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<b>let </b><font color="Maroon" title="c1">X</font> be   <a href="lopban_2.html#NM3" title="LOPBAN_2:NM.3">Banach_Algebra</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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="lopban_3.html#K6" title="LOPBAN_3: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="lopban_4.html#K5" title="LOPBAN_4:func.5">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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font></span><br/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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="lopban_3.html#K6" title="LOPBAN_3: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="lopban_4.html#K5" title="LOPBAN_4:func.5">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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c2">n</font></span><br/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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="lopban_3.html#K6" title="LOPBAN_3: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="lopban_4.html#K5" title="LOPBAN_4:func.5">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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c2">n</font> )</span><br/>







<b>assume </b><a NAME="E1:29"/><i><font color="Green" title="E29">A1</font></i>: 
<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:()"><i><font color="Red">::  thesis: </font></i></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="lopban_3.html#K6" title="LOPBAN_3: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="lopban_4.html#K5" title="LOPBAN_4:func.5">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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c2">n</font></span><br/>

<i><font color="Green" title="E30">A2</font></i>: <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> 
<b>by </b><i><a class="ref" href="xreal_0.html#D2" title="XREAL_0:def.2">XREAL_0:def 2</a></i>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 

;<br/>
<b>defpred </b><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> ] <b>means </b><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="lopban_3.html#K6" title="LOPBAN_3: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="lopban_4.html#K5" title="LOPBAN_4:func.5">Expan</a> $1,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> $1;<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#K5" title="LOPBAN_4:func.5">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="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"><a href="lopban_4.html#K5" title="LOPBAN_4:func.5">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="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="ref" href="bhsp_4.html#D1" title="BHSP_4:def.1">BHSP_4:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lopban_4.html#K2" title="LOPBAN_4:func.2">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="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="lopban_3.html#K6" title="LOPBAN_3: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="lopban_3.html#K6" title="LOPBAN_3:func.6">#N</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
<b>by </b><i><a class="txt" href="#E2:29"><i><font color="Green" title="E30">A2</font></i></a>, <a class="ref" href="lopban_4.html#D6" target="_self" title="LOPBAN_4:def.6">Def6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default">1 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="lopban_3.html#K6" title="LOPBAN_3: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="lopban_3.html#K6" title="LOPBAN_3:func.6">#N</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
<b>by </b><i><a class="txt" href="#E2:29"><i><font color="Green" title="E30">A2</font></i></a>, <a class="ref" href="lopban_4.html#D3" target="_self" title="LOPBAN_4:def.3">Def3</a>, <a class="ref" href="newton.html#T18" title="NEWTON:th.18">NEWTON:18</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="lopban_3.html#K5" title="LOPBAN_3:func.5">GeoSeq</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="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="lopban_3.html#K6" title="LOPBAN_3:func.6">#N</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9" title="RLVECT_1:def.9">RLVECT_1:def 9</a></i>
<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="lopban_3.html#K5" title="LOPBAN_3:func.5">GeoSeq</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>
<b>by </b><i><a class="ref" href="lopban_3.html#D13" title="LOPBAN_3:def.13">LOPBAN_3:def 13</a></i>
<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>
<b>by </b><i><a class="ref" href="lopban_3.html#D13" title="LOPBAN_3:def.13">LOPBAN_3:def 13</a></i>
<br/>.= 
 <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">X</font>
<b>by </b><i><a class="ref" href="lopban_3.html#T43" title="LOPBAN_3:th.43">LOPBAN_3:43</a></i>
;<br/>
<a NAME="E4:29"/><b>then </b><i><font color="Green" title="E31">A3</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<b>by </b><i><a class="ref" href="lopban_3.html#D13" title="LOPBAN_3:def.13">LOPBAN_3:def 13</a></i>;<br/>
<a NAME="E5:29"/><i><font color="Green" title="E32">A4</font></i>: 
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="29_3"><b>proof </b></a><div class="add">

<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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/>

<b>assume </b><a NAME="E1:29_3"/><i><font color="Green" title="E33">A5</font></i>: 
<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="lopban_3.html#K6" title="LOPBAN_3: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="lopban_4.html#K5" title="LOPBAN_4:func.5">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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c5">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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/>

<i><font color="Green" title="E34">A6</font></i>: <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="lopban_3.html#K6" title="LOPBAN_3: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"><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="lopban_3.html#K5" title="LOPBAN_3:func.5">GeoSeq</a> </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="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>
<b>by </b><i><a class="ref" href="lopban_3.html#D13" title="LOPBAN_3:def.13">LOPBAN_3:def 13</a></i>
<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#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3: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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c5">n</font>
<b>by </b><i><a class="txt" href="#E1:29_3"><i><font color="Green" title="E33">A5</font></i></a>, <a class="ref" href="lopban_3.html#D10" title="LOPBAN_3:def.10">LOPBAN_3:def 10</a></i>
<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="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3: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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c5">n</font>
<b>by </b><i><a class="ref" href="lopban_4.html#T9" target="_self" title="LOPBAN_4:th.9">Th9</a></i>
;<br/>
<a NAME="E3:29_3"/>
<span class="p1">(<span class="default"><a href="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3: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="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3: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="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3:func.3">*</a> <font color="Maroon" title="c4">w</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_3_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_3_2_1"><b>now </b></a><div class="add"><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3: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="nat_1.html#K6" title="NAT_1:func.6">.</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="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3: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="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3:func.3">*</a> <font color="Maroon" title="c4">w</font></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c6">k</font></span><br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3: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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c6">k</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_4.html#K5" title="LOPBAN_4:func.5">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="nat_1.html#K6" title="NAT_1:func.6">.</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>
<b>by </b><i><a class="ref" href="lopban_3.html#D10" title="LOPBAN_3:def.10">LOPBAN_3:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lopban_4.html#K5" title="LOPBAN_4:func.5">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="nat_1.html#K6" title="NAT_1:func.6">.</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="lopban_4.html#K5" title="LOPBAN_4:func.5">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="nat_1.html#K6" title="NAT_1:func.6">.</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>
<b>by </b><i><a class="ref" href="lopban_3.html#T43" title="LOPBAN_3:th.43">LOPBAN_3:43</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3:func.3">*</a> <font color="Maroon" title="c3">z</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="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="lopban_4.html#K5" title="LOPBAN_4:func.5">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="nat_1.html#K6" title="NAT_1:func.6">.</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>
<b>by </b><i><a class="ref" href="lopban_3.html#D10" title="LOPBAN_3:def.10">LOPBAN_3:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3:func.3">*</a> <font color="Maroon" title="c3">z</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="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="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3:func.3">*</a> <font color="Maroon" title="c4">w</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c6">k</font></span>)</span>
<b>by </b><i><a class="ref" href="lopban_3.html#D10" title="LOPBAN_3:def.10">LOPBAN_3:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3: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="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3:func.3">*</a> <font color="Maroon" title="c4">w</font></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c6">k</font>
<b>by </b><i><a class="ref" href="normsp_1.html#D5" title="NORMSP_1:def.5">NORMSP_1:def 5</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:29_3_2"/>
<span class="p1">(<span class="default"><a href="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3: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="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3: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="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3:func.3">*</a> <font color="Maroon" title="c4">w</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>then </b><i><font color="Green" title="E35">A7</font></i>: <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="lopban_3.html#K6" title="LOPBAN_3: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="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3: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="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3:func.3">*</a> <font color="Maroon" title="c4">w</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c5">n</font>
<b>by </b><i><a class="txt" href="#E2:29_3"><i><font color="Green" title="E34">A6</font></i></a>, <a class="ref" href="lopban_3.html#T20" title="LOPBAN_3:th.20">LOPBAN_3:20</a></i>
<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="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span></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="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="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3:func.3">*</a> <font color="Maroon" title="c4">w</font></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c5">n</font></span>)</span>
<b>by </b><i><a class="ref" href="normsp_1.html#D5" title="NORMSP_1:def.5">NORMSP_1:def 5</a></i>
;<br/>
<a NAME="E5:29_3"/><i><font color="Green" title="E36">A8</font></i>: 
<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="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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"><span class="p3">(<span class="default"><a href="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3:func.3">*</a> <font color="Maroon" title="c3">z</font></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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_3_4"><b>proof </b></a><div class="add">

