A37</font></span>: 
<font color="Maroon" title="c7">n2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c11">l</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_4_1"><span class="lab"><font color="Green" title="E53">A32</font></span></a>, <a class="txt" href="#E3:43_4_1"><span class="lab"><font color="Green" title="E54">A33</font></span></a>, <a class="txt" href="#E6:43_4_1"><span class="lab"><font color="Green" title="E57">A36</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E8:43_4_1"/><span class="lab"><font color="Green" title="E59">A38</font></span>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">n2</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">n2</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">n2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><a NAME="E9:43_4_1"/><span class="lab"><font color="Green" title="E60">A39</font></span>: 
<font color="Maroon" title="c7">n2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">n4</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43_4_1"><span class="lab"><font color="Green" title="E54">A33</font></span></a>, <a class="txt" href="#E8:43_4_1"><span class="lab"><font color="Green" title="E59">A38</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E10:43_4_1"/><span class="lab"><font color="Green" title="E61">A40</font></span>: 
<font color="Maroon" title="c7">n2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_4"><span class="lab"><font color="Green" title="E46">A25</font></span></a>, <a class="txt" href="#E9:43_4_1"><span class="lab"><font color="Green" title="E60">A39</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E11:43_4_1"/><span class="lab"><font color="Green" title="E62">A41</font></span>: 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">w</font> <a href="sin_cos.html#K7" title="SIN_COS:func.7">ExpSeq</a> </span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">w</font> <a href="sin_cos.html#K7" title="SIN_COS:func.7">ExpSeq</a> </span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c11">l</font></span>)</span></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">pp</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default">4 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p4">(<span class="default"><span class="p5"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">pp</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default">4 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p4">(<span class="default"><span class="p5"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c2">w</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:43"><span class="lab"><font color="Green" title="E42">A9</font></span></a>, <a class="txt" href="#E7:43_4_1"><span class="lab"><font color="Green" title="E58">A37</font></span></a>, <a class="txt" href="#E10:43_4_1"><span class="lab"><font color="Green" title="E61">A40</font></span></a></span>;<br/><a NAME="E12:43_4_1"/><span class="lab"><font color="Green" title="E63">A42</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c11">l</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="sin_cos.html#T19" target="_self" title="SIN_COS:th.19">Th19</a></span>;<br/><a NAME="E13:43_4_1"/><span class="lab"><font color="Green" title="E64">A43</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c11">l</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">w</font> <a href="sin_cos.html#K7" title="SIN_COS:func.7">ExpSeq</a> </span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">w</font> <a href="sin_cos.html#K7" title="SIN_COS:func.7">ExpSeq</a> </span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c11">l</font></span>)</span></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c11">l</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">pp</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default">4 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p5">(<span class="default"><span class="p0"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">pp</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default">4 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p5">(<span class="default"><span class="p0"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c2">w</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:43_4_1"><span class="lab"><font color="Green" title="E62">A41</font></span></a>, <a class="txt" href="#E12:43_4_1"><span class="lab"><font color="Green" title="E63">A42</font></span></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></span>;<br/><span class="kw">thus </span><a NAME="E14:43_4_1"/>
<span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K16" title="SIN_COS:func.16">Conj</a> <font color="Maroon" title="c10">k</font>,<font color="Maroon" title="c1">z</font>,<font color="Maroon" title="c2">w</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c11">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">pp</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default">4 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p5">(<span class="default"><span class="p0"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">pp</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default">4 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p5">(<span class="default"><span class="p0"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c2">w</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c11">l</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:43"><span class="lab"><font color="Green" title="E43">A10</font></span></a>, <a class="txt" href="#E3:43_4"><span class="lab"><font color="Green" title="E48">A27</font></span></a>, <a class="txt" href="#E1:43_4_1"><span class="lab"><font color="Green" title="E52">A31</font></span></a>, <a class="txt" href="#E13:43_4_1"><span class="lab"><font color="Green" title="E64">A43</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E7:43_4"/><span class="lab"><font color="Green" title="E52">A44</font></span>: 
<span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K16" title="SIN_COS:func.16">Conj</a> <font color="Maroon" title="c10">k</font>,<font color="Maroon" title="c1">z</font>,<font color="Maroon" title="c2">w</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">n2</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">n2</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">pp</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default">4 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p5">(<span class="default"><span class="p0"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">pp</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default">4 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p5">(<span class="default"><span class="p0"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c2">w</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:43_4"><span class="lab"><font color="Green" title="E51">A30</font></span></a>, <a class="ref" href="comseq_3.html#T7" title="COMSEQ_3:th.7">COMSEQ_3:7</a></span>;<br/><a NAME="E8:43_4"/><span class="lab"><font color="Green" title="E53">A45</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">n2</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">pp</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default">4 