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

<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>;<br/><b>let </b><font color="Maroon" title="c2">x</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>;<br/>



<b>assume </b><a NAME="E1:75"/><i><font color="Green" title="E47">A1</font></i>: 
<span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1
 ;<br/>

<b>set </b><font color="Maroon" title="c3">seq</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> ;<br/>
<a NAME="E2:75"/><i><font color="Green" title="E48">A2</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a>  is <a href="lopban_3.html#V1" title="LOPBAN_3:attr.1">summable</a>
 
<b>by </b><i><a class="txt" href="lopban_3.html#E1:75"><i><font color="Green" title="E47">A1</font></i></a>, <a class="ref" href="lopban_3.html#T45" target="_self" title="LOPBAN_3:th.45">Th45</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">y</font> =  <a href="lopban_3.html#K2" title="LOPBAN_3:func.2">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font> ;<br/>
<a NAME="E4:75"/><i><font color="Green" title="E49">A3</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>  holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_3.html#K1" title="LOPBAN_3:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <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_3.html#K1" title="LOPBAN_3:func.1">Partial_Sums</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span></span>)</span> <a href="lopban_3.html#K3" title="LOPBAN_3:func.3">^\</a> 1</span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</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="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> 0</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1"><b>proof </b></a><div class="add">

<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"><span class="p2">(<span class="default"><a href="lopban_3.html#K1" title="LOPBAN_3:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> $1</span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <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_3.html#K1" title="LOPBAN_3:func.1">Partial_Sums</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span></span>)</span> <a href="lopban_3.html#K3" title="LOPBAN_3:func.3">^\</a> 1</span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> $1</span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</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="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> 0</span>)</span>;<br/>
<i><font color="Green" title="E50">A4</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_3.html#K1" title="LOPBAN_3:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> 0</span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> = 
<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="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> 0</span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span>
<b>by </b><i><a class="ref" href="lopban_3.html#D1" target="_self" title="LOPBAN_3:def.1">Def1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span>
<b>by </b><i><a class="ref" href="lopban_3.html#D13" target="_self" title="LOPBAN_3:def.13">Def13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font>
<b>by </b><i><a class="ref" href="lopban_3.html#T43" target="_self" title="LOPBAN_3:th.43">Th43</a></i>
;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lopban_3.html#K1" title="LOPBAN_3:func.1">Partial_Sums</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span></span>)</span> <a href="lopban_3.html#K3" title="LOPBAN_3:func.3">^\</a> 1</span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> 0</span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</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="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> 0</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_3.html#K1" title="LOPBAN_3:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p2">(<span class="default">0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</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="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="lopban_3.html#D6" target="_self" title="LOPBAN_3: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_3.html#K1" title="LOPBAN_3:func.1">Partial_Sums</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> 0</span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</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="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="lopban_3.html#D1" target="_self" title="LOPBAN_3:def.1">Def1</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="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> 0</span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</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="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="lopban_3.html#D1" target="_self" title="LOPBAN_3:def.1">Def1</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="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> 1</span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</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="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> 0</span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> 0</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lopban_3.html#T43" target="_self" title="LOPBAN_3:th.43">Th43</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="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> 1</span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">X</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T28" title="RLVECT_1:th.28">RLVECT_1:28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</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="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> 0</span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span>
<b>by </b><i><a class="ref" href="lopban_3.html#D13" target="_self" title="LOPBAN_3:def.13">Def13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span>
<b>by </b><i><a class="ref" href="lopban_3.html#D13" target="_self" title="LOPBAN_3:def.13">Def13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font>
<b>by </b><i><a class="ref" href="lopban_3.html#T43" target="_self" title="LOPBAN_3:th.43">Th43</a></i>
;<br/>
<a NAME="E3:75_1"/><b>then </b><i><font color="Green" title="E51">A5</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i><a class="txt" href="lopban_3.html#E1:75_1"><i><font color="Green" title="E50">A4</font></i></a></i>;<br/>
<a NAME="E4:75_1"/><i><font color="Green" title="E52">A6</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1_3"><b>proof </b></a><div class="add">

<b>let </b><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> ;<br/>

<b>assume </b><a NAME="E1:75_1_3"/><i><font color="Green" title="E53">A7</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font>]
 ;<br/>

