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<span class="kw">let </span><font color="Maroon" title="c1">X</font> be   <a href="lopban_2.html#NM3" title="LOPBAN_2:NM.3">Banach_Algebra</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">z</font>, <font color="Olive" title="b2">w</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b3">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="b3">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="b4">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="b5">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="b4">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b5">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#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p3"><a href="normsp_0.html#K4" title="NORMSP_0:func.4">||.</a><span class="default"><span class="p4">(<span class="default"><a href="lopban_4.html#K8" title="LOPBAN_4:func.8">Conj</a> (<font color="Olive" title="b5">k</font>,<font color="Olive" title="b1">z</font>,<font color="Olive" title="b2">w</font>)</span>)</span></span><a href="normsp_0.html#K4" title="NORMSP_0:func.4">.||</a></span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b5">k</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">p</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">z</font>, <font color="Maroon" title="c3">w</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">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#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p3"><a href="normsp_0.html#K4" title="NORMSP_0:func.4">||.</a><span class="default"><span class="p4">(<span class="default"><a href="lopban_4.html#K8" title="LOPBAN_4:func.8">Conj</a> (<font color="Olive" title="b3">k</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">w</font>)</span>)</span></span><a href="normsp_0.html#K4" title="NORMSP_0:func.4">.||</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="c4">p</font> be   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">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#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p3"><a href="normsp_0.html#K4" title="NORMSP_0:func.4">||.</a><span class="default"><span class="p4">(<span class="default"><a href="lopban_4.html#K8" title="LOPBAN_4:func.8">Conj</a> (<font color="Olive" title="b2">k</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">w</font>)</span>)</span></span><a href="normsp_0.html#K4" title="NORMSP_0:func.4">.||</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="c4">p</font> )</span><br/>







<span class="kw">assume </span><a NAME="E1:45"/><span class="lab"><font color="Green" title="E44">A1</font></span>: 
<font color="Maroon" title="c4">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/> for <font color="Olive" title="b2">k</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">k</font> holds <br/> <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p3"><a href="normsp_0.html#K4" title="NORMSP_0:func.4">||.</a><span class="default"><span class="p4">(<span class="default"><a href="lopban_4.html#K8" title="LOPBAN_4:func.8">Conj</a> (<font color="Olive" title="b2">k</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">w</font>)</span>)</span></span><a href="normsp_0.html#K4" title="NORMSP_0:func.4">.||</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="c4">p</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c5">pp</font> = <font color="Maroon" title="c4">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/>
<span class="kw">set </span><font color="Maroon" title="c6">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="c5">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#K3" title="SERIES_1:func.3">Sum</a> <span class="p4">(<span class="default"><span class="p5"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">rExpSeq</a></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">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#K3" title="SERIES_1:func.3">Sum</a> <span class="p4">(<span class="default"><span class="p5"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c3">w</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">rExpSeq</a></span>)</span></span>)</span></span>)</span></span>)</span>);<br/>
<a NAME="E3:45"/><span class="lab"><font color="Green" title="E45">A2</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">rExpSeq</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lopban_4.html#T29" target="_self" title="LOPBAN_4:th.29">Th29</a></span>;<br/>
<a NAME="E4:45"/><span class="kw">then </span>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p2">(<span class="default"><span class="p3"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">rExpSeq</a></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 4 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p2">(<span class="default"><span class="p3"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">rExpSeq</a></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T68" title="XREAL_1:th.68">XREAL_1:68</a></span>;<br/>
<a NAME="E5:45"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A3</font></span>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <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#K3" title="SERIES_1:func.3">Sum</a> <span class="p3">(<span class="default"><span class="p4"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">rExpSeq</a></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">p</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</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#K3" title="SERIES_1:func.3">Sum</a> <span class="p3">(<span class="default"><span class="p4"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">rExpSeq</a></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45"><span class="lab"><font color="Green" title="E44">A1</font></span></a>, <a class="ref" href="xreal_1.html#T74" title="XREAL_1:th.74">XREAL_1:74</a></span>;<br/>
<a NAME="E6:45"/><span class="lab"><font color="Green" title="E47">A4</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c3">w</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">rExpSeq</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lopban_4.html#T29" target="_self" title="LOPBAN_4:th.29">Th29</a></span>;<br/>
<a NAME="E7:45"/><span class="kw">then </span>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p2">(<span class="default"><span class="p3"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c3">w</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">rExpSeq</a></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 4 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p2">(<span class="default"><span class="p3"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c3">w</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">rExpSeq</a></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T68" title="XREAL_1:th.68">XREAL_1:68</a></span>;<br/>
<a NAME="E8:45"/><span class="kw">then </span>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <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#K3" title="SERIES_1:func.3">Sum</a> <span class="p3">(<span class="default"><span class="p4"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c3">w</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">rExpSeq</a></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">p</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</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#K3" title="SERIES_1:func.3">Sum</a> <span class="p3">(<span class="default"><span class="p4"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c3">w</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">rExpSeq</a></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45"><span class="lab"><font color="Green" title="E44">A1</font></span></a>, <a class="ref" href="xreal_1.html#T74" title="XREAL_1:th.74">XREAL_1:74</a></span>;<br/>
<a NAME="E9:45"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A5</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="c5">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#K3" title="SERIES_1:func.3">Sum</a> <span class="p4">(<span class="default"><span class="p5"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">rExpSeq</a></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">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#K3" title="SERIES_1:func.3">Sum</a> <span class="p4">(<span class="default"><span class="p5"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c3">w</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">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="#E5:45"><span class="lab"><font color="Green" title="E46">A3</font></span></a>, <a class="ref" href="xxreal_0.html#T15" title="XXREAL_0:th.15">XXREAL_0:15</a></span>;<br/>
<a NAME="E10:45"/>
 <a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</a></span>)</span> is  <a href="normsp_1.html#V3" title="NORMSP_1:attr.3">convergent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lopban_3.html#D1" title="LOPBAN_3:def.1">LOPBAN_3:def 1</a></span>;<br/>
<a NAME="E11:45"/><span class="kw">then </span>
 for <font color="Olive" title="b1">s</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">s</font> 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">m</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">m</font> holds <br/><span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</a></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b3">m</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</a></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">n</font></span>)</span></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lopban_3.html#T4" title="LOPBAN_3:th.4">LOPBAN_3:4</a></span>;<br/>
<a NAME="E12:45"/><span class="kw">then </span>
 <a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</a></span>)</span> is  <a href="rsspace3.html#V1" title="RSSPACE3:attr.1">CCauchy</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lopban_3.html#T5" title="LOPBAN_3:th.5">LOPBAN_3:5</a></span>;<br/>
<span class="kw">then </span><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="E14:45"/><span class="lab"><font color="Green" title="E49">A6</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="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">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="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">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></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</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="c5">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#K3" title="SERIES_1:func.3">Sum</a> <span class="p4">(<span class="default"><span class="p5"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">rExpSeq</a></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">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#K3" title="SERIES_1:func.3">Sum</a> <span class="p4">(<span class="default"><span class="p5"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c3">w</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">rExpSeq</a></span>)</span></span>)</span></span>)</span></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:45"><span class="lab"><font color="Green" title="E48">A5</font></span></a>, <a class="ref" href="rsspace3.html#T8" title="RSSPACE3:th.8">RSSPACE3:8</a></span>;<br/>
