<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">X</font> be   <a href="lopban_2.html#NM3" title="LOPBAN_2:NM.3">Banach_Algebra</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">z</font>, <font color="Olive" title="b2">w</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b3">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   <a href="lopban_4.html#K6" title="LOPBAN_4:func.6">Expan_e</a> (<font color="Olive" title="b3">n</font>,<font color="Olive" title="b1">z</font>,<font color="Olive" title="b2">w</font>) <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">n</font> <a href="newton.html#K5" title="NEWTON:func.5">!</a></span>)</span></span>)</span> <a href="normsp_1.html#K5" title="NORMSP_1:func.5">*</a> <span class="p1">(<span class="default"><a href="lopban_4.html#K5" title="LOPBAN_4:func.5">Expan</a> (<font color="Olive" title="b3">n</font>,<font color="Olive" title="b1">z</font>,<font color="Olive" title="b2">w</font>)</span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">z</font>, <font color="Maroon" title="c3">w</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">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   <a href="lopban_4.html#K6" title="LOPBAN_4:func.6">Expan_e</a> (<font color="Olive" title="b1">n</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">w</font>) <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="newton.html#K5" title="NEWTON:func.5">!</a></span>)</span></span>)</span> <a href="normsp_1.html#K5" title="NORMSP_1:func.5">*</a> <span class="p1">(<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="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">w</font>)</span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c4">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="lopban_4.html#K6" title="LOPBAN_4:func.6">Expan_e</a> (<font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">w</font>) <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="newton.html#K5" title="NEWTON:func.5">!</a></span>)</span></span>)</span> <a href="normsp_1.html#K5" title="NORMSP_1:func.5">*</a> <span class="p1">(<span class="default"><a href="lopban_4.html#K5" title="LOPBAN_4:func.5">Expan</a> (<font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">w</font>)</span>)</span></span><br/>







<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lopban_4.html#K6" title="LOPBAN_4:func.6">Expan_e</a> (<font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">w</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c5">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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"><font color="Maroon" title="c4">n</font> <a href="newton.html#K5" title="NEWTON:func.5">!</a></span>)</span></span>)</span> <a href="normsp_1.html#K5" title="NORMSP_1:func.5">*</a> <span class="p2">(<span class="default"><a href="lopban_4.html#K5" title="LOPBAN_4:func.5">Expan</a> (<font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">w</font>)</span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c5">k</font></span><br/><div><span class="lab"><font color="Green" title="E30">A1</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1"><span class="kw">now </span></a><div class="add"><a NAME="E1:30_1_1"/>
<font color="Maroon" title="c4">n</font> <a href="newton.html#K5" title="NEWTON:func.5">!</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T17" title="NEWTON:th.17">NEWTON:17</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A2</font></span>: 1 <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">k</font> <a href="newton.html#K5" title="NEWTON:func.5">!</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="c4">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="newton.html#K5" title="NEWTON:func.5">!</a></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="c4">n</font> <a href="newton.html#K5" title="NEWTON:func.5">!</a></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> 1</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="newton.html#K5" title="NEWTON:func.5">!</a></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="c5">k</font> <a href="newton.html#K5" title="NEWTON:func.5">!</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="c4">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="newton.html#K5" title="NEWTON:func.5">!</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T60" title="XCMPLX_1:th.60">XCMPLX_1:60</a></span>
<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"><font color="Maroon" title="c4">n</font> <a href="newton.html#K5" title="NEWTON:func.5">!</a></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="newton.html#K5" title="NEWTON:func.5">!</a></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="c5">k</font> <a href="newton.html#K5" title="NEWTON:func.5">!</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="c4">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="newton.html#K5" title="NEWTON:func.5">!</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T74" title="XCMPLX_1:th.74">XCMPLX_1:74</a></span>
;<br/><span class="kw">assume </span><a NAME="E3:30_1_1"/><span class="lab"><font color="Green" title="E32">A3</font></span>: 
<font color="Maroon" title="c5">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="lopban_4.html#K6" title="LOPBAN_4:func.6">Expan_e</a> (<font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">w</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c5">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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"><span class="p4">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="newton.html#K5" title="NEWTON:func.5">!</a></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="newton.html#K5" title="NEWTON:func.5">!</a></span>)</span></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="c2">z</font> <a href="lopban_3.html#K6" title="LOPBAN_3:func.6">#N</a> <font color="Maroon" title="c5">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="c3">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="c4">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c5">k</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="lopban_4.html#K6" title="LOPBAN_4:func.6">Expan_e</a> (<font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">w</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c5">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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"><font color="Maroon" title="c4">n</font> <a href="newton.html#K5" title="NEWTON:func.5">!