<i><font color="Green" title="E54">A8</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_3.html#K1" title="LOPBAN_3:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lopban_3.html#K1" title="LOPBAN_3:func.1">Partial_Sums</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span>
<b>by </b><i><a class="ref" href="lopban_3.html#D1" target="_self" title="LOPBAN_3:def.1">Def1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lopban_3.html#K1" title="LOPBAN_3:func.1">Partial_Sums</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</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="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lopban_3.html#T43" target="_self" title="LOPBAN_3:th.43">Th43</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_3.html#K1" title="LOPBAN_3:func.1">Partial_Sums</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span></span>)</span> <a href="lopban_3.html#K3" title="LOPBAN_3:func.3">^\</a> 1</span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> 0</span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</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="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="lopban_3.html#E1:75_1_3"><i><font color="Green" title="E53">A7</font></i></a>, <a class="ref" href="lopban_3.html#D13" target="_self" title="LOPBAN_3:def.13">Def13</a></i>
;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lopban_3.html#K1" title="LOPBAN_3:func.1">Partial_Sums</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span></span>)</span> <a href="lopban_3.html#K3" title="LOPBAN_3:func.3">^\</a> 1</span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</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="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> 0</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_3.html#K1" title="LOPBAN_3:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</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="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="lopban_3.html#D6" target="_self" title="LOPBAN_3: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_3.html#K1" title="LOPBAN_3:func.1">Partial_Sums</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</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="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="lopban_3.html#D1" target="_self" title="LOPBAN_3:def.1">Def1</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_3.html#K1" title="LOPBAN_3:func.1">Partial_Sums</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span></span>)</span> <a href="lopban_3.html#K3" title="LOPBAN_3:func.3">^\</a> 1</span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> 0</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lopban_3.html#D6" target="_self" title="LOPBAN_3: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_3.html#K1" title="LOPBAN_3:func.1">Partial_Sums</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span></span>)</span> <a href="lopban_3.html#K3" title="LOPBAN_3:func.3">^\</a> 1</span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> 0</span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</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="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a></i>
;<br/>
<b>hence </b><a NAME="E4:75_1_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_3.html#K1" title="LOPBAN_3:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <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_3.html#K1" title="LOPBAN_3:func.1">Partial_Sums</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span></span>)</span> <a href="lopban_3.html#K3" title="LOPBAN_3:func.3">^\</a> 1</span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</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="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> 0</span>)</span>
 <b>by </b><i><a class="txt" href="lopban_3.html#E2:75_1_3"><i><font color="Green" title="E54">A8</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:75_1"/>
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="lopban_3.html#E3:75_1"><i><font color="Green" title="E51">A5</font></i></a>, <a class="txt" href="lopban_3.html#E4:75_1"><i><font color="Green" title="E52">A6</font></i></a>);<br/></i>
<b>hence </b><a NAME="E6:75_1"/>
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  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_3.html#K1" title="LOPBAN_3:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <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_3.html#K1" title="LOPBAN_3:func.1">Partial_Sums</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span></span>)</span> <a href="lopban_3.html#K3" title="LOPBAN_3:func.3">^\</a> 1</span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</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="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> 0</span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E5:75"/><i><font color="Green" title="E50">A9</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>  holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="group_1.html#K1" title="GROUP_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="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b1">n</font></span>)</span> <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="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_2"><b>proof </b></a><div class="add">

<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"><span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="group_1.html#K1" title="GROUP_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="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_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"><span class="p3">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p1">(<span class="default">$1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>;<br/>
<i><font color="Green" title="E51">A10</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="group_1.html#K1" title="GROUP_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="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> 0</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span>
<b>by </b><i><a class="ref" href="lopban_3.html#D13" target="_self" title="LOPBAN_3:def.13">Def13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font>
<b>by </b><i><a class="ref" href="lopban_3.html#T43" target="_self" title="LOPBAN_3:th.43">Th43</a></i>
;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
<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="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> 0</span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span>
<b>by </b><i><a class="ref" href="lopban_3.html#D13" target="_self" title="LOPBAN_3:def.13">Def13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span>
<b>by </b><i><a class="ref" href="lopban_3.html#D13" target="_self" title="LOPBAN_3:def.13">Def13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font>
<b>by </b><i><a class="ref" href="lopban_3.html#T43" target="_self" title="LOPBAN_3:th.43">Th43</a></i>
;<br/>
<a NAME="E3:75_2"/><b>then </b><i><font color="Green" title="E52">A11</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i><a class="txt" href="lopban_3.html#E1:75_2"><i><font color="Green" title="E51">A10</font></i></a></i>;<br/>
<a NAME="E4:75_2"/><i><font color="Green" title="E53">A12</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_2_3"><b>proof </b></a><div class="add">

