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<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">X</font> be   <a href="clopban2.html#NM3" title="CLOPBAN2:NM.3">Complex_Banach_Algebra</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">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> <br/> for <font color="Olive" title="b2">z</font>, <font color="Olive" title="b3">w</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b2">z</font>,<font color="Olive" title="b3">w</font> <a href="lopban_4.html#R1" title="LOPBAN_4:pred.1">are_commutative</a>  holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b2">z</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b3">w</font></span>)</span> <a href="clopban3.html#K6" title="CLOPBAN3:func.6">#N</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Olive" title="b1">n</font>,<font color="Olive" title="b2">z</font>,<font color="Olive" title="b3">w</font></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b1">n</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">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>  st <font color="Olive" title="b1">z</font>,<font color="Olive" title="b2">w</font> <a href="lopban_4.html#R1" title="LOPBAN_4:pred.1">are_commutative</a>  holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b1">z</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b2">w</font></span>)</span> <a href="clopban3.html#K6" title="CLOPBAN3:func.6">#N</a> <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c2">n</font>,<font color="Olive" title="b1">z</font>,<font color="Olive" title="b2">w</font></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c2">n</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">z</font>, <font color="Maroon" title="c4">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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font> <a href="lopban_4.html#R1" title="LOPBAN_4:pred.1">are_commutative</a>  implies <span class="p1">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K6" title="CLOPBAN3:func.6">#N</a> <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c2">n</font> )</span><br/>







<span class="kw">assume </span><a NAME="E1:24"/><span class="lab"><font color="Green" title="E23">A1</font></span>: 
<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font> <a href="lopban_4.html#R1" title="LOPBAN_4:pred.1">are_commutative</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K6" title="CLOPBAN3:func.6">#N</a> <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c2">n</font></span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <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">means </span><span class="p1">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K6" title="CLOPBAN3:func.6">#N</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> $1,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> $1;<br/>
<a NAME="E2:24"/><span class="lab"><font color="Green" title="E24">A2</font></span>: 
for <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 <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">n</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:24_1"/><span class="lab"><font color="Green" title="E25">A3</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K6" title="CLOPBAN3:func.6">#N</a> <font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/>

<a NAME="E2:24_1"/><span class="lab"><font color="Green" title="E26">A4</font></span>: 
<font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</a> <font color="Maroon" title="c4">w</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">+</a> <span class="p2">(<span class="default"><a href="lopban_4.html#K4" title="LOPBAN_4:func.4">Shift</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">k</font></span><br/><div><span class="lab"><font color="Green" title="E27">A5</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_1"><span class="kw">now </span></a><div class="add"><div><span class="lab"><font color="Green" title="E28">A6</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:24_1_1_1_1"/><span class="lab"><font color="Green" title="E29">A7</font></span>: 
<font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</a> <font color="Maroon" title="c4">w</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">+</a> <span class="p2">(<span class="default"><a href="lopban_4.html#K4" title="LOPBAN_4:func.4">Shift</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">k</font></span><br/><div><span class="lab"><font color="Green" title="E30">A8</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_1_1_1"><span class="kw">now </span></a><div class="add"><a NAME="E1:24_1_1_1_1_1"/><span class="lab"><font color="Green" title="E31">A9</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24_1_1_1_1"><span class="lab"><font color="Green" title="E29">A7</font></span></a>, <a class="ref" href="int_1.html#T20" title="INT_1:th.20">INT_1:20</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A10</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c6">k</font>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c6">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24_1_1_1_1"><span class="lab"><font color="Green" title="E29">A7</font></span></a>, <a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></span>
;<br/><a NAME="E3:24_1_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c6">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24_1_1_1_1"><span class="lab"><font color="Green" title="E29">A7</font></span></a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A11</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!c</a> </span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!c</a> </span>)</span> <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!c</a> </span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!c</a> </span>)</span> <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!c</a> </span>)</span> <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!c</a> </span>)</span></span>)</span> <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> </span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T92" title="XCMPLX_1:th.92">XCMPLX_1:92</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!c</a> </span>)</span> <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!c</a> </span>)</span> <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!c</a> </span>)</span> <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p4">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> </span>)</span></span>)</span></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!c</a> </span>)</span> <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!c</a> </span>)</span> <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!c</a> </span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24_1_1_1_1_1"><span class="lab"><font color="Green" title="E31">A9</font></span></a>, <a class="ref" href="sin_cos.html#T3" title="SIN_COS:th.3">SIN_COS:3</a></span>
;<br/><span class="kw">assume </span><a NAME="E5:24_1_1_1_1_1"/><span class="lab"><font color="Green" title="E34">A12</font></span>: 
<font color="Maroon" title="c6">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</a> <font color="Maroon" title="c4">w</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">+</a> <span class="p2">(<span class="default"><a href="lopban_4.html#K4" title="LOPBAN_4:func.4">Shift</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">k</font></span><br/><a NAME="E6:24_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A13</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T3" title="NAT_1:th.3">NAT_1:3</a></span>;<br/><a NAME="E7:24_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A14</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> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T20" title="INT_1:th.20">INT_1:20</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A15</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></span>
<br/>.= 
<font color="Maroon" title="c6">k</font>

;<br/><a NAME="E9:24_1_1_1_1_1"/>