<i><font color="Green" title="E37">A9</font></i>: <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="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3:func.3">*</a> <font color="Maroon" title="c3">z</font></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"><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="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span></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="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="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3:func.3">*</a> <font color="Maroon" title="c3">z</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="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="bhsp_4.html#D1" title="BHSP_4:def.1">BHSP_4:def 1</a></i>
<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="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span></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="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="lopban_4.html#K5" title="LOPBAN_4:func.5">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="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="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">z</font></span>)</span>
<b>by </b><i><a class="ref" href="lopban_3.html#D10" title="LOPBAN_3:def.10">LOPBAN_3:def 10</a></i>
;<br/>
<a NAME="E2:29_3_4"/>
( <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 &amp; <font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 is    <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>  )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>;<br/>
<b>hence </b><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="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3:func.3">*</a> <font color="Maroon" title="c3">z</font></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"><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="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span></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="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>
<b>by </b><i><a class="txt" href="#E1:29_3_4"><i><font color="Green" title="E37">A9</font></i></a>, <a class="ref" href="lopban_4.html#D6" target="_self" title="LOPBAN_4:def.6">Def6</a></i>
<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="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span></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="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>
<b>by </b><i><a class="ref" href="lopban_3.html#T43" title="LOPBAN_3:th.43">LOPBAN_3:43</a></i>
<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="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c5">n</font>
<b>by </b><i><a class="ref" href="lopban_3.html#T43" title="LOPBAN_3:th.43">LOPBAN_3:43</a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E6:29_3"/><i><font color="Green" title="E37">A10</font></i>: 
<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="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3:func.3">*</a> <font color="Maroon" title="c4">w</font></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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"><span class="p3">(<span class="default"><a href="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3:func.3">*</a> <font color="Maroon" title="c4">w</font></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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_3_5"><b>proof </b></a><div class="add">

<i><font color="Green" title="E38">A11</font></i>: <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="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3:func.3">*</a> <font color="Maroon" title="c4">w</font></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"><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="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3:func.3">*</a> <font color="Maroon" title="c4">w</font></span>)</span></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="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="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3:func.3">*</a> <font color="Maroon" title="c4">w</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="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="bhsp_4.html#D1" title="BHSP_4:def.1">BHSP_4:def 1</a></i>
<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="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3:func.3">*</a> <font color="Maroon" title="c4">w</font></span>)</span></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="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="lopban_4.html#K5" title="LOPBAN_4:func.5">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="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="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">w</font></span>)</span>
<b>by </b><i><a class="ref" href="lopban_3.html#D10" title="LOPBAN_3:def.10">LOPBAN_3:def 10</a></i>
;<br/>
<a NAME="E2:29_3_5"/>
( <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 &amp; <font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 is    <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>  )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>;<br/>
<b>hence </b><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="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3:func.3">*</a> <font color="Maroon" title="c4">w</font></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"><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="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3:func.3">*</a> <font color="Maroon" title="c4">w</font></span>)</span></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="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>
<b>by </b><i><a class="txt" href="#E1:29_3_5"><i><font color="Green" title="E38">A11</font></i></a>, <a class="ref" href="lopban_4.html#D6" target="_self" title="LOPBAN_4:def.6">Def6</a></i>
<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="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3:func.3">*</a> <font color="Maroon" title="c4">w</font></span>)</span></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="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>
<b>by </b><i><a class="ref" href="lopban_3.html#T43" title="LOPBAN_3:th.43">LOPBAN_3:43</a></i>
<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="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3:func.3">*</a> <font color="Maroon" title="c4">w</font></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c5">n</font>
<b>by </b><i><a class="ref" href="lopban_3.html#T43" title="LOPBAN_3:th.43">LOPBAN_3:43</a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E7:29_3"/><i><font color="Green" title="E38">A12</font></i>: 
<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="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3:func.3">*</a> <font color="Maroon" title="c3">z</font></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> <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="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="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span></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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_3_6"><b>proof </b></a><div class="add">

<a NAME="E1:29_3_6"/><i><font color="Green" title="E39">A13</font></i>: 
<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="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3:func.3">*</a> <font color="Maroon" title="c3">z</font></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> <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="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="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="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3:func.3">*</a> <font color="Maroon" title="c3">z</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="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="lopban_4.html#T16" target="_self" title="LOPBAN_4:th.16">Th16</a></i>;<br/>
<a NAME="E2:29_3_6"/>
<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
 
<b>by </b><i><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>;<br/>
<a NAME="E3:29_3_6"/><b>then </b><i><font color="Green" title="E40">A14</font></i>: 
<span class="p1">(<span class="default"><a href="lopban_4.html#K5" title="LOPBAN_4:func.5">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="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> <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>
 
<b>by </b><i><a class="ref" href="lopban_4.html#D6" target="_self" title="LOPBAN_4:def.6">Def6</a></i>;<br/>
<b>thus </b><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="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3:func.3">*</a> <font color="Maroon" title="c3">z</font></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"><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="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="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="lopban_4.html#K5" title="LOPBAN_4:func.5">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="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="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">z</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:29_3_6"><i><font color="Green" title="E39">A13</font></i></a>, <a class="ref" href="lopban_3.html#D10" title="LOPBAN_3:def.10">LOPBAN_3:def 10</a></i>
<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="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="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>
<b>by </b><i><a class="txt" href="#E3:29_3_6"><i><font color="Green" title="E40">A14</font></i></a>, <a class="ref" href="lopban_3.html#T43" title="LOPBAN_3:th.43">LOPBAN_3:43</a></i>
<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="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span></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>
<b>by </b><i><a class="ref" href="lopban_3.html#T43" title="LOPBAN_3:th.43">LOPBAN_3:43</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E8:29_3"/><i><font color="Green" title="E39">A15</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3: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="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3: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="lopban_4.html#K5" title="LOPBAN_4:func.5">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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_3_7"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_3_7_1"><b>now </b></a><div class="add"><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3: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="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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="lopban_4.html#K5" title="LOPBAN_4:func.5">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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c6">k</font></span><br/><div><i><font color="Green" title="E40">A16</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="29_3_7_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:29_3_7_1_1"/><i><font color="Green" title="E41">A17</font></i>: 
<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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3: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="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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="lopban_4.html#K5" title="LOPBAN_4:func.5">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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c6">k</font></span><br/><a NAME="E2:29_3_7_1_1"/><i><font color="Green" title="E42">A18</font></i>: 
( <font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">k</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">k</font> &amp; 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">k</font> &amp;  <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> &amp; <font color="Maroon" title="c5">n</font> <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 &amp; <font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_3_7_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:29_3_7_1_1_1"/><i><font color="Green" title="E43">A19</font></i>: 
<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
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>
<a NAME="E2:29_3_7_1_1_1"/><i><font color="Green" title="E44">A20</font></i>: 
<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
 