<a NAME="E15:45"/>
<span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">rExpSeq</a>  is  <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="sin_cos.html#T45" title="SIN_COS:th.45">SIN_COS:45</a></span>;<br/>
<a NAME="E16:45"/><span class="kw">then </span>
 <a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">rExpSeq</a></span>)</span> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="series_1.html#D2" title="SERIES_1:def.2">SERIES_1:def 2</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">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="E18:45"/><span class="lab"><font color="Green" title="E50">A7</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="c8">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="c8">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#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p4">(<span class="default"><span class="p5"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">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#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p4">(<span class="default"><span class="p5"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">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="c5">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#K3" title="SERIES_1:func.3">Sum</a> <span class="p4">(<span class="default"><span class="p5"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">rExpSeq</a></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">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#K3" title="SERIES_1:func.3">Sum</a> <span class="p4">(<span class="default"><span class="p5"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c3">w</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">rExpSeq</a></span>)</span></span>)</span></span>)</span></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:45"><span class="lab"><font color="Green" title="E48">A5</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">set </span><font color="Maroon" title="c9">n3</font> = <font color="Maroon" title="c8">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="c10">n4</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c8">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="c8">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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">k</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c10">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#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p3"><a href="normsp_0.html#K4" title="NORMSP_0:func.4">||.</a><span class="default"><span class="p4">(<span class="default"><a href="lopban_4.html#K8" title="LOPBAN_4:func.8">Conj</a> (<font color="Olive" title="b1">k</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">w</font>)</span>)</span></span><a href="normsp_0.html#K4" title="NORMSP_0:func.4">.||</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="c4">p</font></span><br/>

<div><span class="lab"><font color="Green" title="E51">A8</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c11">n</font>, <font color="Maroon" title="c12">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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="c12">k</font> holds <br/><span class="p1"><a href="normsp_0.html#K4" title="NORMSP_0:func.4">||.</a><span class="default"><span class="p2">(<span class="default"><a href="lopban_4.html#K8" title="LOPBAN_4:func.8">Conj</a> (<font color="Maroon" title="c12">k</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">w</font>)</span>)</span></span><a href="normsp_0.html#K4" title="NORMSP_0:func.4">.||</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="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">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="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</a></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">k</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</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="c12">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="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span></span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c13">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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c13">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">k</font> implies <span class="p1"><a href="normsp_0.html#K4" title="NORMSP_0:func.4">||.</a><span class="default"><span class="p2">(<span class="default"><a href="lopban_4.html#K8" title="LOPBAN_4:func.8">Conj</a> (<font color="Maroon" title="c12">k</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">w</font>)</span>)</span></span><a href="normsp_0.html#K4" title="NORMSP_0:func.4">.||</a></span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">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="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">rExpSeq</a></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">l</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</a></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">k</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</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="c12">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c13">l</font></span>)</span></span>)</span></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> )</span><br/><span class="kw">assume </span><a NAME="E1:45_1_1"/><span class="lab"><font color="Green" title="E52">A9</font></span>: 
<font color="Maroon" title="c13">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="normsp_0.html#K4" title="NORMSP_0:func.4">||.</a><span class="default"><span class="p2">(<span class="default"><a href="lopban_4.html#K8" title="LOPBAN_4:func.8">Conj</a> (<font color="Maroon" title="c12">k</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">w</font>)</span>)</span></span><a href="normsp_0.html#K4" title="NORMSP_0:func.4">.||</a></span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">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="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">rExpSeq</a></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">l</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</a></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">k</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</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="c12">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c13">l</font></span>)</span></span>)</span></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span></span><br/><a NAME="E2:45_1_1"/><span class="lab"><font color="Green" title="E53">A10</font></span>: 
<span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</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="c2">z</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</a></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">l</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</a></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">k</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</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="c12">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c13">l</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</a></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">l</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</a></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">k</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</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="c12">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c13">l</font></span>)</span></span>)</span></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lopban_3.html#T38" title="LOPBAN_3:th.38">LOPBAN_3:38</a></span>;<br/><a NAME="E3:45_1_1"/>
 <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"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</a></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">k</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</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="c12">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c13">l</font></span>)</span></span>)</span></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="normsp_1.html#T4" title="NORMSP_1:th.4">NORMSP_1:4</a></span>;<br/><a NAME="E4:45_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A11</font></span>: 
<span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</a></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">l</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</a></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">k</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</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="c12">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c13">l</font></span>)</span></span>)</span></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</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="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">rExpSeq</a></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">l</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</a></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">k</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</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="c12">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c13">l</font></span>)</span></span>)</span></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lopban_4.html#T14" target="_self" title="LOPBAN_4:th.14">Th14</a>, <a class="ref" href="xreal_1.html#T64" title="XREAL_1:th.64">XREAL_1:64</a></span>;<br/><span class="p1"><a href="normsp_0.html#K4" title="NORMSP_0:func.4">||.</a><span class="default"><span class="p2">(<span class="default"><a href="lopban_4.html#K8" title="LOPBAN_4:func.8">Conj</a> (<font color="Maroon" title="c12">k</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">w</font>)</span>)</span></span><a href="normsp_0.html#K4" title="NORMSP_0:func.4">.||</a></span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">l</font> = 