<font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></span>;<br/><a NAME="E10:24_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">k</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></span>;<br/><a NAME="E11:24_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">k</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24_1_1_1_1_1"><span class="lab"><font color="Green" title="E31">A9</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E12:24_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A16</font></span>: 
<font color="Maroon" title="c6">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:24_1_1_1_1_1"><span class="lab"><font color="Green" title="E36">A14</font></span></a>, <a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A17</font></span>: <font color="Maroon" title="c5">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> = 
<font color="Maroon" title="c5">n</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></span>
<br/>.= 
<font color="Maroon" title="c5">n</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:24_1_1_1_1_1"><span class="lab"><font color="Green" title="E36">A14</font></span></a>, <a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c6">k</font>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c6">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24_1_1_1_1"><span class="lab"><font color="Green" title="E29">A7</font></span></a>, <a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></span>
;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!c</a> </span>)</span> <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!c</a> </span>)</span></span>)</span> <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <font color="Maroon" title="c6">k</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!c</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!c</a> </span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!c</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!c</a> </span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!c</a> </span>)</span> <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!c</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:24_1_1_1_1_1"><span class="lab"><font color="Green" title="E35">A13</font></span></a>, <a class="txt" href="#E13:24_1_1_1_1_1"><span class="lab"><font color="Green" title="E39">A17</font></span></a>, <a class="ref" href="sin_cos.html#T3" title="SIN_COS:th.3">SIN_COS:3</a></span>
;<br/><a NAME="E15:24_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A18</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!c</a> </span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!c</a> </span>)</span> <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!c</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!c</a> </span>)</span> <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!c</a> </span>)</span> <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!c</a> </span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:24_1_1_1_1_1"><span class="lab"><font color="Green" title="E34">A12</font></span></a>, <a class="ref" href="xcmplx_1.html#T92" title="XCMPLX_1:th.92">XCMPLX_1:92</a></span>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K9" title="SIN_COS:func.9">Coef</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K9" title="SIN_COS:func.9">Coef</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!c</a> </span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!c</a> </span>)</span> <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!c</a> </span>)</span></span>)</span></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K9" title="SIN_COS:func.9">Coef</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24_1_1_1_1_1"><span class="lab"><font color="Green" title="E31">A9</font></span></a>, <a class="ref" href="sin_cos.html#D10" title="SIN_COS:def.10">SIN_COS:def 10</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!c</a> </span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!c</a> </span>)</span> <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!c</a> </span>)</span></span>)</span></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!c</a> </span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!c</a> </span>)</span> <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p5">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!c</a> </span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:24_1_1_1_1_1"><span class="lab"><font color="Green" title="E38">A16</font></span></a>, <a class="ref" href="sin_cos.html#D10" title="SIN_COS:def.10">SIN_COS:def 10</a></span>
;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A19</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K9" title="SIN_COS:func.9">Coef</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K9" title="SIN_COS:func.9">Coef</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!c</a> </span>)</span> <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!c</a> </span>)</span> <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!c</a> </span>)</span> <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!c</a> </span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:24_1_1_1_1_1"><span class="lab"><font color="Green" title="E33">A11</font></span></a>, <a class="txt" href="#E15:24_1_1_1_1_1"><span class="lab"><font color="Green" title="E40">A18</font></span></a>, <a class="ref" href="xcmplx_1.html#T63" title="XCMPLX_1:th.63">XCMPLX_1:63</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!c</a> </span>)</span> <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!c</a> </span>)</span> <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!c</a> </span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!c</a> </span>)</span> <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> </span>)</span></span>)</span></span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!c</a> </span>)</span> <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!c</a> </span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!c</a> </span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!c</a> </span>)</span> <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!c</a> </span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="sin_cos.html#T1" title="SIN_COS:th.1">SIN_COS:1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="sin_cos.html#K9" title="SIN_COS:func.9">Coef</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c6">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24_1_1_1_1"><span class="lab"><font color="Green" title="E29">A7</font></span></a>, <a class="ref" href="sin_cos.html#D10" title="SIN_COS:def.10">SIN_COS:def 10</a></span>
;<br/><span class="lab"><font color="Green" title="E42">A20</font></span>: <font color="Maroon" title="c5">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> = 
<font color="Maroon" title="c5">n</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:24_1_1_1_1_1"><span class="lab"><font color="Green" title="E38">A16</font></span></a>, <a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></span>
<br/>.= 
<font color="Maroon" title="c5">n</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:24_1_1_1_1_1"><span class="lab"><font color="Green" title="E36">A14</font></span></a>, <a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c6">k</font>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c6">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24_1_1_1_1"><span class="lab"><font color="Green" title="E29">A7</font></span></a>, <a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></span>