<b>by </b><i><a class="txt" href="#E1:29_3_7_1_1"><i><font color="Green" title="E41">A17</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/>
<a NAME="E3:29_3_7_1_1_1"/><b>then </b>
<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
 
<b>by </b><i><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/>
<b>hence </b><a NAME="E4:29_3_7_1_1_1"/>
( <font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">k</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">k</font> &amp; 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">k</font> &amp;  <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> &amp; <font color="Maroon" title="c5">n</font> <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 &amp; <font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 )
 <b>by </b><i><a class="txt" href="#E1:29_3_7_1_1"><i><font color="Green" title="E41">A17</font></i></a>, <a class="txt" href="#E1:29_3_7_1_1_1"><i><font color="Green" title="E43">A19</font></i></a>, <a class="txt" href="#E2:29_3_7_1_1_1"><i><font color="Green" title="E44">A20</font></i></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></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3: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="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span></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="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3:func.3">*</a> <font color="Maroon" title="c4">w</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="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="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3:func.3">*</a> <font color="Maroon" title="c3">z</font></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>
<b>by </b><i><a class="ref" href="normsp_1.html#D5" title="NORMSP_1:def.5">NORMSP_1:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lopban_4.html#K5" title="LOPBAN_4:func.5">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="nat_1.html#K6" title="NAT_1:func.6">.</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="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3:func.3">*</a> <font color="Maroon" title="c3">z</font></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>
<b>by </b><i><a class="ref" href="lopban_3.html#D10" title="LOPBAN_3:def.10">LOPBAN_3:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lopban_4.html#K5" title="LOPBAN_4:func.5">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="nat_1.html#K6" title="NAT_1:func.6">.</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="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3:func.3">*</a> <font color="Maroon" title="c3">z</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>
<b>by </b><i><a class="txt" href="#E2:29_3_7_1_1"><i><font color="Green" title="E42">A18</font></i></a>, <a class="ref" href="lopban_4.html#T15" target="_self" title="LOPBAN_4:th.15">Th15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lopban_4.html#K5" title="LOPBAN_4:func.5">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="nat_1.html#K6" title="NAT_1:func.6">.</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="lopban_4.html#K5" title="LOPBAN_4:func.5">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="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="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">z</font></span>)</span>
<b>by </b><i><a class="ref" href="lopban_3.html#D10" title="LOPBAN_3:def.10">LOPBAN_3:def 10</a></i>
<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="lopban_4.html#K5" title="LOPBAN_4:func.5">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="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="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">z</font></span>)</span>
<b>by </b><i><a class="txt" href="#E2:29_3_7_1_1"><i><font color="Green" title="E42">A18</font></i></a>, <a class="ref" href="lopban_4.html#D6" target="_self" title="LOPBAN_4:def.6">Def6</a></i>
<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="lopban_4.html#K5" title="LOPBAN_4:func.5">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="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="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">z</font></span>)</span>
<b>by </b><i><a class="ref" href="lopban_3.html#T43" title="LOPBAN_3:th.43">LOPBAN_3:43</a></i>
<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>
<b>by </b><i><a class="txt" href="#E2:29_3_7_1_1"><i><font color="Green" title="E42">A18</font></i></a>, <a class="ref" href="lopban_4.html#D6" target="_self" title="LOPBAN_4:def.6">Def6</a></i>
<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>
<b>by </b><i><a class="ref" href="lopban_3.html#T43" title="LOPBAN_3:th.43">LOPBAN_3:43</a></i>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font>
<b>by </b><i><a class="ref" href="lopban_3.html#T43" title="LOPBAN_3:th.43">LOPBAN_3:43</a></i>
;<br/><b>hence </b><a NAME="E4:29_3_7_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3: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="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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="lopban_4.html#K5" title="LOPBAN_4:func.5">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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c6">k</font>
 <b>by </b><i><a class="txt" href="#E1:29_3_7_1_1"><i><font color="Green" title="E41">A17</font></i></a>, <a class="ref" href="lopban_4.html#D6" target="_self" title="LOPBAN_4:def.6">Def6</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_3_7_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:29_3_7_1_2"/><i><font color="Green" title="E41">A21</font></i>: 
<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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3: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="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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="lopban_4.html#K5" title="LOPBAN_4:func.5">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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c6">k</font></span><br/><div><i><font color="Green" title="E42">A22</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="29_3_7_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:29_3_7_1_2_1"/><i><font color="Green" title="E43">A23</font></i>: 
<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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3: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="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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="lopban_4.html#K5" title="LOPBAN_4:func.5">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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c6">k</font></span><br/><b>thus </b><a NAME="E2:29_3_7_1_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3: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="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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="lopban_4.html#K5" title="LOPBAN_4:func.5">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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c6">k</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_3_7_1_2_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:29_3_7_1_2_1_1"/><i><font color="Green" title="E44">A24</font></i>: 
<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
 
<b>by </b><i><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>;<br/>
<i><font color="Green" title="E45">A25</font></i>: <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>
<b>by </b><i><a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></i>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 

;<br/>
<i><font color="Green" title="E46">A26</font></i>: <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>
<b>by </b><i><a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></i>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 