<span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lopban_4.html#K8" title="LOPBAN_4:func.8">Conj</a> (<font color="Maroon" title="c12">k</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">w</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">l</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="normsp_0.html#D4" title="NORMSP_0:def.4">NORMSP_0:def 4</a></span>
<br/>.= 
<span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</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="c2">z</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</a></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">l</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</a></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">k</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</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="c12">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c13">l</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45_1_1"><span class="lab"><font color="Green" title="E52">A9</font></span></a>, <a class="ref" href="lopban_4.html#D9" target="_self" title="LOPBAN_4:def.9">Def9</a></span>
;<br/><span class="kw">hence </span><a NAME="E6:45_1_1"/>
<span class="p1"><a href="normsp_0.html#K4" title="NORMSP_0:func.4">||.</a><span class="default"><span class="p2">(<span class="default"><a href="lopban_4.html#K8" title="LOPBAN_4:func.8">Conj</a> (<font color="Maroon" title="c12">k</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">w</font>)</span>)</span></span><a href="normsp_0.html#K4" title="NORMSP_0:func.4">.||</a></span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">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="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">rExpSeq</a></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">l</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</a></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">k</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</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="c12">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c13">l</font></span>)</span></span>)</span></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:45_1_1"><span class="lab"><font color="Green" title="E53">A10</font></span></a>, <a class="txt" href="#E4:45_1_1"><span class="lab"><font color="Green" title="E54">A11</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="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E2:45_1"/>
 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="c12">k</font> holds <br/><span class="p1"><a href="normsp_0.html#K4" title="NORMSP_0:func.4">||.</a><span class="default"><span class="p2">(<span class="default"><a href="lopban_4.html#K8" title="LOPBAN_4:func.8">Conj</a> (<font color="Maroon" title="c12">k</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">w</font>)</span>)</span></span><a href="normsp_0.html#K4" title="NORMSP_0:func.4">.||</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="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">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="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</a></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">k</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</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="c12">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="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E52">A12</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="45_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><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="firebrick">::  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="normsp_0.html#K4" title="NORMSP_0:func.4">||.</a><span class="default"><span class="p2">(<span class="default"><a href="lopban_4.html#K8" title="LOPBAN_4:func.8">Conj</a> (<font color="Maroon" title="c11">k</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">w</font>)</span>)</span></span><a href="normsp_0.html#K4" title="NORMSP_0:func.4">.||</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="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">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#K3" title="SERIES_1:func.3">Sum</a> <span class="p3">(<span class="default"><span class="p4"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c3">w</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">rExpSeq</a></span>)</span></span>)</span></span>)</span></span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_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="firebrick">::  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="normsp_0.html#K4" title="NORMSP_0:func.4">||.</a><span class="default"><span class="p2">(<span class="default"><a href="lopban_4.html#K8" title="LOPBAN_4:func.8">Conj</a> (<font color="Maroon" title="c11">k</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">w</font>)</span>)</span></span><a href="normsp_0.html#K4" title="NORMSP_0:func.4">.||</a></span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">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="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">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">(<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#K3" title="SERIES_1:func.3">Sum</a> <span class="p3">(<span class="default"><span class="p4"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c3">w</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">rExpSeq</a></span>)</span></span>)</span></span>)</span> )</span><br/><a NAME="E1:45_2_1"/><span class="lab"><font color="Green" title="E53">A13</font></span>: 
<span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</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="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</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="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</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></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">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="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><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="normsp_1.html#T3" title="NORMSP_1:th.3">NORMSP_1:3</a></span>;<br/><a NAME="E2:45_2_1"/>
<span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">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="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><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c3">w</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">rExpSeq</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lopban_4.html#T28" target="_self" title="LOPBAN_4:th.28">Th28</a></span>;<br/><a NAME="E3:45_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A14</font></span>: 
<span class="p1">(<span class="default"><a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p2">(<span class="default"><span class="p3"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c3">w</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">rExpSeq</a></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">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="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><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</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#K3" title="SERIES_1:func.3">Sum</a> <span class="p2">(<span class="default"><span class="p3"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c3">w</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">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#K3" title="SERIES_1:func.3">Sum</a> <span class="p2">(<span class="default"><span class="p3"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c3">w</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">rExpSeq</a></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><a NAME="E4:45_2_1"/>
<span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</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></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c3">w</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">rExpSeq</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lopban_4.html#T28" target="_self" title="LOPBAN_4:th.28">Th28</a></span>;<br/><a NAME="E5:45_2_1"/><span class="kw">then </span>
<span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</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></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">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="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><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</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#K3" title="SERIES_1:func.3">Sum</a> <span class="p2">(<span class="default"><span class="p3"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c3">w</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">rExpSeq</a></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">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="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><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><a NAME="E6:45_2_1"/><span class="kw">then </span>
<span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</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></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">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="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><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</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#K3" title="SERIES_1:func.3">Sum</a> <span class="p2">(<span class="default"><span class="p3"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c3">w</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">rExpSeq</a></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:45_2_1"><span class="lab"><font color="Green" title="E54">A14</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E7:45_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A15</font></span>: 
<span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</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="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</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="normsp_0.html#K1" title="NORMSP_0:func.1">.||</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#K3" title="SERIES_1:func.3">Sum</a> <span class="p2">(<span class="default"><span class="p3"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c3">w</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">rExpSeq</a></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45_2_1"><span class="lab"><font color="Green" title="E53">A13</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><span class="kw">assume </span><a NAME="E8:45_2_1"/>