;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</a> <font color="Maroon" title="c4">w</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">+</a> <span class="p2">(<span class="default"><a href="lopban_4.html#K4" title="LOPBAN_4:func.4">Shift</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">k</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</a> <font color="Maroon" title="c4">w</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_4.html#K4" title="LOPBAN_4:func.4">Shift</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="normsp_1.html#D5" title="NORMSP_1:def.5">NORMSP_1:def 5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">w</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_4.html#K4" title="LOPBAN_4:func.4">Shift</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="clopban3.html#D9" title="CLOPBAN3:def.9">CLOPBAN3:def 9</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">w</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:24_1_1_1_1_1"><span class="lab"><font color="Green" title="E34">A12</font></span></a>, <a class="ref" href="lopban_4.html#T15" title="LOPBAN_4:th.15">LOPBAN_4:15</a>, <a class="ref" href="nat_1.html#T3" title="NAT_1:th.3">NAT_1:3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">w</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">z</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="clopban3.html#D9" title="CLOPBAN3:def.9">CLOPBAN3:def 9</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="sin_cos.html#K9" title="SIN_COS:func.9">Coef</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="clopban3.html#K6" title="CLOPBAN3:func.6">#N</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="clopban3.html#K6" title="CLOPBAN3:func.6">#N</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">w</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">z</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24_1_1_1_1_1"><span class="lab"><font color="Green" title="E31">A9</font></span></a>, <a class="ref" href="clopban4.html#D4" target="_self" title="CLOPBAN4:def.4">Def4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="sin_cos.html#K9" title="SIN_COS:func.9">Coef</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="clopban3.html#K6" title="CLOPBAN3:func.6">#N</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="clopban3.html#K6" title="CLOPBAN3:func.6">#N</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">w</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="sin_cos.html#K9" title="SIN_COS:func.9">Coef</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="clopban3.html#K6" title="CLOPBAN3:func.6">#N</a> <span class="p5">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="clopban3.html#K6" title="CLOPBAN3:func.6">#N</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p5">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">z</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:24_1_1_1_1_1"><span class="lab"><font color="Green" title="E38">A16</font></span></a>, <a class="ref" href="clopban4.html#D4" target="_self" title="CLOPBAN4:def.4">Def4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K9" title="SIN_COS:func.9">Coef</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="clopban3.html#K6" title="CLOPBAN3:func.6">#N</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="clopban3.html#K6" title="CLOPBAN3:func.6">#N</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">w</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="sin_cos.html#K9" title="SIN_COS:func.9">Coef</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="clopban3.html#K6" title="CLOPBAN3:func.6">#N</a> <span class="p5">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="clopban3.html#K6" title="CLOPBAN3:func.6">#N</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p5">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">z</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="clopban3.html#T42" title="CLOPBAN3:th.42">CLOPBAN3:42</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K9" title="SIN_COS:func.9">Coef</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="clopban3.html#K6" title="CLOPBAN3:func.6">#N</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="clopban3.html#K6" title="CLOPBAN3:func.6">#N</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="sin_cos.html#K9" title="SIN_COS:func.9">Coef</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="clopban3.html#K6" title="CLOPBAN3:func.6">#N</a> <span class="p5">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="clopban3.html#K6" title="CLOPBAN3:func.6">#N</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p5">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">z</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="clopban4.html#E2"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K9" title="SIN_COS:func.9">Coef</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="clopban3.html#K6" title="CLOPBAN3:func.6">#N</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="clopban3.html#K6" title="CLOPBAN3:func.6">#N</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K9" title="SIN_COS:func.9">Coef</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="clopban3.html#K6" title="CLOPBAN3:func.6">#N</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="clopban3.html#K6" title="CLOPBAN3:func.6">#N</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p5">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">z</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="clopban3.html#T42" title="CLOPBAN3:th.42">CLOPBAN3:42</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K9" title="SIN_COS:func.9">Coef</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="clopban3.html#K6" title="CLOPBAN3:func.6">#N</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="clopban3.html#K6" title="CLOPBAN3:func.6">#N</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K9" title="SIN_COS:func.9">Coef</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="clopban3.html#K6" title="CLOPBAN3:func.6">#N</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="clopban3.html#K6" title="CLOPBAN3:func.6">#N</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p5">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="txt" href="clopban4.html#E3"><span class="lab"><font color="Green" title="E2">Lm2</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K9" title="SIN_COS:func.9">Coef</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="clopban3.html#K6" title="CLOPBAN3:func.6">#N</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="clopban3.html#K6" title="CLOPBAN3:func.6">#N</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="sin_cos.html#K9" title="SIN_COS:func.9">Coef</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="clopban3.html#K6" title="CLOPBAN3:func.6">#N</a> <span class="p5">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">z</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="clopban3.html#K6" title="CLOPBAN3:func.6">#N</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="clopban3.html#T42" title="CLOPBAN3:th.42">CLOPBAN3:42</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K9" title="SIN_COS:func.9">Coef</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="clopban3.html#K6" title="CLOPBAN3:func.6">#N</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="clopban3.html#K6" title="CLOPBAN3:func.6">#N</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K9" title="SIN_COS:func.9">Coef</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="clopban3.html#K6" title="CLOPBAN3:func.6">#N</a> <span class="p5">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">z</font></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="clopban3.html#K6" title="CLOPBAN3:func.6">#N</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="clopban3.html#T42" title="CLOPBAN3:th.42">CLOPBAN3:42</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K9" title="SIN_COS:func.9">Coef</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="clopban3.html#K6" title="CLOPBAN3:func.6">#N</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="clopban3.html#K6" title="CLOPBAN3:func.6">#N</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K9" title="SIN_COS:func.9">Coef</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="clopban3.html#K6" title="CLOPBAN3:func.6">#N</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="clopban3.html#K6" title="CLOPBAN3:func.6">#N</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="clopban4.html#E2"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a></span>