</a></span>)</span></span>)</span> <a href="normsp_1.html#K5" title="NORMSP_1:func.5">*</a> <span class="p2">(<span class="default"><a href="lopban_4.html#K5" title="LOPBAN_4:func.5">Expan</a> (<font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">w</font>)</span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c5">k</font> )</span><br/><span class="kw">hence </span><span class="p1">(<span class="default"><a href="lopban_4.html#K6" title="LOPBAN_4:func.6">Expan_e</a> (<font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">w</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c5">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#K3" title="LOPBAN_4:func.3">Coef_e</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="lopban_3.html#K6" title="LOPBAN_3:func.6">#N</a> <font color="Maroon" title="c5">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="c3">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="c4">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c5">k</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lopban_4.html#D7" target="_self" title="LOPBAN_4:def.7">Def7</a></span>
<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"><span class="p4">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="newton.html#K5" title="NEWTON:func.5">!</a></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="newton.html#K5" title="NEWTON:func.5">!</a></span>)</span></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="c2">z</font> <a href="lopban_3.html#K6" title="LOPBAN_3:func.6">#N</a> <font color="Maroon" title="c5">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="c3">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="c4">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c5">k</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:30_1_1"><span class="lab"><font color="Green" title="E32">A3</font></span></a>, <a class="ref" href="lopban_4.html#D4" target="_self" title="LOPBAN_4:def.4">Def4</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lopban_4.html#K6" title="LOPBAN_4:func.6">Expan_e</a> (<font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">w</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c5">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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"><font color="Maroon" title="c4">n</font> <a href="newton.html#K5" title="NEWTON:func.5">!</a></span>)</span></span>)</span> <a href="normsp_1.html#K5" title="NORMSP_1:func.5">*</a> <span class="p2">(<span class="default"><a href="lopban_4.html#K5" title="LOPBAN_4:func.5">Expan</a> (<font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">w</font>)</span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c5">k</font></span><br/><span class="kw">hence </span><span class="p1">(<span class="default"><a href="lopban_4.html#K6" title="LOPBAN_4:func.6">Expan_e</a> (<font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">w</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c5">k</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="newton.html#K5" title="NEWTON:func.5">!</a></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="newton.html#K5" title="NEWTON:func.5">!</a></span>)</span></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="c5">k</font> <a href="newton.html#K5" title="NEWTON:func.5">!</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="c4">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="newton.html#K5" title="NEWTON:func.5">!</a></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="c2">z</font> <a href="lopban_3.html#K6" title="LOPBAN_3:func.6">#N</a> <font color="Maroon" title="c5">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="c3">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="c4">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c5">k</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:30_1_1"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="ref" href="lopban_3.html#T38" title="LOPBAN_3:th.38">LOPBAN_3:38</a></span>
<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"><font color="Maroon" title="c4">n</font> <a href="newton.html#K5" title="NEWTON:func.5">!</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="c4">n</font> <a href="newton.html#K5" title="NEWTON:func.5">!</a></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="newton.html#K5" title="NEWTON:func.5">!</a></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="newton.html#K5" title="NEWTON:func.5">!</a></span>)</span></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="c2">z</font> <a href="lopban_3.html#K6" title="LOPBAN_3:func.6">#N</a> <font color="Maroon" title="c5">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="c3">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="c4">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c5">k</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T74" title="XCMPLX_1:th.74">XCMPLX_1:74</a></span>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="newton.html#K5" title="NEWTON:func.5">!</a></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"><span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="newton.html#K5" title="NEWTON:func.5">!</a></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="newton.html#K5" title="NEWTON:func.5">!</a></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="newton.html#K5" title="NEWTON:func.5">!</a></span>)</span></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="c2">z</font> <a href="lopban_3.html#K6" title="LOPBAN_3:func.6">#N</a> <font color="Maroon" title="c5">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="c3">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="c4">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c5">k</font></span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lopban_3.html#T38" title="LOPBAN_3:th.38">LOPBAN_3:38</a></span>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="newton.html#K5" title="NEWTON:func.5">!</a></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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="newton.html#K5" title="NEWTON:func.5">!</a></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="newton.html#K5" title="NEWTON:func.5">!</a></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="newton.html#K5" title="NEWTON:func.5">!