<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="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="normsp_0.html#K4" title="NORMSP_0:func.4">||.</a><span class="default"><span class="p2">(<span class="default"><a href="lopban_4.html#K8" title="LOPBAN_4:func.8">Conj</a> (<font color="Maroon" title="c11">k</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">w</font>)</span>)</span></span><a href="normsp_0.html#K4" title="NORMSP_0:func.4">.||</a></span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">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="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">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">(<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#K3" title="SERIES_1:func.3">Sum</a> <span class="p3">(<span class="default"><span class="p4"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c3">w</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">rExpSeq</a></span>)</span></span>)</span></span>)</span></span><br/><a NAME="E9:45_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A16</font></span>: 
<span class="p1"><a href="normsp_0.html#K4" title="NORMSP_0:func.4">||.</a><span class="default"><span class="p2">(<span class="default"><a href="lopban_4.html#K8" title="LOPBAN_4:func.8">Conj</a> (<font color="Maroon" title="c11">k</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">w</font>)</span>)</span></span><a href="normsp_0.html#K4" title="NORMSP_0:func.4">.||</a></span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">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="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">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="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</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="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</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="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:45"><span class="lab"><font color="Green" title="E51">A8</font></span></a></span>;<br/><a NAME="E10:45_2_1"/>
 <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="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">rExpSeq</a></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">l</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lopban_4.html#T27" target="_self" title="LOPBAN_4:th.27">Th27</a></span>;<br/><a NAME="E11:45_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">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="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</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="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</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="normsp_0.html#K1" title="NORMSP_0:func.1">.||</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="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">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">(<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#K3" title="SERIES_1:func.3">Sum</a> <span class="p3">(<span class="default"><span class="p4"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c3">w</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">rExpSeq</a></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:45_2_1"><span class="lab"><font color="Green" title="E55">A15</font></span></a>, <a class="ref" href="xreal_1.html#T64" title="XREAL_1:th.64">XREAL_1:64</a></span>;<br/><span class="kw">hence </span><a NAME="E12:45_2_1"/>
<span class="p1"><a href="normsp_0.html#K4" title="NORMSP_0:func.4">||.</a><span class="default"><span class="p2">(<span class="default"><a href="lopban_4.html#K8" title="LOPBAN_4:func.8">Conj</a> (<font color="Maroon" title="c11">k</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">w</font>)</span>)</span></span><a href="normsp_0.html#K4" title="NORMSP_0:func.4">.||</a></span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">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="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">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">(<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#K3" title="SERIES_1:func.3">Sum</a> <span class="p3">(<span class="default"><span class="p4"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c3">w</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">rExpSeq</a></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:45_2_1"><span class="lab"><font color="Green" title="E56">A16</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="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E2:45_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="normsp_0.html#K4" title="NORMSP_0:func.4">||.</a><span class="default"><span class="p2">(<span class="default"><a href="lopban_4.html#K8" title="LOPBAN_4:func.8">Conj</a> (<font color="Maroon" title="c11">k</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">w</font>)</span>)</span></span><a href="normsp_0.html#K4" title="NORMSP_0:func.4">.||</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="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">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#K3" title="SERIES_1:func.3">Sum</a> <span class="p3">(<span class="default"><span class="p4"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c3">w</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">rExpSeq</a></span>)</span></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_3"><span class="kw">now </span></a><div class="add"><a NAME="E1:45_3"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">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="#E1:45"><span class="lab"><font color="Green" title="E44">A1</font></span></a>, <a class="ref" href="xreal_1.html#T224" title="XREAL_1:th.224">XREAL_1:224</a></span>;<br/><a NAME="E2:45_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A17</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="c4">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="c4">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="c4">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="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><a NAME="E3:45_3"/><span class="lab"><font color="Green" title="E54">A18</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#K3" title="SERIES_1:func.3">Sum</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">rExpSeq</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lopban_4.html#T29" target="_self" title="LOPBAN_4:th.29">Th29</a></span>;<br/><span class="lab"><font color="Green" title="E55">A19</font></span>: <span class="p1">(<span class="default"><a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p2">(<span class="default"><span class="p3"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">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="c4">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#K3" title="SERIES_1:func.3">Sum</a> <span class="p4">(<span class="default"><span class="p5"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">rExpSeq</a></span>)</span></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p2">(<span class="default"><span class="p3"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">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="c4">p</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><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#K3" title="SERIES_1:func.3">Sum</a> <span class="p5">(<span class="default"><span class="p0"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">rExpSeq</a></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_0.html#D9" title="XCMPLX_0:def.9">XCMPLX_0:def 9</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p3">(<span class="default"><span class="p4"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">rExpSeq</a></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c4">p</font></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">4 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p4">(<span class="default"><span class="p5"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">rExpSeq</a></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p3">(<span class="default"><span class="p4"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">rExpSeq</a></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c4">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#K3" title="SERIES_1:func.3">Sum</a> <span class="p3">(<span class="default"><span class="p4"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">rExpSeq</a></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_0.html#D9" title="XCMPLX_0:def.9">XCMPLX_0:def 9</a></span>
<br/>.= 
<font color="Maroon" title="c4">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="#E3:45_3"><span class="lab"><font color="Green" title="E54">A18</font></span></a>, <a class="ref" href="xcmplx_1.html#T91" title="XCMPLX_1:th.91">XCMPLX_1:91</a></span>
;<br/><span class="kw">let </span><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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c10">n4</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">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#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p3"><a href="normsp_0.html#K4" title="NORMSP_0:func.4">||.</a><span class="default"><span class="p4">(<span class="default"><a href="lopban_4.html#K8" title="LOPBAN_4:func.8">Conj</a> (<font color="Maroon" title="c11">k</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">w</font>)</span>)</span></span><a href="normsp_0.html#K4" title="NORMSP_0:func.4">.||</a></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="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">p</font> )</span><br/><span class="kw">assume </span><a NAME="E5:45_3"/><span class="lab"><font color="Green" title="E56">A20</font></span>: 
<font color="Maroon" title="c10">n4</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="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p3"><a href="normsp_0.html#K4" title="NORMSP_0:func.4">||.</a><span class="default"><span class="p4">(<span class="default"><a href="lopban_4.html#K8" title="LOPBAN_4:func.8">Conj</a> (<font color="Maroon" title="c11">k</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">w</font>)</span>)</span></span><a href="normsp_0.html#K4" title="NORMSP_0:func.4">.||</a></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="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">p</font></span><br/><a NAME="E6:45_3"/><span class="lab"><font color="Green" title="E57">A21</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="c8">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="c8">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="c8">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#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><a NAME="E7:45_3"/><span class="kw">then </span>