;<br/><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="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</a> <font color="Maroon" title="c4">w</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">+</a> <span class="p2">(<span class="default"><a href="lopban_4.html#K4" title="LOPBAN_4:func.4">Shift</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">k</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K9" title="SIN_COS:func.9">Coef</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="clopban3.html#K6" title="CLOPBAN3:func.6">#N</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="clopban3.html#K6" title="CLOPBAN3:func.6">#N</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K9" title="SIN_COS:func.9">Coef</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="clopban3.html#K6" title="CLOPBAN3:func.6">#N</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="clopban3.html#K6" title="CLOPBAN3:func.6">#N</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:24_1_1_1_1_1"><span class="lab"><font color="Green" title="E32">A10</font></span></a>, <a class="txt" href="#E18:24_1_1_1_1_1"><span class="lab"><font color="Green" title="E42">A20</font></span></a>, <a class="txt" href="#E8:24_1_1_1_1_1"><span class="lab"><font color="Green" title="E37">A15</font></span></a>, <a class="ref" href="clopban3.html#T42" title="CLOPBAN3:th.42">CLOPBAN3:42</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K9" title="SIN_COS:func.9">Coef</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="clopban3.html#K6" title="CLOPBAN3:func.6">#N</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="clopban3.html#K6" title="CLOPBAN3:func.6">#N</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K9" title="SIN_COS:func.9">Coef</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="clopban3.html#K6" title="CLOPBAN3:func.6">#N</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="clopban3.html#K6" title="CLOPBAN3:func.6">#N</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="clopban3.html#T42" title="CLOPBAN3:th.42">CLOPBAN3:42</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K9" title="SIN_COS:func.9">Coef</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K9" title="SIN_COS:func.9">Coef</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="clopban3.html#K6" title="CLOPBAN3:func.6">#N</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="clopban3.html#K6" title="CLOPBAN3:func.6">#N</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="clopban3.html#T42" title="CLOPBAN3:th.42">CLOPBAN3:42</a></span>
;<br/><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="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</a> <font color="Maroon" title="c4">w</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">+</a> <span class="p2">(<span class="default"><a href="lopban_4.html#K4" title="LOPBAN_4:func.4">Shift</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">k</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K9" title="SIN_COS:func.9">Coef</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="clopban3.html#K6" title="CLOPBAN3:func.6">#N</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="clopban3.html#K6" title="CLOPBAN3:func.6">#N</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:24_1_1_1_1_1"><span class="lab"><font color="Green" title="E41">A19</font></span></a>, <a class="ref" href="clopban3.html#T42" title="CLOPBAN3:th.42">CLOPBAN3:42</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24_1_1_1_1"><span class="lab"><font color="Green" title="E29">A7</font></span></a>, <a class="ref" href="clopban4.html#D4" target="_self" title="CLOPBAN4:def.4">Def4</a></span>
;<br/><span class="kw">hence </span><a NAME="E22:24_1_1_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</a> <font color="Maroon" title="c4">w</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">+</a> <span class="p2">(<span class="default"><a href="lopban_4.html#K4" title="LOPBAN_4:func.4">Shift</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_1_1_2"><span class="kw">now </span></a><div class="add"><a NAME="E1:24_1_1_1_1_2"/><span class="lab"><font color="Green" title="E31">A21</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T2" title="NAT_1:th.2">NAT_1:2</a>, <a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A22</font></span>: <span class="p1">(<span class="default"><a href="sin_cos.html#K9" title="SIN_COS:func.9">Coef</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!c</a> </span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!c</a> </span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="sin_cos.html#T1" title="SIN_COS:th.1">SIN_COS:1</a>, <a class="ref" href="sin_cos.html#D10" title="SIN_COS:def.10">SIN_COS:def 10</a></span>
<br/>.= 
1
<span class="kw">by</span> <span class="lab"><a class="ref" href="sin_cos.html#T1" title="SIN_COS:th.1">SIN_COS:1</a>, <a class="ref" href="xcmplx_1.html#T60" title="XCMPLX_1:th.60">XCMPLX_1:60</a></span>
;<br/><a NAME="E3:24_1_1_1_1_2"/><span class="lab"><font color="Green" title="E33">A23</font></span>: 
<font color="Maroon" title="c5">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">n</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T2" title="NAT_1:th.2">NAT_1:2</a>, <a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A24</font></span>: <span class="p1">(<span class="default"><a href="sin_cos.html#K9" title="SIN_COS:func.9">Coef</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!c</a> </span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!c</a> </span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="sin_cos.html#T1" title="SIN_COS:th.1">SIN_COS:1</a>, <a class="ref" href="sin_cos.html#D10" title="SIN_COS:def.10">SIN_COS:def 10</a></span>
<br/>.= 
1
<span class="kw">by</span> <span class="lab"><a class="ref" href="sin_cos.html#T1" title="SIN_COS:th.1">SIN_COS:1</a>, <a class="ref" href="xcmplx_1.html#T60" title="XCMPLX_1:th.60">XCMPLX_1:60</a></span>
;<br/><a NAME="E5:24_1_1_1_1_2"/><span class="lab"><font color="Green" title="E35">A25</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> <font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T2" title="NAT_1:th.2">NAT_1:2</a></span>;<br/><a NAME="E6:24_1_1_1_1_2"/><span class="lab"><font color="Green" title="E36">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> <font color="Maroon" title="c5">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T2" title="NAT_1:th.2">NAT_1:2</a></span>;<br/><span class="kw">assume </span><a NAME="E7:24_1_1_1_1_2"/><span class="lab"><font color="Green" title="E37">A27</font></span>: 
<font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</a> <font color="Maroon" title="c4">w</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">+</a> <span class="p2">(<span class="default"><a href="lopban_4.html#K4" title="LOPBAN_4:func.4">Shift</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">k</font></span><br/><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="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</a> <font color="Maroon" title="c4">w</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">+</a> <span class="p2">(<span class="default"><a href="lopban_4.html#K4" title="LOPBAN_4:func.4">Shift</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">k</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</a> <font color="Maroon" title="c4">w</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_4.html#K4" title="LOPBAN_4:func.4">Shift</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="normsp_1.html#D5" title="NORMSP_1:def.5">NORMSP_1:def 5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">w</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_4.html#K4" title="LOPBAN_4:func.4">Shift</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="clopban3.html#D9" title="CLOPBAN3:def.9">CLOPBAN3:def 9</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">w</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lopban_4.html#D5" title="LOPBAN_4:def.5">LOPBAN_4:def 