;<br/>
<a NAME="E4:29_3_7_1_2_1_1"/><i><font color="Green" title="E47">A27</font></i>: 
( <font color="Maroon" title="c5">n</font> <a href="sin_cos.html#K6" title="SIN_COS:func.6">!</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <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="sin_cos.html#K6" title="SIN_COS:func.6">!</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<b>by </b><i><a class="ref" href="newton.html#T23" title="NEWTON:th.23">NEWTON:23</a></i>;<br/>
<i><font color="Green" title="E48">A28</font></i>: <span class="p1">(<span class="default"><a href="lopban_4.html#K2" title="LOPBAN_4:func.2">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#K6" title="SIN_COS:func.6">!</a> </span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</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#K6" title="SIN_COS:func.6">!</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="sin_cos.html#K6" title="SIN_COS:func.6">!</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E2:29_3_7_1_2_1_1"><i><font color="Green" title="E45">A25</font></i></a>, <a class="ref" href="lopban_4.html#D3" target="_self" title="LOPBAN_4:def.3">Def3</a></i>
<br/>.= 
1
<b>by </b><i><a class="txt" href="#E4:29_3_7_1_2_1_1"><i><font color="Green" title="E47">A27</font></i></a>, <a class="ref" href="newton.html#T18" title="NEWTON:th.18">NEWTON:18</a>, <a class="ref" href="xcmplx_1.html#T60" title="XCMPLX_1:th.60">XCMPLX_1:60</a></i>
;<br/>
<i><font color="Green" title="E49">A29</font></i>: <span class="p1">(<span class="default"><a href="lopban_4.html#K2" title="LOPBAN_4:func.2">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#K6" title="SIN_COS:func.6">!</a> </span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</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#K6" title="SIN_COS:func.6">!</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="sin_cos.html#K6" title="SIN_COS:func.6">!</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E3:29_3_7_1_2_1_1"><i><font color="Green" title="E46">A26</font></i></a>, <a class="ref" href="lopban_4.html#D3" target="_self" title="LOPBAN_4:def.3">Def3</a></i>
<br/>.= 
1
<b>by </b><i><a class="txt" href="#E4:29_3_7_1_2_1_1"><i><font color="Green" title="E47">A27</font></i></a>, <a class="ref" href="newton.html#T18" title="NEWTON:th.18">NEWTON:18</a>, <a class="ref" href="xcmplx_1.html#T60" title="XCMPLX_1:th.60">XCMPLX_1:60</a></i>
;<br/>
<b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3: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="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span></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="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3:func.3">*</a> <font color="Maroon" title="c4">w</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="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="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E1:29_3_7_1_2_1"><i><font color="Green" title="E43">A23</font></i></a>, <a class="ref" href="normsp_1.html#D5" title="NORMSP_1:def.5">NORMSP_1:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lopban_4.html#K5" title="LOPBAN_4:func.5">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="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="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="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lopban_3.html#D10" title="LOPBAN_3:def.10">LOPBAN_3:def 10</a></i>
<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="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E1:29_3_7_1_2_1_1"><i><font color="Green" title="E44">A24</font></i></a>, <a class="ref" href="lopban_4.html#D6" target="_self" title="LOPBAN_4:def.6">Def6</a></i>
<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="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lopban_3.html#T43" title="LOPBAN_3:th.43">LOPBAN_3:43</a></i>
<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="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3:func.3">*</a> <font color="Maroon" title="c3">z</font></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>
<b>by </b><i><a class="ref" href="lopban_3.html#T43" title="LOPBAN_3:th.43">LOPBAN_3:43</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c5">n</font>
<b>by </b><i><a class="ref" href="lopban_4.html#D5" target="_self" title="LOPBAN_4:def.5">Def5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_4.html#K5" title="LOPBAN_4:func.5">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="nat_1.html#K6" title="NAT_1:func.6">.</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>
<b>by </b><i><a class="ref" href="lopban_3.html#D10" title="LOPBAN_3:def.10">LOPBAN_3:def 10</a></i>
<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="lopban_4.html#K2" title="LOPBAN_4:func.2">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="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="lopban_3.html#K6" title="LOPBAN_3: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="lopban_3.html#K6" title="LOPBAN_3: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>
<b>by </b><i><a class="ref" href="lopban_4.html#D6" target="_self" title="LOPBAN_4:def.6">Def6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lopban_4.html#K2" title="LOPBAN_4:func.2">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="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="lopban_3.html#K6" title="LOPBAN_3: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="lopban_3.html#K6" title="LOPBAN_3: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>
<b>by </b><i><a class="ref" href="lopban_3.html#T43" title="LOPBAN_3:th.43">LOPBAN_3:43</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lopban_4.html#K2" title="LOPBAN_4:func.2">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="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="lopban_3.html#K6" title="LOPBAN_3: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="lopban_3.html#K6" title="LOPBAN_3: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>
<b>by </b><i><a class="txt" href="#E1:29"><i><font color="Green" title="E29">A1</font></i></a>, <a class="txt" href="lopban_4.html#E3"><i><font color="Green" title="E3">Lm2</font></i></a></i>
<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="lopban_4.html#K2" title="LOPBAN_4:func.2">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="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="lopban_3.html#K6" title="LOPBAN_3: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="lopban_3.html#K6" title="LOPBAN_3: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>
<b>by </b><i><a class="ref" href="lopban_3.html#T43" title="LOPBAN_3:th.43">LOPBAN_3:43</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lopban_4.html#K2" title="LOPBAN_4:func.2">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="rlvect_1.html#K1" title="RLVECT_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="lopban_3.html#K6" title="LOPBAN_3: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="lopban_3.html#K6" title="LOPBAN_3: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>
<b>by </b><i><a class="ref" href="lopban_3.html#T43" title="LOPBAN_3:th.43">LOPBAN_3:43</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lopban_4.html#K2" title="LOPBAN_4:func.2">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="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="lopban_3.html#K6" title="LOPBAN_3: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="lopban_3.html#K6" title="LOPBAN_3: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>
<b>by </b><i><a class="txt" href="#E5:29_3_7_1_2_1_1"><i><font color="Green" title="E48">A28</font></i></a>, <a class="txt" href="#E6:29_3_7_1_2_1_1"><i><font color="Green" title="E49">A29</font></i></a>, <a class="txt" href="lopban_4.html#E2"><i><font color="Green" title="E2">Lm1</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lopban_4.html#K5" title="LOPBAN_4:func.5">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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c6">k</font>
<b>by </b><i><a class="txt" href="#E1:29_3_7_1_2_1"><i><font color="Green" title="E43">A23</font></i></a>, <a class="txt" href="#E2:29_3_7_1_2_1_1"><i><font color="Green" title="E45">A25</font></i></a>, <a class="txt" href="#E3:29_3_7_1_2_1_1"><i><font color="Green" title="E46">A26</font></i></a>, <a class="ref" href="lopban_4.html#D6" target="_self" title="LOPBAN_4:def.6">Def6</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_3_7_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:29_3_7_1_2_2"/><i><font color="Green" title="E43">A30</font></i>: 
<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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3: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="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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="lopban_4.html#K5" title="LOPBAN_4:func.5">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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c6">k</font></span><br/><div><i><font color="Green" title="E44">A31</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="29_3_7_1_2_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:29_3_7_1_2_2_1"/><i><font color="Green" title="E45">A32</font></i>: 
<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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3: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="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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="lopban_4.html#K5" title="LOPBAN_4:func.5">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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c6">k</font></span><br/><b>thus </b><a NAME="E2:29_3_7_1_2_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3: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="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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="lopban_4.html#K5" title="LOPBAN_4:func.5">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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c6">k</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_3_7_1_2_2_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:29_3_7_1_2_2_1_1"/><i><font color="Green" title="E46">A33</font></i>: 
<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> 
 
<b>by </b><i><a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></i>;<br/>
<a NAME="E2:29_3_7_1_2_2_1_1"/><i><font color="Green" title="E47">A34</font></i>: 
<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> 
 