<b>let </b><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> ;<br/>

<b>assume </b><a NAME="E1:75_2_3"/><i><font color="Green" title="E54">A13</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font>]
 ;<br/>

<b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="group_1.html#K1" title="GROUP_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="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="group_1.html#K1" title="GROUP_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"><span class="p5">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lopban_3.html#D13" target="_self" title="LOPBAN_3:def.13">Def13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">k</font></span>)</span></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span>
<b>by </b><i><a class="ref" href="lopban_3.html#T43" target="_self" title="LOPBAN_3:th.43">Th43</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="lopban_3.html#E1:75_2_3"><i><font color="Green" title="E54">A13</font></i></a>, <a class="ref" href="lopban_3.html#D13" target="_self" title="LOPBAN_3:def.13">Def13</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E5:75_2"/>
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="lopban_3.html#E3:75_2"><i><font color="Green" title="E52">A11</font></i></a>, <a class="txt" href="lopban_3.html#E4:75_2"><i><font color="Green" title="E53">A12</font></i></a>);<br/></i>
<b>hence </b><a NAME="E6:75_2"/>
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  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="group_1.html#K1" title="GROUP_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="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b1">n</font></span>)</span> <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="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E6:75"/><i><font color="Green" title="E51">A14</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>  holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_3.html#K1" title="LOPBAN_3:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b1">n</font></span>)</span> <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_3.html#K1" title="LOPBAN_3:func.1">Partial_Sums</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span></span>)</span> <a href="lopban_3.html#K3" title="LOPBAN_3:func.3">^\</a> 1</span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</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="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> 0</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_3"><b>proof </b></a><div class="add">

<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"><span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_3.html#K1" title="LOPBAN_3:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_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"><span class="p3">(<span class="default"><a href="lopban_3.html#K1" title="LOPBAN_3:func.1">Partial_Sums</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span></span>)</span> <a href="lopban_3.html#K3" title="LOPBAN_3:func.3">^\</a> 1</span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> $1</span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</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="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> 0</span>)</span>;<br/>
<i><font color="Green" title="E52">A15</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_3.html#K1" title="LOPBAN_3:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> 0</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="group_1.html#K1" title="GROUP_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="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="lopban_3.html#D1" target="_self" title="LOPBAN_3:def.1">Def1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span>
<b>by </b><i><a class="ref" href="lopban_3.html#D13" target="_self" title="LOPBAN_3:def.13">Def13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font>
<b>by </b><i><a class="ref" href="lopban_3.html#T43" target="_self" title="LOPBAN_3:th.43">Th43</a></i>
;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lopban_3.html#K1" title="LOPBAN_3:func.1">Partial_Sums</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span></span>)</span> <a href="lopban_3.html#K3" title="LOPBAN_3:func.3">^\</a> 1</span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> 0</span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</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="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> 0</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_3.html#K1" title="LOPBAN_3:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p2">(<span class="default">0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</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="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="lopban_3.html#D6" target="_self" title="LOPBAN_3: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_3.html#K1" title="LOPBAN_3:func.1">Partial_Sums</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> 0</span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</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="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="lopban_3.html#D1" target="_self" title="LOPBAN_3:def.1">Def1</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="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> 0</span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</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="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="lopban_3.html#D1" target="_self" title="LOPBAN_3:def.1">Def1</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="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> 1</span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</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="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> 0</span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> 0</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lopban_3.html#T43" target="_self" title="LOPBAN_3:th.43">Th43</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="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> 1</span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">X</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T28" title="RLVECT_1:th.28">RLVECT_1:28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</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="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> 0</span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span>
<b>by </b><i><a class="ref" href="lopban_3.html#D13" target="_self" title="LOPBAN_3:def.13">Def13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span>
<b>by </b><i><a class="ref" href="lopban_3.html#D13" target="_self" title="LOPBAN_3:def.13">Def13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font>
<b>by </b><i><a class="ref" href="lopban_3.html#T43" target="_self" title="LOPBAN_3:th.43">Th43</a></i>
;<br/>
<a NAME="E3:75_3"/><b>then </b><i><font color="Green" title="E53">A16</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i><a class="txt" href="lopban_3.html#E1:75_3"><i><font color="Green" title="E52">A15</font></i></a></i>;<br/>
<a NAME="E4:75_3"/><i><font color="Green" title="E54">A17</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_3_3"><b>proof </b></a><div class="add">