<font color="Maroon" title="c11">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">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="c11">k</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">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="#E5:45_3"><span class="lab"><font color="Green" title="E56">A20</font></span></a>, <a class="ref" href="xreal_1.html#T233" title="XREAL_1:th.233">XREAL_1:233</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E8:45_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A22</font></span>: 
<font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">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="c8">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">n2</font></span>)</span>
 ;<br/><a NAME="E9:45_3"/><span class="lab"><font color="Green" title="E59">A23</font></span>: 
<font color="Maroon" title="c8">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="c11">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:45_3"><span class="lab"><font color="Green" title="E56">A20</font></span></a>, <a class="txt" href="#E6:45_3"><span class="lab"><font color="Green" title="E57">A21</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_3_2"><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="firebrick">::  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="c8">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="normsp_0.html#K4" title="NORMSP_0:func.4">||.</a><span class="default"><span class="p2">(<span class="default"><a href="lopban_4.html#K8" title="LOPBAN_4:func.8">Conj</a> (<font color="Maroon" title="c11">k</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">w</font>)</span>)</span></span><a href="normsp_0.html#K4" title="NORMSP_0:func.4">.||</a></span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">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="c5">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#K3" title="SERIES_1:func.3">Sum</a> <span class="p5">(<span class="default"><span class="p0"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">rExpSeq</a></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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#K3" title="SERIES_1:func.3">Sum</a> <span class="p5">(<span class="default"><span class="p0"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c3">w</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">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="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">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> )</span><br/><a NAME="E1:45_3_2"/><span class="lab"><font color="Green" title="E60">A24</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="c8">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#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><a NAME="E2:45_3_2"/>
<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="c8">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="c8">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="c8">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#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><a NAME="E3:45_3_2"/><span class="kw">then </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">n4</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45_3_2"><span class="lab"><font color="Green" title="E60">A24</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E4:45_3_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A25</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="c11">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:45_3"><span class="lab"><font color="Green" title="E56">A20</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E5:45_3_2"/><span class="lab"><font color="Green" title="E62">A26</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="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">rExpSeq</a></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">l</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lopban_4.html#T27" target="_self" title="LOPBAN_4:th.27">Th27</a></span>;<br/><span class="kw">assume </span><a NAME="E6:45_3_2"/><span class="lab"><font color="Green" title="E63">A27</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="c8">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="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="normsp_0.html#K4" title="NORMSP_0:func.4">||.</a><span class="default"><span class="p2">(<span class="default"><a href="lopban_4.html#K8" title="LOPBAN_4:func.8">Conj</a> (<font color="Maroon" title="c11">k</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">w</font>)</span>)</span></span><a href="normsp_0.html#K4" title="NORMSP_0:func.4">.||</a></span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">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="c5">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#K3" title="SERIES_1:func.3">Sum</a> <span class="p5">(<span class="default"><span class="p0"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">rExpSeq</a></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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#K3" title="SERIES_1:func.3">Sum</a> <span class="p5">(<span class="default"><span class="p0"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c3">w</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">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="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">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></span><br/><a NAME="E7:45_3_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A28</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">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="c8">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="c8">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="c8">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="c8">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="c12">l</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></span>;<br/><a NAME="E8:45_3_2"/>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:45_3"><span class="lab"><font color="Green" title="E59">A23</font></span></a>, <a class="txt" href="#E6:45_3_2"><span class="lab"><font color="Green" title="E63">A27</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E9:45_3_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A29</font></span>: 
<span class="p1"><a href="normsp_0.html#K4" title="NORMSP_0:func.4">||.</a><span class="default"><span class="p2">(<span class="default"><a href="lopban_4.html#K8" title="LOPBAN_4:func.8">Conj</a> (<font color="Maroon" title="c11">k</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">w</font>)</span>)</span></span><a href="normsp_0.html#K4" title="NORMSP_0:func.4">.||</a></span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">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="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">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="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</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="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</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="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:45"><span class="lab"><font color="Green" title="E51">A8</font></span></a></span>;<br/><a NAME="E10:45_3_2"/>
<font color="Maroon" title="c10">n4</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <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 href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c12">l</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:45_3"><span class="lab"><font color="Green" title="E56">A20</font></span></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/><a NAME="E11:45_3_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A30</font></span>: 
<font color="Maroon" title="c8">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="c11">k</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c12">l</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:45_3_2"><span class="lab"><font color="Green" title="E64">A28</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E12:45_3_2"/>
<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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">k</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c12">l</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:45_3"><span class="lab"><font color="Green" title="E59">A23</font></span></a>, <a class="txt" href="#E6:45_3_2"><span class="lab"><font color="Green" title="E63">A27</font></span></a>, <a class="ref" href="xreal_1.html#T233" title="XREAL_1:th.233">XREAL_1:233</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E13:45_3_2"/><span class="kw">then </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="c11">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c12">l</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45_3_2"><span class="lab"><font color="Green" title="E60">A24</font></span></a>, <a class="txt" href="#E11:45_3_2"><span class="lab"><font color="Green" title="E66">A30</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E14:45_3_2"/><span class="kw">then </span>