<b>by </b><i><a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></i>;<br/>
<a NAME="E3:29_3_7_1_2_2_1_1"/><i><font color="Green" title="E48">A35</font></i>: 
( <font color="Maroon" title="c5">n</font> <a href="sin_cos.html#K6" title="SIN_COS:func.6">!</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <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="sin_cos.html#K6" title="SIN_COS:func.6">!</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<b>by </b><i><a class="ref" href="newton.html#T23" title="NEWTON:th.23">NEWTON:23</a></i>;<br/>
<i><font color="Green" title="E49">A36</font></i>: <span class="p1">(<span class="default"><a href="lopban_4.html#K2" title="LOPBAN_4:func.2">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#K6" title="SIN_COS:func.6">!</a> </span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="sin_cos.html#K6" title="SIN_COS:func.6">!</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="sin_cos.html#K6" title="SIN_COS:func.6">!</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E1:29_3_7_1_2_2_1_1"><i><font color="Green" title="E46">A33</font></i></a>, <a class="ref" href="lopban_4.html#D3" target="_self" title="LOPBAN_4:def.3">Def3</a></i>
<br/>.= 
1
<b>by </b><i><a class="txt" href="#E3:29_3_7_1_2_2_1_1"><i><font color="Green" title="E48">A35</font></i></a>, <a class="ref" href="newton.html#T18" title="NEWTON:th.18">NEWTON:18</a>, <a class="ref" href="xcmplx_1.html#T60" title="XCMPLX_1:th.60">XCMPLX_1:60</a></i>
;<br/>
<i><font color="Green" title="E50">A37</font></i>: <span class="p1">(<span class="default"><a href="lopban_4.html#K2" title="LOPBAN_4:func.2">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#K6" title="SIN_COS:func.6">!</a> </span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="sin_cos.html#K6" title="SIN_COS:func.6">!</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</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#K6" title="SIN_COS:func.6">!</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E2:29_3_7_1_2_2_1_1"><i><font color="Green" title="E47">A34</font></i></a>, <a class="ref" href="lopban_4.html#D3" target="_self" title="LOPBAN_4:def.3">Def3</a></i>
<br/>.= 
1
<b>by </b><i><a class="txt" href="#E3:29_3_7_1_2_2_1_1"><i><font color="Green" title="E48">A35</font></i></a>, <a class="ref" href="newton.html#T18" title="NEWTON:th.18">NEWTON:18</a>, <a class="ref" href="xcmplx_1.html#T60" title="XCMPLX_1:th.60">XCMPLX_1:60</a></i>
;<br/>
<b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3: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="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span></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="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3:func.3">*</a> <font color="Maroon" title="c4">w</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="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="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3:func.3">*</a> <font color="Maroon" title="c3">z</font></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>
<b>by </b><i><a class="txt" href="#E1:29_3_7_1_2_2_1"><i><font color="Green" title="E45">A32</font></i></a>, <a class="ref" href="normsp_1.html#D5" title="NORMSP_1:def.5">NORMSP_1:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lopban_4.html#K5" title="LOPBAN_4:func.5">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="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="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="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3:func.3">*</a> <font color="Maroon" title="c3">z</font></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>
<b>by </b><i><a class="ref" href="lopban_3.html#D10" title="LOPBAN_3:def.10">LOPBAN_3:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lopban_4.html#K5" title="LOPBAN_4:func.5">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="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="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>
<b>by </b><i><a class="ref" href="lopban_4.html#D5" target="_self" title="LOPBAN_4:def.5">Def5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_4.html#K5" title="LOPBAN_4:func.5">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="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="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">w</font>
<b>by </b><i><a class="ref" href="lopban_3.html#T43" title="LOPBAN_3:th.43">LOPBAN_3:43</a></i>
<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="lopban_4.html#K2" title="LOPBAN_4:func.2">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="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="lopban_3.html#K6" title="LOPBAN_3: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="lopban_3.html#K6" title="LOPBAN_3: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>
<b>by </b><i><a class="ref" href="lopban_4.html#D6" target="_self" title="LOPBAN_4:def.6">Def6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lopban_4.html#K2" title="LOPBAN_4:func.2">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="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="lopban_3.html#K6" title="LOPBAN_3: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="lopban_3.html#K6" title="LOPBAN_3: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>
<b>by </b><i><a class="ref" href="lopban_3.html#T43" title="LOPBAN_3:th.43">LOPBAN_3:43</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lopban_4.html#K2" title="LOPBAN_4:func.2">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="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="lopban_3.html#K6" title="LOPBAN_3: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="lopban_3.html#K6" title="LOPBAN_3: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>
<b>by </b><i><a class="txt" href="#E1:29_3_7_1_2_2_1_1"><i><font color="Green" title="E46">A33</font></i></a>, <a class="txt" href="#E2:29_3_7_1_2_2_1_1"><i><font color="Green" title="E47">A34</font></i></a>, <a class="txt" href="#E4:29_3_7_1_2_2_1_1"><i><font color="Green" title="E49">A36</font></i></a>, <a class="txt" href="#E5:29_3_7_1_2_2_1_1"><i><font color="Green" title="E50">A37</font></i></a>, <a class="txt" href="lopban_4.html#E2"><i><font color="Green" title="E2">Lm1</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lopban_4.html#K5" title="LOPBAN_4:func.5">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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c6">k</font>
<b>by </b><i><a class="txt" href="#E1:29_3_7_1_2_2_1"><i><font color="Green" title="E45">A32</font></i></a>, <a class="ref" href="lopban_4.html#D6" target="_self" title="LOPBAN_4:def.6">Def6</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_3_7_1_2_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:29_3_7_1_2_2_2"/><i><font color="Green" title="E45">A38</font></i>: 
<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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3: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="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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="lopban_4.html#K5" title="LOPBAN_4:func.5">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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c6">k</font></span><br/><a NAME="E2:29_3_7_1_2_2_2"/><b>then </b><i><font color="Green" title="E46">A39</font></i>: 
 <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>
 ;<br/><a NAME="E3:29_3_7_1_2_2_2"/><b>then </b><i><font color="Green" title="E47">A40</font></i>: 
<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>
 <b>by </b><i><a class="ref" href="int_1.html#T20" title="INT_1:th.20">INT_1:20</a></i>;<br/><a NAME="E4:29_3_7_1_2_2_2"/><i><font color="Green" title="E48">A41</font></i>: 
<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
 <b>by </b><i><a class="txt" href="#E1:29_3_7_1_2_2"><i><font color="Green" title="E43">A30</font></i></a>, <a class="ref" href="int_1.html#T20" title="INT_1:th.20">INT_1:20</a></i>;<br/><a NAME="E5:29_3_7_1_2_2_2"/>
<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
 <b>by </b><i><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>;<br/><a NAME="E6:29_3_7_1_2_2_2"/><b>then </b>
<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
 <b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E7:29_3_7_1_2_2_2"/><b>then </b>
<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>
 <b>by </b><i><a class="txt" href="#E4:29_3_7_1_2_2_2"><i><font color="Green" title="E48">A41</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E8:29_3_7_1_2_2_2"/><b>then </b><i><font color="Green" title="E49">A42</font></i>: 
<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>
 <b>by </b><i><a class="txt" href="#E3:29_3_7_1_2_2_2"><i><font color="Green" title="E47">A40</font></i></a>, <a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></i>;<br/><b>thus </b><a NAME="E9:29_3_7_1_2_2_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3: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="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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="lopban_4.html#K5" title="LOPBAN_4:func.5">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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c6">k</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_3_7_1_2_2_2_1"><b>proof </b></a><div class="add">