5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">w</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="clopban3.html#T42" title="CLOPBAN3:th.42">CLOPBAN3:42</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K9" title="SIN_COS:func.9">Coef</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="clopban3.html#K6" title="CLOPBAN3:func.6">#N</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="clopban3.html#K6" title="CLOPBAN3:func.6">#N</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">w</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:24_1_1_1_1_2"><span class="lab"><font color="Green" title="E36">A26</font></span></a>, <a class="ref" href="clopban4.html#D4" target="_self" title="CLOPBAN4:def.4">Def4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K9" title="SIN_COS:func.9">Coef</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="clopban3.html#K6" title="CLOPBAN3:func.6">#N</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="clopban3.html#K6" title="CLOPBAN3:func.6">#N</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">w</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="clopban3.html#T42" title="CLOPBAN3:th.42">CLOPBAN3:42</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K9" title="SIN_COS:func.9">Coef</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="clopban3.html#K6" title="CLOPBAN3:func.6">#N</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="clopban3.html#K6" title="CLOPBAN3:func.6">#N</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:24_1_1_1_1_2"><span class="lab"><font color="Green" title="E33">A23</font></span></a>, <a class="txt" href="#E1:24_1_1_1_1_2"><span class="lab"><font color="Green" title="E31">A21</font></span></a>, <a class="txt" href="#E4:24_1_1_1_1_2"><span class="lab"><font color="Green" title="E34">A24</font></span></a>, <a class="txt" href="#E2:24_1_1_1_1_2"><span class="lab"><font color="Green" title="E32">A22</font></span></a>, <a class="txt" href="clopban4.html#E2"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:24_1_1_1_1_2"><span class="lab"><font color="Green" title="E37">A27</font></span></a>, <a class="txt" href="#E5:24_1_1_1_1_2"><span class="lab"><font color="Green" title="E35">A25</font></span></a>, <a class="ref" href="clopban4.html#D4" target="_self" title="CLOPBAN4:def.4">Def4</a></span>
;<br/><span class="kw">hence </span><a NAME="E9:24_1_1_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</a> <font color="Maroon" title="c4">w</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">+</a> <span class="p2">(<span class="default"><a href="lopban_4.html#K4" title="LOPBAN_4:func.4">Shift</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:24_1_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</a> <font color="Maroon" title="c4">w</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">+</a> <span class="p2">(<span class="default"><a href="lopban_4.html#K4" title="LOPBAN_4:func.4">Shift</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:24_1_1_1_1"><span class="lab"><font color="Green" title="E30">A8</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><span class="lab"><font color="Green" title="E29">A28</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_1_2"><span class="kw">now </span></a><div class="add"><span class="lab"><font color="Green" title="E30">A29</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></span>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 

;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A30</font></span>: <span class="p1">(<span class="default"><a href="sin_cos.html#K9" title="SIN_COS:func.9">Coef</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</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="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!c</a> </span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!c</a> </span>)</span> <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="sin_cos.html#T1" title="SIN_COS:th.1">SIN_COS:1</a>, <a class="ref" href="sin_cos.html#D10" title="SIN_COS:def.10">SIN_COS:def 10</a></span>
<br/>.= 
1
<span class="kw">by</span> <span class="lab"><a class="ref" href="sin_cos.html#T1" title="SIN_COS:th.1">SIN_COS:1</a>, <a class="ref" href="xcmplx_1.html#T60" title="XCMPLX_1:th.60">XCMPLX_1:60</a></span>
;<br/><a NAME="E3:24_1_1_1_2"/><span class="lab"><font color="Green" title="E32">A31</font></span>: 
<font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></span>;<br/><span class="lab"><font color="Green" title="E33">A32</font></span>: <font color="Maroon" title="c5">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c5">n</font> = 
<font color="Maroon" title="c5">n</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c5">n</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></span>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 

;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A33</font></span>: <span class="p1">(<span class="default"><a href="sin_cos.html#K9" title="SIN_COS:func.9">Coef</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c5">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!c</a> </span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!c</a> </span>)</span> <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="sin_cos.html#T1" title="SIN_COS:th.1">SIN_COS:1</a>, <a class="ref" href="sin_cos.html#D10" title="SIN_COS:def.10">SIN_COS:def 10</a></span>
<br/>.= 
1
<span class="kw">by</span> <span class="lab"><a class="ref" href="sin_cos.html#T1" title="SIN_COS:th.1">SIN_COS:1</a>, <a class="ref" href="xcmplx_1.html#T60" title="XCMPLX_1:th.60">XCMPLX_1:60</a></span>
;<br/><span class="kw">assume </span><a NAME="E6:24_1_1_1_2"/><span class="lab"><font color="Green" title="E35">A34</font></span>: 
<font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</a> <font color="Maroon" title="c4">w</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">+</a> <span class="p2">(<span class="default"><a href="lopban_4.html#K4" title="LOPBAN_4:func.4">Shift</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">k</font></span><br/><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="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</a> <font color="Maroon" title="c4">w</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">+</a> <span class="p2">(<span class="default"><a href="lopban_4.html#K4" title="LOPBAN_4:func.4">Shift</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">k</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</a> <font color="Maroon" title="c4">w</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_4.html#K4" title="LOPBAN_4:func.4">Shift</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="normsp_1.html#D5" title="NORMSP_1:def.5">NORMSP_1:def 5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">w</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_4.html#K4" title="LOPBAN_4:func.4">Shift</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="clopban3.html#D9" title="CLOPBAN3:def.9">CLOPBAN3:def 9</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">w</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_4.html#K4" title="LOPBAN_4:func.4">Shift</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:24_1_1_1_2"><span class="lab"><font color="Green" title="E32">A31</font></span></a>, <a class="ref" href="clopban4.html#D4" target="_self" title="CLOPBAN4:def.4">Def4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_4.html#K4" title="LOPBAN_4:func.4">Shift</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="clopban3.html#T42" title="CLOPBAN3:th.42">CLOPBAN3:42</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="lopban_4.html#K4" title="LOPBAN_4:func.4">Shift</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="clopban3.html#T42" title="CLOPBAN3:th.42">CLOPBAN3:42</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lopban_4.html#D5" title="LOPBAN_4:def.5">LOPBAN_4:def 5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">z</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="clopban3.html#D9" title="CLOPBAN3:def.9">CLOPBAN3:def 