<span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</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="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</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="normsp_0.html#K1" title="NORMSP_0:func.1">.||</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="c5">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#K3" title="SERIES_1:func.3">Sum</a> <span class="p4">(<span class="default"><span class="p5"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">rExpSeq</a></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">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#K3" title="SERIES_1:func.3">Sum</a> <span class="p4">(<span class="default"><span class="p5"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c3">w</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">rExpSeq</a></span>)</span></span>)</span></span>)</span></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:45"><span class="lab"><font color="Green" title="E49">A6</font></span></a>, <a class="txt" href="#E4:45_3_2"><span class="lab"><font color="Green" title="E61">A25</font></span></a></span>;<br/><a NAME="E15:45_3_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">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="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</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="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</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="normsp_0.html#K1" title="NORMSP_0:func.1">.||</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="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">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">(<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="c5">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#K3" title="SERIES_1:func.3">Sum</a> <span class="p5">(<span class="default"><span class="p0"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">rExpSeq</a></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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#K3" title="SERIES_1:func.3">Sum</a> <span class="p5">(<span class="default"><span class="p0"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c3">w</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">rExpSeq</a></span>)</span></span>)</span></span>)</span></span>)</span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:45_3_2"><span class="lab"><font color="Green" title="E62">A26</font></span></a>, <a class="ref" href="xreal_1.html#T64" title="XREAL_1:th.64">XREAL_1:64</a></span>;<br/><span class="kw">hence </span><a NAME="E16:45_3_2"/>
<span class="p1"><a href="normsp_0.html#K4" title="NORMSP_0:func.4">||.</a><span class="default"><span class="p2">(<span class="default"><a href="lopban_4.html#K8" title="LOPBAN_4:func.8">Conj</a> (<font color="Maroon" title="c11">k</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">w</font>)</span>)</span></span><a href="normsp_0.html#K4" title="NORMSP_0:func.4">.||</a></span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">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="c5">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#K3" title="SERIES_1:func.3">Sum</a> <span class="p5">(<span class="default"><span class="p0"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">rExpSeq</a></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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#K3" title="SERIES_1:func.3">Sum</a> <span class="p5">(<span class="default"><span class="p0"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c3">w</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">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="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:45_3_2"><span class="lab"><font color="Green" title="E65">A29</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="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E11:45_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A31</font></span>: 
<span class="p1">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p2"><a href="normsp_0.html#K4" title="NORMSP_0:func.4">||.</a><span class="default"><span class="p3">(<span class="default"><a href="lopban_4.html#K8" title="LOPBAN_4:func.8">Conj</a> (<font color="Maroon" title="c11">k</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">w</font>)</span>)</span></span><a href="normsp_0.html#K4" title="NORMSP_0:func.4">.||</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="c8">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#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">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="c8">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="c5">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#K3" title="SERIES_1:func.3">Sum</a> <span class="p5">(<span class="default"><span class="p0"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">rExpSeq</a></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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#K3" title="SERIES_1:func.3">Sum</a> <span class="p5">(<span class="default"><span class="p0"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c3">w</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">rExpSeq</a></span>)</span></span>)</span></span>)</span></span>)</span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="comseq_3.html#T7" title="COMSEQ_3:th.7">COMSEQ_3:7</a></span>;<br/><a NAME="E12:45_3"/><span class="lab"><font color="Green" title="E61">A32</font></span>: 
<font color="Maroon" title="c8">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="c8">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#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><a NAME="E13:45_3"/><span class="kw">then </span>
<font color="Maroon" title="c8">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:45_3"><span class="lab"><font color="Green" title="E59">A23</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E14:45_3"/><span class="kw">then </span>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p4">(<span class="default"><span class="p5"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">rExpSeq</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="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#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p4">(<span class="default"><span class="p5"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">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="c8">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="c5">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#K3" title="SERIES_1:func.3">Sum</a> <span class="p4">(<span class="default"><span class="p5"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">rExpSeq</a></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">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#K3" title="SERIES_1:func.3">Sum</a> <span class="p4">(<span class="default"><span class="p5"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c3">w</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">rExpSeq</a></span>)</span></span>)</span></span>)</span></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:45"><span class="lab"><font color="Green" title="E50">A7</font></span></a>, <a class="txt" href="#E12:45_3"><span class="lab"><font color="Green" title="E61">A32</font></span></a></span>;<br/><a NAME="E15:45_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">rExpSeq</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="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#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">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="c8">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="c5">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#K3" title="SERIES_1:func.3">Sum</a> <span class="p4">(<span class="default"><span class="p5"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">rExpSeq</a></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">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#K3" title="SERIES_1:func.3">Sum</a> <span class="p4">(<span class="default"><span class="p5"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c3">w</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">rExpSeq</a></span>)</span></span>)</span></span>)</span></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:45_3"><span class="lab"><font color="Green" title="E56">A20</font></span></a>, <a class="txt" href="#E6:45_3"><span class="lab"><font color="Green" title="E57">A21</font></span></a>, <a class="ref" href="lopban_4.html#T30" target="_self" title="LOPBAN_4:th.30">Th30</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E16:45_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A33</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#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p4">(<span class="default"><span class="p5"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">rExpSeq</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="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#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p4">(<span class="default"><span class="p5"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">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="c8">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#K3" title="SERIES_1:func.3">Sum</a> <span class="p3">(<span class="default"><span class="p4"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c3">w</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">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="c5">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#K3" title="SERIES_1:func.3">Sum</a> <span class="p5">(<span class="default"><span class="p0"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">rExpSeq</a></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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#K3" title="SERIES_1:func.3">Sum</a> <span class="p5">(<span class="default"><span class="p0"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c3">w</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">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#K3" title="SERIES_1:func.3">Sum</a> <span class="p3">(<span class="default"><span class="p4"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c3">w</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">rExpSeq</a></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:45"><span class="lab"><font color="Green" title="E47">A4</font></span></a>, <a class="ref" href="xreal_1.html#T64" title="XREAL_1:th.64">XREAL_1:64</a></span>;<br/><a NAME="E17:45_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="c11">k</font> holds <br/><span class="p1"><a href="normsp_0.html#K4" title="NORMSP_0:func.4">||.</a><span class="default"><span class="p2">(<span class="default"><a href="lopban_4.html#K8" title="LOPBAN_4:func.8">Conj</a> (<font color="Maroon" title="c11">k</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">w</font>)</span>)</span></span><a href="normsp_0.html#K4" title="NORMSP_0:func.4">.||</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#K3" title="SERIES_1:func.3">Sum</a> <span class="p3">(<span class="default"><span class="p4"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c3">w</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">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="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">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="#E20:45"><span class="lab"><font color="Green" title="E52">A12</font></span></a></span>;<br/><a NAME="E18:45_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p3"><a href="normsp_0.html#K4" title="NORMSP_0:func.4">||.