<i><font color="Green" title="E50">A43</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3: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="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span></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="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3:func.3">*</a> <font color="Maroon" title="c4">w</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="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="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3:func.3">*</a> <font color="Maroon" title="c3">z</font></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>
<b>by </b><i><a class="ref" href="normsp_1.html#D5" title="NORMSP_1:def.5">NORMSP_1:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lopban_4.html#K5" title="LOPBAN_4:func.5">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="nat_1.html#K6" title="NAT_1:func.6">.</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="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3:func.3">*</a> <font color="Maroon" title="c3">z</font></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>
<b>by </b><i><a class="ref" href="lopban_3.html#D10" title="LOPBAN_3:def.10">LOPBAN_3:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lopban_4.html#K5" title="LOPBAN_4:func.5">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="nat_1.html#K6" title="NAT_1:func.6">.</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="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3:func.3">*</a> <font color="Maroon" title="c3">z</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>
<b>by </b><i><a class="txt" href="#E2:29_3_7_1_2_2_2"><i><font color="Green" title="E46">A39</font></i></a>, <a class="ref" href="lopban_4.html#T15" target="_self" title="LOPBAN_4:th.15">Th15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lopban_4.html#K5" title="LOPBAN_4:func.5">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="nat_1.html#K6" title="NAT_1:func.6">.</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="lopban_4.html#K5" title="LOPBAN_4:func.5">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="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="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">z</font></span>)</span>
<b>by </b><i><a class="ref" href="lopban_3.html#D10" title="LOPBAN_3:def.10">LOPBAN_3:def 10</a></i>
<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="lopban_4.html#K2" title="LOPBAN_4:func.2">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="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="lopban_3.html#K6" title="LOPBAN_3: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="lopban_3.html#K6" title="LOPBAN_3: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="lopban_4.html#K5" title="LOPBAN_4:func.5">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="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="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">z</font></span>)</span>
<b>by </b><i><a class="txt" href="#E4:29_3_7_1_2_2_2"><i><font color="Green" title="E48">A41</font></i></a>, <a class="ref" href="lopban_4.html#D6" target="_self" title="LOPBAN_4:def.6">Def6</a></i>
<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="lopban_4.html#K2" title="LOPBAN_4:func.2">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="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="lopban_3.html#K6" title="LOPBAN_3: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="lopban_3.html#K6" title="LOPBAN_3: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="lopban_4.html#K2" title="LOPBAN_4:func.2">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="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="lopban_3.html#K6" title="LOPBAN_3: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="lopban_3.html#K6" title="LOPBAN_3: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>
<b>by </b><i><a class="txt" href="#E8:29_3_7_1_2_2_2"><i><font color="Green" title="E49">A42</font></i></a>, <a class="ref" href="lopban_4.html#D6" target="_self" title="LOPBAN_4:def.6">Def6</a></i>
<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="lopban_4.html#K2" title="LOPBAN_4:func.2">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="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="lopban_3.html#K6" title="LOPBAN_3: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="lopban_3.html#K6" title="LOPBAN_3: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="lopban_4.html#K2" title="LOPBAN_4:func.2">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="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="lopban_3.html#K6" title="LOPBAN_3: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="lopban_3.html#K6" title="LOPBAN_3: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>
<b>by </b><i><a class="ref" href="lopban_3.html#T43" title="LOPBAN_3:th.43">LOPBAN_3:43</a></i>
<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="lopban_4.html#K2" title="LOPBAN_4:func.2">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="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="lopban_3.html#K6" title="LOPBAN_3: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="lopban_3.html#K6" title="LOPBAN_3: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="lopban_4.html#K2" title="LOPBAN_4:func.2">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="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="lopban_3.html#K6" title="LOPBAN_3: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="lopban_3.html#K6" title="LOPBAN_3: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>
<b>by </b><i><a class="txt" href="lopban_4.html#E2"><i><font color="Green" title="E2">Lm1</font></i></a></i>
<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="lopban_4.html#K2" title="LOPBAN_4:func.2">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="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="lopban_3.html#K6" title="LOPBAN_3: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="lopban_3.html#K6" title="LOPBAN_3: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="lopban_4.html#K2" title="LOPBAN_4:func.2">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="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="lopban_3.html#K6" title="LOPBAN_3: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="lopban_3.html#K6" title="LOPBAN_3: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>
<b>by </b><i><a class="ref" href="lopban_3.html#T43" title="LOPBAN_3:th.43">LOPBAN_3:43</a></i>
<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="lopban_4.html#K2" title="LOPBAN_4:func.2">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="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="lopban_3.html#K6" title="LOPBAN_3: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="lopban_3.html#K6" title="LOPBAN_3: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="lopban_4.html#K2" title="LOPBAN_4:func.2">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="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="lopban_3.html#K6" title="LOPBAN_3: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="lopban_3.html#K6" title="LOPBAN_3: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>
<b>by </b><i><a class="txt" href="#E1:29"><i><font color="Green" title="E29">A1</font></i></a>, <a class="txt" href="lopban_4.html#E3"><i><font color="Green" title="E3">Lm2</font></i></a></i>
<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="lopban_4.html#K2" title="LOPBAN_4:func.2">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="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="lopban_3.html#K6" title="LOPBAN_3: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="lopban_3.html#K6" title="LOPBAN_3: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="lopban_4.html#K2" title="LOPBAN_4:func.2">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="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="lopban_3.html#K6" title="LOPBAN_3: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="lopban_3.html#K6" title="LOPBAN_3: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>
<b>by </b><i><a class="ref" href="lopban_3.html#T43" title="LOPBAN_3:th.43">LOPBAN_3:43</a></i>
<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="lopban_4.html#K2" title="LOPBAN_4:func.2">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="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="lopban_3.html#K6" title="LOPBAN_3: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="lopban_3.html#K6" title="LOPBAN_3: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="lopban_4.html#K2" title="LOPBAN_4:func.2">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="rlvect_1.html#K1" title="RLVECT_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="lopban_3.html#K6" title="LOPBAN_3: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="lopban_3.html#K6" title="LOPBAN_3: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>
<b>by </b><i><a class="ref" href="lopban_3.html#T43" title="LOPBAN_3:th.43">LOPBAN_3:43</a></i>
<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="lopban_4.html#K2" title="LOPBAN_4:func.2">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="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="lopban_3.html#K6" title="LOPBAN_3: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="lopban_3.html#K6" title="LOPBAN_3: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="lopban_4.html#K2" title="LOPBAN_4:func.2">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="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="lopban_3.html#K6" title="LOPBAN_3: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="lopban_3.html#K6" title="LOPBAN_3: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>
<b>by </b><i><a class="txt" href="lopban_4.html#E2"><i><font color="Green" title="E2">Lm1</font></i></a></i>
;<br/>
<a NAME="E2:29_3_7_1_2_2_2_1"/><i><font color="Green" title="E51">A44</font></i>: 
<font color="Maroon" title="c4">w</font> <a href="lopban_3.html#K6" title="LOPBAN_3:func.6">#N</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> <font color="Maroon" title="c6">k</font></span>)</span> <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> <font color="Maroon" title="c4">w</font> <a href="lopban_3.html#K6" title="LOPBAN_3:func.6">#N</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="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c6">k</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_3_7_1_2_2_2_1_2"><b>proof </b></a><div class="add">

<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
<b>by </b><i><a class="txt" href="#E4:29_3_7_1_2_2_2"><i><font color="Green" title="E48">A41</font></i></a>, <a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></i>
<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>
<b>by </b><i><a class="txt" href="#E1:29_3_7_1_2_2"><i><font color="Green" title="E43">A30</font></i></a>, <a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></i>
;<br/>
<b>hence </b><a NAME="E2:29_3_7_1_2_2_2_1_2"/>
<font color="Maroon" title="c4">w</font> <a href="lopban_3.html#K6" title="LOPBAN_3:func.6">#N</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> <font color="Maroon" title="c6">k</font></span>)</span> <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> <font color="Maroon" title="c4">w</font> <a href="lopban_3.html#K6" title="LOPBAN_3:func.6">#N</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="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c6">k</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E3:29_3_7_1_2_2_2_1"/><i><font color="Green" title="E52">A45</font></i>: 
<font color="Maroon" title="c4">w</font> <a href="lopban_3.html#K6" title="LOPBAN_3:func.6">#N</a> <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> <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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">w</font> <a href="lopban_3.html#K6" title="LOPBAN_3:func.6">#N</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="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c6">k</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_3_7_1_2_2_2_1_3"><b>proof </b></a><div class="add">