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p5">(<span class="default"><span class="p0"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">pp</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default">4 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p5">(<span class="default"><span class="p0"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c2">w</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">pp</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default">4 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p5">(<span class="default"><span class="p0"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">pp</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default">4 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p5">(<span class="default"><span class="p0"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c2">w</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:43"><span class="lab"><font color="Green" title="E38">A4</font></span></a>, <a class="ref" href="sin_cos.html#T17" target="_self" title="SIN_COS:th.17">Th17</a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></span>;<br/><a NAME="E9:43_4"/><span class="lab"><font color="Green" title="E54">A46</font></span>: 
<span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K16" title="SIN_COS:func.16">Conj</a> <font color="Maroon" title="c10">k</font>,<font color="Maroon" title="c1">z</font>,<font color="Maroon" title="c2">w</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">n2</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">pp</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default">4 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p5">(<span class="default"><span class="p0"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">pp</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default">4 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p5">(<span class="default"><span class="p0"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c2">w</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:43_4"><span class="lab"><font color="Green" title="E52">A44</font></span></a>, <a class="txt" href="#E8:43_4"><span class="lab"><font color="Green" title="E53">A45</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E10:43_4"/><span class="lab"><font color="Green" title="E55">A47</font></span>: 
<span class="p1">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">pp</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default">4 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p5">(<span class="default"><span class="p0"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">pp</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default">4 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p5">(<span class="default"><span class="p0"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c2">w</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default">4 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p4">(<span class="default"><span class="p5"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43"><span class="lab"><font color="Green" title="E36">A2</font></span></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a>, <a class="ref" href="xxreal_0.html#T17" title="XXREAL_0:th.17">XXREAL_0:17</a></span>;<br/><a NAME="E11:43_4"/><span class="lab"><font color="Green" title="E56">A48</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="sin_cos.html#T18" target="_self" title="SIN_COS:th.18">Th18</a></span>;<br/><span class="lab"><font color="Green" title="E57">A49</font></span>: <span class="p1">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default">4 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p4">(<span class="default"><span class="p5"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p3">(<span class="default"><span class="p4"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">4 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p3">(<span class="default"><span class="p4"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span></span>)</span></span>)</span>

<br/>.= 
<font color="Maroon" title="c3">p</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 4
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:43_4"><span class="lab"><font color="Green" title="E56">A48</font></span></a>, <a class="ref" href="xcmplx_1.html#T92" title="XCMPLX_1:th.92">XCMPLX_1:92</a></span>
;<br/><a NAME="E13:43_4"/><span class="lab"><font color="Green" title="E58">A50</font></span>: 
<span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K16" title="SIN_COS:func.16">Conj</a> <font color="Maroon" title="c10">k</font>,<font color="Maroon" title="c1">z</font>,<font color="Maroon" title="c2">w</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">n2</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">p</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 4
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:43_4"><span class="lab"><font color="Green" title="E54">A46</font></span></a>, <a class="txt" href="#E10:43_4"><span class="lab"><font color="Green" title="E55">A47</font></span></a>, <a class="txt" href="#E12:43_4"><span class="lab"><font color="Green" title="E57">A49</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E14:43_4"/><span class="lab"><font color="Green" title="E59">A51</font></span>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 4</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 4</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 4</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><a NAME="E15:43_4"/><span class="lab"><font color="Green" title="E60">A52</font></span>: 
<span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K16" title="SIN_COS:func.16">Conj</a> <font color="Maroon" title="c10">k</font>,<font color="Maroon" title="c1">z</font>,<font color="Maroon" title="c2">w</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">n2</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">p</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:43_4"><span class="lab"><font color="Green" title="E58">A50</font></span></a>, <a class="txt" href="#E14:43_4"><span class="lab"><font color="Green" title="E59">A51</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E16:43_4"/><span class="lab"><font color="Green" title="E61">A53</font></span>: 
<font color="Maroon" title="c10">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">n2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">k</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">n2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_4"><span class="lab"><font color="Green" title="E46">A25</font></span></a>, <a class="txt" href="#E2:43_4"><span class="lab"><font color="Green" title="E47">A26</font></span></a>, <a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E17:43_4"/><span class="lab"><font color="Green" title="E62">A54</font></span>: 
<font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">n2</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">n2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:43_4"><span class="lab"><font color="Green" title="E61">A53</font></span></a></span>;<br/><a NAME="E18:43_4"/><span class="lab"><font color="Green" title="E63">A55</font></span>: 