</a><span class="default"><span class="p4">(<span class="default"><a href="lopban_4.html#K8" title="LOPBAN_4:func.8">Conj</a> (<font color="Maroon" title="c11">k</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">w</font>)</span>)</span></span><a href="normsp_0.html#K4" title="NORMSP_0:func.4">.||</a></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="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#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p3"><a href="normsp_0.html#K4" title="NORMSP_0:func.4">||.</a><span class="default"><span class="p4">(<span class="default"><a href="lopban_4.html#K8" title="LOPBAN_4:func.8">Conj</a> (<font color="Maroon" title="c11">k</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">w</font>)</span>)</span></span><a href="normsp_0.html#K4" title="NORMSP_0:func.4">.||</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="c8">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#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p4">(<span class="default"><span class="p5"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">rExpSeq</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="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#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p4">(<span class="default"><span class="p5"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">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="c8">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#K3" title="SERIES_1:func.3">Sum</a> <span class="p3">(<span class="default"><span class="p4"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c3">w</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">rExpSeq</a></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:45_3"><span class="lab"><font color="Green" title="E58">A22</font></span></a>, <a class="ref" href="comseq_3.html#T8" title="COMSEQ_3:th.8">COMSEQ_3:8</a></span>;<br/><a NAME="E19:45_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A34</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p3"><a href="normsp_0.html#K4" title="NORMSP_0:func.4">||.</a><span class="default"><span class="p4">(<span class="default"><a href="lopban_4.html#K8" title="LOPBAN_4:func.8">Conj</a> (<font color="Maroon" title="c11">k</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">w</font>)</span>)</span></span><a href="normsp_0.html#K4" title="NORMSP_0:func.4">.||</a></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="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#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p3"><a href="normsp_0.html#K4" title="NORMSP_0:func.4">||.</a><span class="default"><span class="p4">(<span class="default"><a href="lopban_4.html#K8" title="LOPBAN_4:func.8">Conj</a> (<font color="Maroon" title="c11">k</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">w</font>)</span>)</span></span><a href="normsp_0.html#K4" title="NORMSP_0:func.4">.||</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="c8">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="c5">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#K3" title="SERIES_1:func.3">Sum</a> <span class="p5">(<span class="default"><span class="p0"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">rExpSeq</a></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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#K3" title="SERIES_1:func.3">Sum</a> <span class="p5">(<span class="default"><span class="p0"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c3">w</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">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#K3" title="SERIES_1:func.3">Sum</a> <span class="p3">(<span class="default"><span class="p4"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c3">w</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">rExpSeq</a></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:45_3"><span class="lab"><font color="Green" title="E62">A33</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E20:45_3"/><span class="lab"><font color="Green" title="E64">A35</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#K3" title="SERIES_1:func.3">Sum</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c3">w</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">rExpSeq</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lopban_4.html#T29" target="_self" title="LOPBAN_4:th.29">Th29</a></span>;<br/><a NAME="E21:45_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">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="c8">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="c5">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#K3" title="SERIES_1:func.3">Sum</a> <span class="p5">(<span class="default"><span class="p0"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">rExpSeq</a></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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#K3" title="SERIES_1:func.3">Sum</a> <span class="p5">(<span class="default"><span class="p0"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c3">w</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">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#K3" title="SERIES_1:func.3">Sum</a> <span class="p2">(<span class="default"><span class="p3"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">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="c5">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#K3" title="SERIES_1:func.3">Sum</a> <span class="p5">(<span class="default"><span class="p0"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">rExpSeq</a></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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#K3" title="SERIES_1:func.3">Sum</a> <span class="p5">(<span class="default"><span class="p0"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c3">w</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">rExpSeq</a></span>)</span></span>)</span></span>)</span></span>)</span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:45"><span class="lab"><font color="Green" title="E48">A5</font></span></a>, <a class="ref" href="lopban_4.html#T28" target="_self" title="LOPBAN_4:th.28">Th28</a>, <a class="ref" href="xreal_1.html#T64" title="XREAL_1:th.64">XREAL_1:64</a></span>;<br/><a NAME="E22:45_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A36</font></span>: 
<span class="p1">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p2"><a href="normsp_0.html#K4" title="NORMSP_0:func.4">||.</a><span class="default"><span class="p3">(<span class="default"><a href="lopban_4.html#K8" title="LOPBAN_4:func.8">Conj</a> (<font color="Maroon" title="c11">k</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">w</font>)</span>)</span></span><a href="normsp_0.html#K4" title="NORMSP_0:func.4">.||</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="c8">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#K3" title="SERIES_1:func.3">Sum</a> <span class="p2">(<span class="default"><span class="p3"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">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="c5">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#K3" title="SERIES_1:func.3">Sum</a> <span class="p5">(<span class="default"><span class="p0"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">rExpSeq</a></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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#K3" title="SERIES_1:func.3">Sum</a> <span class="p5">(<span class="default"><span class="p0"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c3">w</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">rExpSeq</a></span>)</span></span>)</span></span>)</span></span>)</span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:45_3"><span class="lab"><font color="Green" title="E60">A31</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E23:45_3"/>
<span class="p1">(<span class="default"><a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p2">(<span class="default"><span class="p3"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">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="c5">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#K3" title="SERIES_1:func.3">Sum</a> <span class="p5">(<span class="default"><span class="p0"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">rExpSeq</a></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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#K3" title="SERIES_1:func.3">Sum</a> <span class="p5">(<span class="default"><span class="p0"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c3">w</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">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#K3" title="SERIES_1:func.3">Sum</a> <span class="p2">(<span class="default"><span class="p3"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">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="c4">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#K3" title="SERIES_1:func.3">Sum</a> <span class="p4">(<span class="default"><span class="p5"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">rExpSeq</a></span>)</span></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:45"><span class="lab"><font color="Green" title="E45">A2</font></span></a>, <a class="ref" href="xreal_1.html#T64" title="XREAL_1:th.64">XREAL_1:64</a>, <a class="ref" href="xxreal_0.html#T17" title="XXREAL_0:th.17">XXREAL_0:17</a></span>;<br/><a NAME="E24:45_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p2"><a href="normsp_0.html#K4" title="NORMSP_0:func.4">||.</a><span class="default"><span class="p3">(<span class="default"><a href="lopban_4.html#K8" title="LOPBAN_4:func.8">Conj</a> (<font color="Maroon" title="c11">k</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">w</font>)</span>)</span></span><a href="normsp_0.html#K4" title="NORMSP_0:func.4">.||</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="c8">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="c4">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="#E22:45_3"><span class="lab"><font color="Green" title="E65">A36</font></span></a>, <a class="txt" href="#E4:45_3"><span class="lab"><font color="Green" title="E55">A19</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E25:45_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A37</font></span>: 