<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>
<b>by </b><i><a class="txt" href="#E8:29_3_7_1_2_2_2"><i><font color="Green" title="E49">A42</font></i></a>, <a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></i>
<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>
<b>by </b><i><a class="txt" href="#E3:29_3_7_1_2_2_2"><i><font color="Green" title="E47">A40</font></i></a>, <a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></i>
<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>
<b>by </b><i><a class="txt" href="#E1:29_3_7_1_2_2"><i><font color="Green" title="E43">A30</font></i></a>, <a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></i>
;<br/>
<b>hence </b><a NAME="E2:29_3_7_1_2_2_2_1_3"/>
<font color="Maroon" title="c4">w</font> <a href="lopban_3.html#K6" title="LOPBAN_3:func.6">#N</a> <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> <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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">w</font> <a href="lopban_3.html#K6" title="LOPBAN_3:func.6">#N</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="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c6">k</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E4:29_3_7_1_2_2_2_1"/>
<font color="Maroon" title="c3">z</font> <a href="lopban_3.html#K6" title="LOPBAN_3:func.6">#N</a> <span class="p1">(<span class="default"><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> <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> <font color="Maroon" title="c3">z</font> <a href="lopban_3.html#K6" title="LOPBAN_3:func.6">#N</a> <font color="Maroon" title="c6">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_3_7_1_2_2_2_1_4"><b>proof </b></a><div class="add">

<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
<b>by </b><i><a class="txt" href="#E3:29_3_7_1_2_2_2"><i><font color="Green" title="E47">A40</font></i></a>, <a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></i>
<br/>.= 
<font color="Maroon" title="c6">k</font>

;<br/>
<b>hence </b><a NAME="E2:29_3_7_1_2_2_2_1_4"/>
<font color="Maroon" title="c3">z</font> <a href="lopban_3.html#K6" title="LOPBAN_3:func.6">#N</a> <span class="p1">(<span class="default"><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> <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> <font color="Maroon" title="c3">z</font> <a href="lopban_3.html#K6" title="LOPBAN_3:func.6">#N</a> <font color="Maroon" title="c6">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>then </b><i><font color="Green" title="E53">A46</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3: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="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span></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="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lopban_4.html#K2" title="LOPBAN_4:func.2">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="rlvect_1.html#K1" title="RLVECT_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="lopban_3.html#K6" title="LOPBAN_3: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="lopban_3.html#K6" title="LOPBAN_3: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="lopban_4.html#K2" title="LOPBAN_4:func.2">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="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="lopban_3.html#K6" title="LOPBAN_3: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="lopban_3.html#K6" title="LOPBAN_3: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>
<b>by </b><i><a class="txt" href="#E1:29_3_7_1_2_2_2_1"><i><font color="Green" title="E50">A43</font></i></a>, <a class="txt" href="#E2:29_3_7_1_2_2_2_1"><i><font color="Green" title="E51">A44</font></i></a>, <a class="txt" href="#E3:29_3_7_1_2_2_2_1"><i><font color="Green" title="E52">A45</font></i></a>, <a class="ref" href="lopban_3.html#T43" title="LOPBAN_3:th.43">LOPBAN_3:43</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lopban_4.html#K2" title="LOPBAN_4:func.2">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="rlvect_1.html#K1" title="RLVECT_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="lopban_3.html#K6" title="LOPBAN_3: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="lopban_3.html#K6" title="LOPBAN_3: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="lopban_4.html#K2" title="LOPBAN_4:func.2">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="rlvect_1.html#K1" title="RLVECT_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="lopban_3.html#K6" title="LOPBAN_3: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="lopban_3.html#K6" title="LOPBAN_3: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>
<b>by </b><i><a class="ref" href="lopban_3.html#T43" title="LOPBAN_3:th.43">LOPBAN_3:43</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lopban_4.html#K2" title="LOPBAN_4:func.2">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="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lopban_4.html#K2" title="LOPBAN_4:func.2">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="rlvect_1.html#K1" title="RLVECT_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="lopban_3.html#K6" title="LOPBAN_3: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="lopban_3.html#K6" title="LOPBAN_3: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>
<b>by </b><i><a class="ref" href="lopban_3.html#T43" title="LOPBAN_3:th.43">LOPBAN_3:43</a></i>
;<br/>
<i><font color="Green" title="E54">A47</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_4.html#K2" title="LOPBAN_4:func.2">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="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_4.html#K2" title="LOPBAN_4:func.2">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#K6" title="SIN_COS:func.6">!</a> </span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</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#K6" title="SIN_COS:func.6">!</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</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#K6" title="SIN_COS:func.6">!</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_4.html#K2" title="LOPBAN_4:func.2">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>
<b>by </b><i><a class="txt" href="#E4:29_3_7_1_2_2_2"><i><font color="Green" title="E48">A41</font></i></a>, <a class="ref" href="lopban_4.html#D3" target="_self" title="LOPBAN_4:def.3">Def3</a></i>
<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#K6" title="SIN_COS:func.6">!</a> </span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</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#K6" title="SIN_COS:func.6">!</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</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#K6" title="SIN_COS:func.6">!</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="sin_cos.html#K6" title="SIN_COS:func.6">!</a> </span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</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#K6" title="SIN_COS:func.6">!</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</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#K6" title="SIN_COS:func.6">!</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E8:29_3_7_1_2_2_2"><i><font color="Green" title="E49">A42</font></i></a>, <a class="ref" href="lopban_4.html#D3" target="_self" title="LOPBAN_4:def.3">Def3</a></i>
;<br/>
<a NAME="E7:29_3_7_1_2_2_2_1"/><i><font color="Green" title="E55">A48</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="sin_cos.html#K6" title="SIN_COS:func.6">!</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</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#K6" title="SIN_COS:func.6">!</a> </span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="sin_cos.html#K6" title="SIN_COS:func.6">!</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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#K6" title="SIN_COS:func.6">!</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</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>
 
;<br/>
<i><font color="Green" title="E56">A49</font></i>: <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>
<b>by </b><i><a class="txt" href="#E8:29_3_7_1_2_2_2"><i><font color="Green" title="E49">A42</font></i></a>, <a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></i>
<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>
<b>by </b><i><a class="txt" href="#E3:29_3_7_1_2_2_2"><i><font color="Green" title="E47">A40</font></i></a>, <a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></i>
<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>
<b>by </b><i><a class="txt" href="#E1:29_3_7_1_2_2"><i><font color="Green" title="E43">A30</font></i></a>, <a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></i>
;<br/>
<i><font color="Green" title="E57">A50</font></i>: <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#K6" title="SIN_COS:func.6">!</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</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#K6" title="SIN_COS:func.6">!</a> </span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c6">k</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</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#K6" title="SIN_COS:func.6">!</a> </span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><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#K6" title="SIN_COS:func.6">!</a> </span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="sin_cos.html#K6" title="SIN_COS:func.6">!</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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#K6" title="SIN_COS:func.6">!</a> </span>)</span>
<b>by </b><i><a class="txt" href="#E2:29_3_7_1_2_2_2"><i><font color="Green" title="E46">A39</font></i></a>, <a class="txt" href="#E8:29_3_7_1_2_2_2_1"><i><font color="Green" title="E56">A49</font></i></a>, <a class="ref" href="lopban_4.html#T13" target="_self" title="LOPBAN_4:th.13">Th13</a></i>
;<br/>
<a NAME="E10:29_3_7_1_2_2_2_1"/>
( <font color="Maroon" title="c6">k</font> <a href="sin_cos.html#K6" title="SIN_COS:func.6">!</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <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="sin_cos.html#K6" title="SIN_COS:func.6">!</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <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="sin_cos.html#K6" title="SIN_COS:func.6">!</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <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> <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> <a href="sin_cos.html#K6" title="SIN_COS:func.6">!</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <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="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="sin_cos.html#K6" title="SIN_COS:func.6">!</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <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>  &amp; <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>  )
 