9</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K9" title="SIN_COS:func.9">Coef</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="clopban3.html#K6" title="CLOPBAN3:func.6">#N</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="clopban3.html#K6" title="CLOPBAN3:func.6">#N</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">z</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="clopban4.html#D4" target="_self" title="CLOPBAN4:def.4">Def4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K9" title="SIN_COS:func.9">Coef</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="clopban3.html#K6" title="CLOPBAN3:func.6">#N</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="clopban3.html#K6" title="CLOPBAN3:func.6">#N</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">z</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="clopban3.html#T42" title="CLOPBAN3:th.42">CLOPBAN3:42</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K9" title="SIN_COS:func.9">Coef</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="clopban3.html#K6" title="CLOPBAN3:func.6">#N</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="clopban3.html#K6" title="CLOPBAN3:func.6">#N</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="txt" href="clopban4.html#E3"><span class="lab"><font color="Green" title="E2">Lm2</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K9" title="SIN_COS:func.9">Coef</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="clopban3.html#K6" title="CLOPBAN3:func.6">#N</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">z</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="clopban3.html#K6" title="CLOPBAN3:func.6">#N</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="clopban3.html#T42" title="CLOPBAN3:th.42">CLOPBAN3:42</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K9" title="SIN_COS:func.9">Coef</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="clopban3.html#K6" title="CLOPBAN3:func.6">#N</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">z</font></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="clopban3.html#K6" title="CLOPBAN3:func.6">#N</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="clopban3.html#T42" title="CLOPBAN3:th.42">CLOPBAN3:42</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K9" title="SIN_COS:func.9">Coef</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</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="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="clopban3.html#K6" title="CLOPBAN3:func.6">#N</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="clopban3.html#K6" title="CLOPBAN3:func.6">#N</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:24_1_1_1_2"><span class="lab"><font color="Green" title="E34">A33</font></span></a>, <a class="txt" href="#E2:24_1_1_1_2"><span class="lab"><font color="Green" title="E31">A30</font></span></a>, <a class="txt" href="clopban4.html#E2"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:24_1_1_1_2"><span class="lab"><font color="Green" title="E35">A34</font></span></a>, <a class="txt" href="#E4:24_1_1_1_2"><span class="lab"><font color="Green" title="E33">A32</font></span></a>, <a class="txt" href="#E1:24_1_1_1_2"><span class="lab"><font color="Green" title="E30">A29</font></span></a>, <a class="ref" href="clopban4.html#D4" target="_self" title="CLOPBAN4:def.4">Def4</a></span>
;<br/><span class="kw">hence </span><a NAME="E8:24_1_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</a> <font color="Maroon" title="c4">w</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">+</a> <span class="p2">(<span class="default"><a href="lopban_4.html#K4" title="LOPBAN_4:func.4">Shift</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">assume </span><a NAME="E3:24_1_1_1"/>
<font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</a> <font color="Maroon" title="c4">w</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">+</a> <span class="p2">(<span class="default"><a href="lopban_4.html#K4" title="LOPBAN_4:func.4">Shift</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">k</font></span><br/><span class="kw">hence </span><a NAME="E4:24_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</a> <font color="Maroon" title="c4">w</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">+</a> <span class="p2">(<span class="default"><a href="lopban_4.html#K4" title="LOPBAN_4:func.4">Shift</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:24_1_1_1"><span class="lab"><font color="Green" title="E29">A28</font></span></a>, <a class="txt" href="#E1:24_1_1_1"><span class="lab"><font color="Green" title="E28">A6</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:24_1_1_2"/><span class="lab"><font color="Green" title="E28">A35</font></span>: 
<font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</a> <font color="Maroon" title="c4">w</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">+</a> <span class="p2">(<span class="default"><a href="lopban_4.html#K4" title="LOPBAN_4:func.4">Shift</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">k</font></span><br/><a NAME="E2:24_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A36</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">k</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></span>;<br/><a NAME="E3:24_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A37</font></span>: 
<font color="Maroon" title="c5">n</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#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
 <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="E4:24_1_1_2"/>
 <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="c5">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T2" title="NAT_1:th.2">NAT_1:2</a></span>;<br/><a NAME="E5:24_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A38</font></span>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><a NAME="E6:24_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A39</font></span>: 
<font color="Maroon" title="c6">k</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24_1_1_2"><span class="lab"><font color="Green" title="E28">A35</font></span></a>, <a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</a> <font color="Maroon" title="c4">w</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">+</a> <span class="p2">(<span class="default"><a href="lopban_4.html#K4" title="LOPBAN_4:func.4">Shift</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">k</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</a> <font color="Maroon" title="c4">w</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_4.html#K4" title="LOPBAN_4:func.4">Shift</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="normsp_1.html#D5" title="NORMSP_1:def.5">NORMSP_1:def 5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">w</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_4.html#K4" title="LOPBAN_4:func.4">Shift</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="clopban3.html#D9" title="CLOPBAN3:def.9">CLOPBAN3:def 9</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">w</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24_1_1_2"><span class="lab"><font color="Green" title="E28">A35</font></span></a>, <a class="txt" href="#E5:24_1_1_2"><span class="lab"><font color="Green" title="E31">A38</font></span></a>, <a class="ref" href="lopban_4.html#T15" title="LOPBAN_4:th.15">LOPBAN_4:15</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">w</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">z</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="clopban3.html#D9" title="CLOPBAN3:def.9">CLOPBAN3:def 9</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">w</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">z</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:24_1_1_2"><span class="lab"><font color="Green" title="E30">A37</font></span></a>, <a class="ref" href="clopban4.html#D4" target="_self" title="CLOPBAN4:def.4">Def4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">z</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="clopban3.html#T42" title="CLOPBAN3:th.42">CLOPBAN3:42</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">z</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:24_1_1_2"><span class="lab"><font color="Green" title="E29">A36</font></span></a>, <a class="txt" href="#E6:24_1_1_2"><span class="lab"><font color="Green" title="E32">A39</font></span></a>, <a class="ref" href="clopban4.html#D4" target="_self" title="CLOPBAN4:def.4">Def4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="clopban3.html#T42" title="CLOPBAN3:th.42">CLOPBAN3:42</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="clopban3.html#T42" title="CLOPBAN3:th.42">CLOPBAN3:42</a></span>