for <font color="Olive" title="b1">l</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">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">k</font> holds <br/><span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K16" title="SIN_COS:func.16">Conj</a> <font color="Maroon" title="c10">k</font>,<font color="Maroon" title="c1">z</font>,<font color="Maroon" title="c2">w</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p3">(<span class="default"><span class="p4"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c2">w</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">l</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:43"><span class="lab"><font color="Green" title="E44">A13</font></span></a></span>;<br/><a NAME="E19:43_4"/><span class="lab"><font color="Green" title="E64">A56</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p3"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K16" title="SIN_COS:func.16">Conj</a> <font color="Maroon" title="c10">k</font>,<font color="Maroon" title="c1">z</font>,<font color="Maroon" title="c2">w</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p3"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K16" title="SIN_COS:func.16">Conj</a> <font color="Maroon" title="c10">k</font>,<font color="Maroon" title="c1">z</font>,<font color="Maroon" title="c2">w</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">n2</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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" title="SERIES_1:func.1">Partial_Sums</a> <span class="p4">(<span class="default"><span class="p5"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p4">(<span class="default"><span class="p5"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">n2</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p3">(<span class="default"><span class="p4"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c2">w</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:43_4"><span class="lab"><font color="Green" title="E62">A54</font></span></a>, <a class="txt" href="#E18:43_4"><span class="lab"><font color="Green" title="E63">A55</font></span></a>, <a class="ref" href="comseq_3.html#T8" title="COMSEQ_3:th.8">COMSEQ_3:8</a></span>;<br/><a NAME="E20:43_4"/><span class="lab"><font color="Green" title="E65">A57</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#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p4">(<span class="default"><span class="p5"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p4">(<span class="default"><span class="p5"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">n2</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">pp</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default">4 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p4">(<span class="default"><span class="p5"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">pp</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default">4 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p4">(<span class="default"><span class="p5"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c2">w</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:43"><span class="lab"><font color="Green" title="E41">A8</font></span></a>, <a class="txt" href="#E4:43_4"><span class="lab"><font color="Green" title="E49">A28</font></span></a>, <a class="txt" href="#E5:43_4"><span class="lab"><font color="Green" title="E50">A29</font></span></a></span>;<br/><a NAME="E21:43_4"/><span class="lab"><font color="Green" title="E66">A58</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">n2</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">pp</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default">4 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p4">(<span class="default"><span class="p5"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">pp</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default">4 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p4">(<span class="default"><span class="p5"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c2">w</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_4"><span class="lab"><font color="Green" title="E46">A25</font></span></a>, <a class="txt" href="#E2:43_4"><span class="lab"><font color="Green" title="E47">A26</font></span></a>, <a class="txt" href="#E20:43_4"><span class="lab"><font color="Green" title="E65">A57</font></span></a>, <a class="ref" href="sin_cos.html#T20" target="_self" title="SIN_COS:th.20">Th20</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E22:43_4"/><span class="lab"><font color="Green" title="E67">A59</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p4">(<span class="default"><span class="p5"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p4">(<span class="default"><span class="p5"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">n2</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p3">(<span class="default"><span class="p4"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c2">w</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">pp</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default">4 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p5">(<span class="default"><span class="p0"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">pp</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default">4 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p5">(<span class="default"><span class="p0"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c2">w</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p3">(<span class="default"><span class="p4"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c2">w</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43"><span class="lab"><font color="Green" title="E37">A3</font></span></a>, <a class="txt" href="#E21:43_4"><span class="lab"><font color="Green" title="E66">A58</font></span></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></span>;<br/><a NAME="E23:43_4"/><span class="lab"><font color="Green" title="E68">A60</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p3"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K16" title="SIN_COS:func.16">Conj</a> <font color="Maroon" title="c10">k</font>,<font color="Maroon" title="c1">z</font>,<font color="Maroon" title="c2">w</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p3"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K16" title="SIN_COS:func.16">Conj</a> <font color="Maroon" title="c10">k</font>,<font color="Maroon" title="c1">z</font>,<font color="Maroon" title="c2">w</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">n2</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">pp</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default">4 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p5">(<span class="default"><span class="p0"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">pp</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default">4 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p5">(<span class="default"><span class="p0"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c2">w</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p3">(<span class="default"><span class="p4"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c2">w</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:43_4"><span class="lab"><font color="Green" title="E64">A56</font></span></a>, <a class="txt" href="#E22:43_4"><span class="lab"><font color="Green" title="E67">A59</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E24:43_4"/><span class="lab"><font color="Green" title="E69">A61</font></span>: 