</a></span>)</span></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="c2">z</font> <a href="lopban_3.html#K6" title="LOPBAN_3:func.6">#N</a> <font color="Maroon" title="c5">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="c3">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="c4">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c5">k</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lopban_3.html#T38" title="LOPBAN_3:th.38">LOPBAN_3:38</a></span>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="newton.html#K5" title="NEWTON:func.5">!</a></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"><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="c4">n</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="lopban_3.html#K6" title="LOPBAN_3:func.6">#N</a> <font color="Maroon" title="c5">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="c3">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="c4">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c5">k</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:30_1_1"><span class="lab"><font color="Green" title="E32">A3</font></span></a>, <a class="ref" href="lopban_4.html#D3" target="_self" title="LOPBAN_4:def.3">Def3</a></span>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="newton.html#K5" title="NEWTON:func.5">!</a></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"><a href="lopban_4.html#K5" title="LOPBAN_4:func.5">Expan</a> (<font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">w</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c5">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:30_1_1"><span class="lab"><font color="Green" title="E32">A3</font></span></a>, <a class="ref" href="lopban_4.html#D6" target="_self" title="LOPBAN_4:def.6">Def6</a></span>
<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"><font color="Maroon" title="c4">n</font> <a href="newton.html#K5" title="NEWTON:func.5">!</a></span>)</span></span>)</span> <a href="normsp_1.html#K5" title="NORMSP_1:func.5">*</a> <span class="p2">(<span class="default"><a href="lopban_4.html#K5" title="LOPBAN_4:func.5">Expan</a> (<font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">w</font>)</span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c5">k</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="normsp_1.html#D5" title="NORMSP_1:def.5">NORMSP_1:def 5</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:30_1_2"/><span class="lab"><font color="Green" title="E31">A4</font></span>: 
<font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lopban_4.html#K6" title="LOPBAN_4:func.6">Expan_e</a> (<font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">w</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c5">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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"><font color="Maroon" title="c4">n</font> <a href="newton.html#K5" title="NEWTON:func.5">!</a></span>)</span></span>)</span> <a href="normsp_1.html#K5" title="NORMSP_1:func.5">*</a> <span class="p2">(<span class="default"><a href="lopban_4.html#K5" title="LOPBAN_4:func.5">Expan</a> (<font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">w</font>)</span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c5">k</font></span><br/><span class="kw">hence </span><span class="p1">(<span class="default"><a href="lopban_4.html#K6" title="LOPBAN_4:func.6">Expan_e</a> (<font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">w</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c5">k</font> = 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lopban_4.html#D7" target="_self" title="LOPBAN_4:def.7">Def7</a></span>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="newton.html#K5" title="NEWTON:func.5">!</a></span>)</span></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lopban_3.html#T38" title="LOPBAN_3:th.38">LOPBAN_3:38</a></span>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="newton.html#K5" title="NEWTON:func.5">!</a></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"><a href="lopban_4.html#K5" title="LOPBAN_4:func.5">Expan</a> (<font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">w</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c5">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1_2"><span class="lab"><font color="Green" title="E31">A4</font></span></a>, <a class="ref" href="lopban_4.html#D6" target="_self" title="LOPBAN_4:def.6">Def6</a></span>
<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"><font color="Maroon" title="c4">n</font> <a href="newton.html#K5" title="NEWTON:func.5">!</a></span>)</span></span>)</span> <a href="normsp_1.html#K5" title="NORMSP_1:func.5">*</a> <span class="p2">(<span class="default"><a href="lopban_4.html#K5" title="LOPBAN_4:func.5">Expan</a> (<font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">w</font>)</span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c5">k</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="normsp_1.html#D5" title="NORMSP_1:def.5">NORMSP_1:def 5</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:30_1"/>
<span class="p1">(<span class="default"><a href="lopban_4.html#K6" title="LOPBAN_4:func.6">Expan_e</a> (<font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">w</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c5">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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"><font color="Maroon" title="c4">n</font> <a href="newton.html#K5" title="NEWTON:func.5">!</a></span>)</span></span>)</span> <a href="normsp_1.html#K5" title="NORMSP_1:func.5">*</a> <span class="p2">(<span class="default"><a href="lopban_4.html#K5" title="LOPBAN_4:func.5">Expan</a> (<font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">w</font>)</span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c5">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1"><span class="lab"><font color="Green" title="E30">A1</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:30"/>
 <a href="lopban_4.html#K6" title="LOPBAN_4:func.6">Expan_e</a> (<font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">w</font>) <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="newton.html#K5" title="NEWTON:func.5">!</a></span>)</span></span>)</span> <a href="normsp_1.html#K5" title="NORMSP_1:func.5">*</a> <span class="p1">(<span class="default"><a href="lopban_4.html#K5" title="LOPBAN_4:func.5">Expan</a> (<font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">w</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T63" title="FUNCT_2:th.63">FUNCT_2:63</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