<b>by </b><i><a class="txt" href="#E1:29_3_7_1_2_2"><i><font color="Green" title="E43">A30</font></i></a>, <a class="txt" href="#E1:29_3_7_1_2_2_2"><i><font color="Green" title="E45">A38</font></i></a>, <a class="ref" href="newton.html#T23" title="NEWTON:th.23">NEWTON:23</a></i>;<br/>
<b>then </b><i><font color="Green" title="E58">A51</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="sin_cos.html#K6" title="SIN_COS:func.6">!</a> </span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</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#K6" title="SIN_COS:func.6">!</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</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#K6" title="SIN_COS:func.6">!</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#K6" title="SIN_COS:func.6">!</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</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="real_1.html#K10" title="REAL_1:func.10">/</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#K6" title="SIN_COS:func.6">!</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</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#K6" title="SIN_COS:func.6">!</a> </span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</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>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T92" title="XCMPLX_1:th.92">XCMPLX_1:92</a></i>
<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#K6" title="SIN_COS:func.6">!</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</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="real_1.html#K10" title="REAL_1:func.10">/</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#K6" title="SIN_COS:func.6">!</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</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#K6" title="SIN_COS:func.6">!</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E4:29_3_7_1_2_2_2"><i><font color="Green" title="E48">A41</font></i></a>, <a class="txt" href="#E7:29_3_7_1_2_2_2_1"><i><font color="Green" title="E55">A48</font></i></a>, <a class="ref" href="lopban_4.html#T13" target="_self" title="LOPBAN_4:th.13">Th13</a></i>
;<br/>
<a NAME="E12:29_3_7_1_2_2_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="sin_cos.html#K6" title="SIN_COS:func.6">!</a> </span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</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#K6" title="SIN_COS:func.6">!</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</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#K6" title="SIN_COS:func.6">!</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#K6" title="SIN_COS:func.6">!</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</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#K6" title="SIN_COS:func.6">!</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</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#K6" title="SIN_COS:func.6">!</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:29_3_7_1_2_2_2"><i><font color="Green" title="E45">A38</font></i></a>, <a class="txt" href="#E9:29_3_7_1_2_2_2_1"><i><font color="Green" title="E57">A50</font></i></a>, <a class="ref" href="xcmplx_1.html#T92" title="XCMPLX_1:th.92">XCMPLX_1:92</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_4.html#K2" title="LOPBAN_4:func.2">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="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_4.html#K2" title="LOPBAN_4:func.2">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#K6" title="SIN_COS:func.6">!</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</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="real_1.html#K7" title="REAL_1:func.7">+</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#K6" title="SIN_COS:func.6">!</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</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#K6" title="SIN_COS:func.6">!</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</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#K6" title="SIN_COS:func.6">!</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E6:29_3_7_1_2_2_2_1"><i><font color="Green" title="E54">A47</font></i></a>, <a class="txt" href="#E11:29_3_7_1_2_2_2_1"><i><font color="Green" title="E58">A51</font></i></a>, <a class="ref" href="xcmplx_1.html#T63" title="XCMPLX_1:th.63">XCMPLX_1:63</a></i>
<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#K6" title="SIN_COS:func.6">!</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</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="real_1.html#K10" title="REAL_1:func.10">/</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#K6" title="SIN_COS:func.6">!</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</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#K6" title="SIN_COS:func.6">!</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#K6" title="SIN_COS:func.6">!</a> </span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</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#K6" title="SIN_COS:func.6">!</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</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#K6" title="SIN_COS:func.6">!</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T21" title="NEWTON:th.21">NEWTON:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lopban_4.html#K2" title="LOPBAN_4:func.2">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>
<b>by </b><i><a class="txt" href="#E1:29_3_7_1_2_2"><i><font color="Green" title="E43">A30</font></i></a>, <a class="ref" href="lopban_4.html#D3" target="_self" title="LOPBAN_4:def.3">Def3</a></i>
;<br/>
<b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3: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="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span></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="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lopban_4.html#K2" title="LOPBAN_4:func.2">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="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="lopban_3.html#K6" title="LOPBAN_3: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="lopban_3.html#K6" title="LOPBAN_3: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>
<b>by </b><i><a class="txt" href="#E5:29_3_7_1_2_2_2_1"><i><font color="Green" title="E53">A46</font></i></a>, <a class="ref" href="lopban_3.html#T43" title="LOPBAN_3:th.43">LOPBAN_3:43</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lopban_4.html#K5" title="LOPBAN_4:func.5">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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c6">k</font>
<b>by </b><i><a class="txt" href="#E1:29_3_7_1_2_2"><i><font color="Green" title="E43">A30</font></i></a>, <a class="ref" href="lopban_4.html#D6" target="_self" title="LOPBAN_4:def.6">Def6</a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:29_3_7_1_2_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3: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="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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="lopban_4.html#K5" title="LOPBAN_4:func.5">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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c6">k</font>
 <b>by </b><i><a class="txt" href="#E2:29_3_7_1_2_2"><i><font color="Green" title="E44">A31</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>hence </b><a NAME="E4:29_3_7_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3: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="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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="lopban_4.html#K5" title="LOPBAN_4:func.5">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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c6">k</font>
 <b>by </b><i><a class="txt" href="#E1:29_3_7_1_2"><i><font color="Green" title="E41">A21</font></i></a>, <a class="txt" href="#E2:29_3_7_1_2"><i><font color="Green" title="E42">A22</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>hence </b><a NAME="E3:29_3_7_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3: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="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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="lopban_4.html#K5" title="LOPBAN_4:func.5">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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c6">k</font>
 <b>by </b><i><a class="txt" href="#E1:29_3_7_1"><i><font color="Green" title="E40">A16</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:29_3_7"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3: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="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3: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="lopban_4.html#K5" title="LOPBAN_4:func.5">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>
 <b>by </b><i><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>thus </b><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="lopban_3.html#K6" title="LOPBAN_3: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="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3: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="lopban_4.html#K5" title="LOPBAN_4:func.5">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="lopban_3.html#K3" title="LOPBAN_3:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span></span>)</span></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>
<b>by </b><i><a class="txt" href="#E4:29_3"><i><font color="Green" title="E35">A7</font></i></a>, <a class="txt" href="#E5:29_3"><i><font color="Green" title="E36">A8</font></i></a>, <a class="txt" href="#E6:29_3"><i><font color="Green" title="E37">A10</font></i></a>, <a class="txt" href="#E7:29_3"><i><font color="Green" title="E38">A12</font></i></a>, <a class="ref" href="normsp_1.html#D5" title="NORMSP_1:def.5">NORMSP_1:def 5</a></i>
<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#K5" title="LOPBAN_4:func.5">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="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>
<b>by </b><i><a class="txt" href="#E8:29_3"><i><font color="Green" title="E39">A15</font></i></a>, <a class="ref" href="lopban_3.html#T20" title="LOPBAN_3:th.20">LOPBAN_3:20</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E6:29"/>
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>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E4:29"><i><font color="Green" title="E31">A3</font></i></a>, <a class="txt" href="#E5:29"><i><font color="Green" title="E32">A4</font></i></a>);<br/></i>
<b>hence </b><a NAME="E7:29"/>
<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="lopban_3.html#K6" title="LOPBAN_3: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="lopban_4.html#K5" title="LOPBAN_4:func.5">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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c2">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