<span class="p1">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p2"><a href="normsp_0.html#K4" title="NORMSP_0:func.4">||.</a><span class="default"><span class="p3">(<span class="default"><a href="lopban_4.html#K8" title="LOPBAN_4:func.8">Conj</a> (<font color="Maroon" title="c11">k</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">w</font>)</span>)</span></span><a href="normsp_0.html#K4" title="NORMSP_0:func.4">.||</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="c8">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="c4">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="#E2:45_3"><span class="lab"><font color="Green" title="E53">A17</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E26:45_3"/>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p2">(<span class="default"><span class="p3"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c3">w</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">rExpSeq</a></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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#K3" title="SERIES_1:func.3">Sum</a> <span class="p2">(<span class="default"><span class="p3"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c3">w</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">rExpSeq</a></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:45"><span class="lab"><font color="Green" title="E47">A4</font></span></a>, <a class="ref" href="xreal_1.html#T68" title="XREAL_1:th.68">XREAL_1:68</a></span>;<br/><a NAME="E27:45_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A38</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#K3" title="SERIES_1:func.3">Sum</a> <span class="p3">(<span class="default"><span class="p4"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c3">w</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">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="c5">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#K3" title="SERIES_1:func.3">Sum</a> <span class="p5">(<span class="default"><span class="p0"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">rExpSeq</a></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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#K3" title="SERIES_1:func.3">Sum</a> <span class="p5">(<span class="default"><span class="p0"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c3">w</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">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#K3" title="SERIES_1:func.3">Sum</a> <span class="p3">(<span class="default"><span class="p4"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c3">w</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">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="c4">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#K3" title="SERIES_1:func.3">Sum</a> <span class="p4">(<span class="default"><span class="p5"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c3">w</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">rExpSeq</a></span>)</span></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T64" title="XREAL_1:th.64">XREAL_1:64</a>, <a class="ref" href="xxreal_0.html#T17" title="XXREAL_0:th.17">XXREAL_0:17</a></span>;<br/><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#K3" title="SERIES_1:func.3">Sum</a> <span class="p3">(<span class="default"><span class="p4"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c3">w</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">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="c4">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#K3" title="SERIES_1:func.3">Sum</a> <span class="p4">(<span class="default"><span class="p5"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c3">w</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">rExpSeq</a></span>)</span></span>)</span></span>)</span></span>)</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#K3" title="SERIES_1:func.3">Sum</a> <span class="p3">(<span class="default"><span class="p4"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c3">w</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">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="c4">p</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><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#K3" title="SERIES_1:func.3">Sum</a> <span class="p5">(<span class="default"><span class="p0"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c3">w</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">rExpSeq</a></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_0.html#D9" title="XCMPLX_0:def.9">XCMPLX_0:def 9</a></span>
<br/>.= 
<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> <span class="p3">(<span class="default"><a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p4">(<span class="default"><span class="p5"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c3">w</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">rExpSeq</a></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c4">p</font></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">4 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p4">(<span class="default"><span class="p5"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c3">w</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">rExpSeq</a></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a></span>)</span>

<br/>.= 
<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="c4">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#K3" title="SERIES_1:func.3">Sum</a> <span class="p3">(<span class="default"><span class="p4"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c3">w</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">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#K3" title="SERIES_1:func.3">Sum</a> <span class="p3">(<span class="default"><span class="p4"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c3">w</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">rExpSeq</a></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_0.html#D9" title="XCMPLX_0:def.9">XCMPLX_0:def 9</a></span>
<br/>.= 
<span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 4
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:45_3"><span class="lab"><font color="Green" title="E64">A35</font></span></a>, <a class="ref" href="xcmplx_1.html#T91" title="XCMPLX_1:th.91">XCMPLX_1:91</a></span>
<br/>.= 
<font color="Maroon" title="c4">p</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2

;<br/><a NAME="E29:45_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p3"><a href="normsp_0.html#K4" title="NORMSP_0:func.4">||.</a><span class="default"><span class="p4">(<span class="default"><a href="lopban_4.html#K8" title="LOPBAN_4:func.8">Conj</a> (<font color="Maroon" title="c11">k</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">w</font>)</span>)</span></span><a href="normsp_0.html#K4" title="NORMSP_0:func.4">.||</a></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="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#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p3"><a href="normsp_0.html#K4" title="NORMSP_0:func.4">||.</a><span class="default"><span class="p4">(<span class="default"><a href="lopban_4.html#K8" title="LOPBAN_4:func.8">Conj</a> (<font color="Maroon" title="c11">k</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">w</font>)</span>)</span></span><a href="normsp_0.html#K4" title="NORMSP_0:func.4">.||</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="c8">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="c4">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="#E19:45_3"><span class="lab"><font color="Green" title="E63">A34</font></span></a>, <a class="txt" href="#E27:45_3"><span class="lab"><font color="Green" title="E67">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="E30:45_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p4"><a href="normsp_0.html#K4" title="NORMSP_0:func.4">||.</a><span class="default"><span class="p5">(<span class="default"><a href="lopban_4.html#K8" title="LOPBAN_4:func.8">Conj</a> (<font color="Maroon" title="c11">k</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">w</font>)</span>)</span></span><a href="normsp_0.html#K4" title="NORMSP_0:func.4">.||</a></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="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#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p4"><a href="normsp_0.html#K4" title="NORMSP_0:func.4">||.</a><span class="default"><span class="p5">(<span class="default"><a href="lopban_4.html#K8" title="LOPBAN_4:func.8">Conj</a> (<font color="Maroon" title="c11">k</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">w</font>)</span>)</span></span><a href="normsp_0.html#K4" title="NORMSP_0:func.4">.||</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="c8">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#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p3"><a href="normsp_0.html#K4" title="NORMSP_0:func.4">||.</a><span class="default"><span class="p4">(<span class="default"><a href="lopban_4.html#K8" title="LOPBAN_4:func.8">Conj</a> (<font color="Maroon" title="c11">k</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">w</font>)</span>)</span></span><a href="normsp_0.html#K4" title="NORMSP_0:func.4">.||</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="c8">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="c4">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="c4">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="#E25:45_3"><span class="lab"><font color="Green" title="E66">A37</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><span class="kw">hence </span><a NAME="E31:45_3"/>
 <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#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p3"><a href="normsp_0.html#K4" title="NORMSP_0:func.4">||.</a><span class="default"><span class="p4">(<span class="default"><a href="lopban_4.html#K8" title="LOPBAN_4:func.8">Conj</a> (<font color="Maroon" title="c11">k</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">w</font>)</span>)</span></span><a href="normsp_0.html#K4" title="NORMSP_0:func.4">.||</a></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="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">p</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lopban_4.html#T31" target="_self" title="LOPBAN_4:th.31">Th31</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E22:45"/>
 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="c10">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#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p3"><a href="normsp_0.html#K4" title="NORMSP_0:func.4">||.</a><span class="default"><span class="p4">(<span class="default"><a href="lopban_4.html#K8" title="LOPBAN_4:func.8">Conj</a> (<font color="Olive" title="b1">k</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">w</font>)</span>)</span></span><a href="normsp_0.html#K4" title="NORMSP_0:func.4">.||</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="c4">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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