;<br/><span class="kw">hence </span><a NAME="E8:24_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</a> <font color="Maroon" title="c4">w</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">+</a> <span class="p2">(<span class="default"><a href="lopban_4.html#K4" title="LOPBAN_4:func.4">Shift</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24_1_1_2"><span class="lab"><font color="Green" title="E28">A35</font></span></a>, <a class="ref" href="clopban4.html#D4" target="_self" title="CLOPBAN4:def.4">Def4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:24_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</a> <font color="Maroon" title="c4">w</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">+</a> <span class="p2">(<span class="default"><a href="lopban_4.html#K4" title="LOPBAN_4:func.4">Shift</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24_1_1"><span class="lab"><font color="Green" title="E27">A5</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E4:24_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A40</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</a> <font color="Maroon" title="c4">w</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">+</a> <span class="p1">(<span class="default"><a href="lopban_4.html#K4" title="LOPBAN_4:func.4">Shift</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></span>;<br/>
<a NAME="E5:24_1"/><span class="lab"><font color="Green" title="E28">A41</font></span>: 
<font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></span>;<br/>
<span class="p1">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</a> <font color="Maroon" title="c4">w</font></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</a> <font color="Maroon" title="c4">w</font></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</a> <font color="Maroon" title="c4">w</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bhsp_4.html#D1" title="BHSP_4:def.1">BHSP_4:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</a> <font color="Maroon" title="c4">w</font></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">w</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="clopban3.html#D9" title="CLOPBAN3:def.9">CLOPBAN3:def 9</a></span>
;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E29">A42</font></span>: <span class="p1">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</a> <font color="Maroon" title="c4">w</font></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</a> <font color="Maroon" title="c4">w</font></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">w</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:24_1"><span class="lab"><font color="Green" title="E28">A41</font></span></a>, <a class="ref" href="clopban4.html#D4" target="_self" title="CLOPBAN4:def.4">Def4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</a> <font color="Maroon" title="c4">w</font></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="clopban3.html#T42" title="CLOPBAN3:th.42">CLOPBAN3:42</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</a> <font color="Maroon" title="c4">w</font></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="clopban3.html#T42" title="CLOPBAN3:th.42">CLOPBAN3:42</a></span>
;<br/>
<span class="p1">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bhsp_4.html#D1" title="BHSP_4:def.1">BHSP_4:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">z</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="clopban3.html#D9" title="CLOPBAN3:def.9">CLOPBAN3:def 9</a></span>
;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E30">A43</font></span>: <span class="p1">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">z</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:24_1"><span class="lab"><font color="Green" title="E26">A4</font></span></a>, <a class="ref" href="clopban4.html#D4" target="_self" title="CLOPBAN4:def.4">Def4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="clopban3.html#T42" title="CLOPBAN3:th.42">CLOPBAN3:42</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="clopban3.html#T42" title="CLOPBAN3:th.42">CLOPBAN3:42</a></span>
;<br/>
<a NAME="E10:24_1"/>
<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="c5">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></span>;<br/>
<a NAME="E11:24_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A44</font></span>: 
<span class="p1">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="clopban4.html#D4" target="_self" title="CLOPBAN4:def.4">Def4</a></span>;<br/>
<a NAME="E12:24_1"/>
<span class="p1">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><a href="lopban_4.html#K4" title="LOPBAN_4:func.4">Shift</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="clopban4.html#T15" target="_self" title="CLOPBAN4:th.15">Th15</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E32">A45</font></span>: <span class="p1">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><a href="lopban_4.html#K4" title="LOPBAN_4:func.4">Shift</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">z</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="clopban3.html#D9" title="CLOPBAN3:def.9">CLOPBAN3:def 9</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><a href="lopban_4.html#K4" title="LOPBAN_4:func.4">Shift</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:24_1"><span class="lab"><font color="Green" title="E31">A44</font></span></a>, <a class="ref" href="clopban3.html#T42" title="CLOPBAN3:th.42">CLOPBAN3:42</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><a href="lopban_4.html#K4" title="LOPBAN_4:func.4">Shift</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="clopban3.html#T42" title="CLOPBAN3:th.42">CLOPBAN3:42</a></span>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_7"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">w</font></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</a> <font color="Maroon" title="c4">w</font></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">k</font></span><br/><span class="kw">thus </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">w</font></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">k</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">w</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="clopban3.html#D9" title="CLOPBAN3:def.9">CLOPBAN3:def 9</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">z</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">w</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="clopban3.html#T42" title="CLOPBAN3:th.42">CLOPBAN3:42</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">w</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="clopban3.html#D9" title="CLOPBAN3:def.9">CLOPBAN3:def 9</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</a> <font color="Maroon" title="c4">w</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="clopban3.html#D9" title="CLOPBAN3:def.9">CLOPBAN3:def 9</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</a> <font color="Maroon" title="c4">w</font></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">k</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="normsp_1.html#D5" title="NORMSP_1:def.5">NORMSP_1:def 5</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E15:24_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A46</font></span>: 