<span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p3">(<span class="default"><span class="p4"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c2">w</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">pp</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default">4 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p5">(<span class="default"><span class="p0"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">pp</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default">4 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p5">(<span class="default"><span class="p0"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c2">w</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p3">(<span class="default"><span class="p4"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c2">w</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default">4 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p4">(<span class="default"><span class="p5"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c2">w</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43"><span class="lab"><font color="Green" title="E37">A3</font></span></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a>, <a class="ref" href="xxreal_0.html#T17" title="XXREAL_0:th.17">XXREAL_0:17</a></span>;<br/><a NAME="E25:43_4"/><span class="lab"><font color="Green" title="E70">A62</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c2">w</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="sin_cos.html#T18" target="_self" title="SIN_COS:th.18">Th18</a></span>;<br/><span class="lab"><font color="Green" title="E71">A63</font></span>: <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p3">(<span class="default"><span class="p4"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c2">w</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default">4 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p4">(<span class="default"><span class="p5"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c2">w</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p3">(<span class="default"><span class="p4"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c2">w</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">4 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p3">(<span class="default"><span class="p4"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c2">w</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="sin_cos.html#K8" title="SIN_COS:func.8">rExpSeq</a> </span>)</span></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 4
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:43_4"><span class="lab"><font color="Green" title="E70">A62</font></span></a>, <a class="ref" href="xcmplx_1.html#T92" title="XCMPLX_1:th.92">XCMPLX_1:92</a></span>
<br/>.= 
<font color="Maroon" title="c3">p</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2

;<br/><a NAME="E27:43_4"/><span class="lab"><font color="Green" title="E72">A64</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p3"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K16" title="SIN_COS:func.16">Conj</a> <font color="Maroon" title="c10">k</font>,<font color="Maroon" title="c1">z</font>,<font color="Maroon" title="c2">w</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p3"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K16" title="SIN_COS:func.16">Conj</a> <font color="Maroon" title="c10">k</font>,<font color="Maroon" title="c1">z</font>,<font color="Maroon" title="c2">w</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">n2</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">p</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:43_4"><span class="lab"><font color="Green" title="E68">A60</font></span></a>, <a class="txt" href="#E24:43_4"><span class="lab"><font color="Green" title="E69">A61</font></span></a>, <a class="txt" href="#E26:43_4"><span class="lab"><font color="Green" title="E71">A63</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E28:43_4"/><span class="lab"><font color="Green" title="E73">A65</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p4"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p5">(<span class="default"><a href="sin_cos.html#K16" title="SIN_COS:func.16">Conj</a> <font color="Maroon" title="c10">k</font>,<font color="Maroon" title="c1">z</font>,<font color="Maroon" title="c2">w</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p4"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p5">(<span class="default"><a href="sin_cos.html#K16" title="SIN_COS:func.16">Conj</a> <font color="Maroon" title="c10">k</font>,<font color="Maroon" title="c1">z</font>,<font color="Maroon" title="c2">w</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">n2</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p3"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K16" title="SIN_COS:func.16">Conj</a> <font color="Maroon" title="c10">k</font>,<font color="Maroon" title="c1">z</font>,<font color="Maroon" title="c2">w</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">n2</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:43_4"><span class="lab"><font color="Green" title="E60">A52</font></span></a>, <a class="txt" href="#E27:43_4"><span class="lab"><font color="Green" title="E72">A64</font></span></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></span>;<br/><span class="kw">thus </span><a NAME="E29:43_4"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p3"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K16" title="SIN_COS:func.16">Conj</a> <font color="Maroon" title="c10">k</font>,<font color="Maroon" title="c1">z</font>,<font color="Maroon" title="c2">w</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:43_4"><span class="lab"><font color="Green" title="E73">A65</font></span></a>, <a class="ref" href="sin_cos.html#T21" target="_self" title="SIN_COS:th.21">Th21</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">thus </span><a NAME="E15:43"/>
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="c9">n4</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"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p3"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K16" title="SIN_COS:func.16">Conj</a> <font color="Olive" title="b1">k</font>,<font color="Maroon" title="c1">z</font>,<font color="Maroon" title="c2">w</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></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="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:43"><span class="lab"><font color="Green" title="E45">A24</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