<span class="p1">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">w</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</a> <font color="Maroon" title="c4">w</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></span>;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K6" title="CLOPBAN3:func.6">#N</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K5" title="CLOPBAN3:func.5">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="clopban3.html#D13" title="CLOPBAN3:def.13">CLOPBAN3:def 13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K5" title="CLOPBAN3:func.5">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">w</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="clopban3.html#D12" title="CLOPBAN3:def.12">CLOPBAN3:def 12</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">w</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24_1"><span class="lab"><font color="Green" title="E25">A3</font></span></a>, <a class="ref" href="clopban3.html#D13" title="CLOPBAN3:def.13">CLOPBAN3:def 13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span></span>)</span> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">w</font></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="clopban3.html#D9" title="CLOPBAN3:def.9">CLOPBAN3:def 9</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">w</font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="clopban4.html#T9" target="_self" title="CLOPBAN4:th.9">Th9</a></span>
;<br/>
<span class="kw">then </span><span class="p1">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K6" title="CLOPBAN3:func.6">#N</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">+</a> <span class="p2">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</a> <font color="Maroon" title="c4">w</font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:24_1"><span class="lab"><font color="Green" title="E33">A46</font></span></a>, <a class="ref" href="clopban3.html#T19" title="CLOPBAN3:th.19">CLOPBAN3:19</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</a> <font color="Maroon" title="c4">w</font></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="normsp_1.html#D5" title="NORMSP_1:def.5">NORMSP_1:def 5</a></span>
;<br/>
<span class="kw">hence </span><span class="p1">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K6" title="CLOPBAN3:func.6">#N</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</a> <font color="Maroon" title="c4">w</font></span>)</span></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">+</a> <span class="p2">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><a href="lopban_4.html#K4" title="LOPBAN_4:func.4">Shift</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:24_1"><span class="lab"><font color="Green" title="E30">A43</font></span></a>, <a class="txt" href="#E7:24_1"><span class="lab"><font color="Green" title="E29">A42</font></span></a>, <a class="txt" href="#E13:24_1"><span class="lab"><font color="Green" title="E32">A45</font></span></a>, <a class="ref" href="normsp_1.html#D5" title="NORMSP_1:def.5">NORMSP_1:def 5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:24_1"><span class="lab"><font color="Green" title="E27">A40</font></span></a>, <a class="ref" href="clopban3.html#T19" title="CLOPBAN3:th.19">CLOPBAN3:19</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="lab"><font color="Green" title="E25">A47</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K6" title="CLOPBAN3:func.6">#N</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K5" title="CLOPBAN3:func.5">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="clopban3.html#D13" title="CLOPBAN3:def.13">CLOPBAN3:def 13</a></span>
<br/>.= 
 <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">X</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="clopban3.html#D12" title="CLOPBAN3:def.12">CLOPBAN3:def 12</a></span>
;<br/>
<span class="lab"><font color="Green" title="E26">A48</font></span>: <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  = 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D2" title="XREAL_0:def.2">XREAL_0:def 2</a></span>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 

;<br/>
<span class="p1">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  = 
<span class="p1">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bhsp_4.html#D1" title="BHSP_4:def.1">BHSP_4:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K9" title="SIN_COS:func.9">Coef</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="clopban3.html#K6" title="CLOPBAN3:func.6">#N</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="clopban3.html#K6" title="CLOPBAN3:func.6">#N</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:24"><span class="lab"><font color="Green" title="E26">A48</font></span></a>, <a class="ref" href="clopban4.html#D4" target="_self" title="CLOPBAN4:def.4">Def4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a>  <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p3">(<span class="default"><a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a>  <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a> </span>)</span></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="clopban3.html#K6" title="CLOPBAN3:func.6">#N</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="clopban3.html#K6" title="CLOPBAN3:func.6">#N</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:24"><span class="lab"><font color="Green" title="E26">A48</font></span></a>, <a class="ref" href="complex1.html#D7" title="COMPLEX1:def.7">COMPLEX1:def 7</a>, <a class="ref" href="sin_cos.html#T1" title="SIN_COS:th.1">SIN_COS:1</a>, <a class="ref" href="sin_cos.html#D10" title="SIN_COS:def.10">SIN_COS:def 10</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="clopban3.html#K6" title="CLOPBAN3:func.6">#N</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="clopban3.html#K6" title="CLOPBAN3:func.6">#N</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="clvect_1.html#D5" title="CLVECT_1:def.5">CLVECT_1:def 5</a>, <a class="ref" href="complex1.html#D7" title="COMPLEX1:def.7">COMPLEX1:def 7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="clopban3.html#K5" title="CLOPBAN3:func.5">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="clopban3.html#K6" title="CLOPBAN3:func.6">#N</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="clopban3.html#D13" title="CLOPBAN3:def.13">CLOPBAN3:def 13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="clopban3.html#K5" title="CLOPBAN3:func.5">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="clopban3.html#K5" title="CLOPBAN3:func.5">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="clopban3.html#D13" title="CLOPBAN3:def.13">CLOPBAN3:def 13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="clopban3.html#K5" title="CLOPBAN3:func.5">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="clopban3.html#D12" title="CLOPBAN3:def.12">CLOPBAN3:def 12</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">X</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="clopban3.html#D12" title="CLOPBAN3:def.12">CLOPBAN3:def 12</a></span>
<br/>.= 
 <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">X</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="clopban3.html#T42" title="CLOPBAN3:th.42">CLOPBAN3:42</a></span>
;<br/>
<a NAME="E6:24"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A49</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:24"><span class="lab"><font color="Green" title="E25">A47</font></span></a></span>;<br/>
<a NAME="E7:24"/>
for <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>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E6:24"><span class="lab"><font color="Green" title="E27">A49</font></span></a>, <a class="txt" href="#E2:24"><span class="lab"><font color="Green" title="E24">A2</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E8:24"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">w</font></span>)</span> <a href="clopban3.html#K6" title="CLOPBAN3:func.6">#N</a> <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c2">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