<b>let </b><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> ;<br/>

<b>assume </b><a NAME="E1:75_3_3"/><i><font color="Green" title="E55">A18</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font>]
 ;<br/>

<i><font color="Green" title="E56">A19</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_3.html#K1" title="LOPBAN_3:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lopban_3.html#K1" title="LOPBAN_3:func.1">Partial_Sums</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lopban_3.html#D1" target="_self" title="LOPBAN_3:def.1">Def1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lopban_3.html#K1" title="LOPBAN_3:func.1">Partial_Sums</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">k</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lopban_3.html#T43" target="_self" title="LOPBAN_3:th.43">Th43</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_3.html#K1" title="LOPBAN_3:func.1">Partial_Sums</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span></span>)</span> <a href="lopban_3.html#K3" title="LOPBAN_3:func.3">^\</a> 1</span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> 0</span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</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="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="lopban_3.html#E5:75"><i><font color="Green" title="E50">A9</font></i></a>, <a class="txt" href="lopban_3.html#E1:75_3_3"><i><font color="Green" title="E55">A18</font></i></a></i>
;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lopban_3.html#K1" title="LOPBAN_3:func.1">Partial_Sums</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span></span>)</span> <a href="lopban_3.html#K3" title="LOPBAN_3:func.3">^\</a> 1</span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</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="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> 0</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_3.html#K1" title="LOPBAN_3:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</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="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="lopban_3.html#D6" target="_self" title="LOPBAN_3: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_3.html#K1" title="LOPBAN_3:func.1">Partial_Sums</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</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="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="lopban_3.html#D1" target="_self" title="LOPBAN_3:def.1">Def1</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_3.html#K1" title="LOPBAN_3:func.1">Partial_Sums</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span></span>)</span> <a href="lopban_3.html#K3" title="LOPBAN_3:func.3">^\</a> 1</span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> 0</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lopban_3.html#D6" target="_self" title="LOPBAN_3: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_3.html#K1" title="LOPBAN_3:func.1">Partial_Sums</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span></span>)</span> <a href="lopban_3.html#K3" title="LOPBAN_3:func.3">^\</a> 1</span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> 0</span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</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="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a></i>
;<br/>
<b>hence </b><a NAME="E4:75_3_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_3.html#K1" title="LOPBAN_3:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</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"><span class="p3">(<span class="default"><a href="lopban_3.html#K1" title="LOPBAN_3:func.1">Partial_Sums</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span></span>)</span> <a href="lopban_3.html#K3" title="LOPBAN_3:func.3">^\</a> 1</span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</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="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> 0</span>)</span>
 <b>by </b><i><a class="txt" href="lopban_3.html#E2:75_3_3"><i><font color="Green" title="E56">A19</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:75_3"/>
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="lopban_3.html#E3:75_3"><i><font color="Green" title="E53">A16</font></i></a>, <a class="txt" href="lopban_3.html#E4:75_3"><i><font color="Green" title="E54">A17</font></i></a>);<br/></i>
<b>hence </b><a NAME="E6:75_3"/>
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  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_3.html#K1" title="LOPBAN_3:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b1">n</font></span>)</span> <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_3.html#K1" title="LOPBAN_3:func.1">Partial_Sums</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span></span>)</span> <a href="lopban_3.html#K3" title="LOPBAN_3:func.3">^\</a> 1</span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</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="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> 0</span>)</span>
 ;<br/>


</div><b>end;</b></div>
<div><i><font color="Green" title="E52">A20</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="75_4"><b>now </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> ;<br/><b>reconsider </b><font color="Maroon" title="c6">yn</font> = <span class="p1">(<span class="default"><a href="lopban_3.html#K1" title="LOPBAN_3:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font> ;<br/><i><font color="Green" title="E53">A21</font></i>: <span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">x</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">x</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T26" title="RLVECT_1:th.26">RLVECT_1:26</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">yn</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">yn</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">x</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T28" title="RLVECT_1:th.28">RLVECT_1:28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">yn</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">yn</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">x</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T43" title="RLVECT_1:th.43">RLVECT_1:43</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">yn</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">yn</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">yn</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">yn</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">x</font></span>)</span>
<b>by </b><i><a class="ref" href="lopban_3.html#T43" target="_self" title="LOPBAN_3:th.43">Th43</a></i>
;<br/><i><font color="Green" title="E54">A22</font></i>: <font color="Maroon" title="c6">yn</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">yn</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">yn</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lopban_3.html#T43" target="_self" title="LOPBAN_3:th.43">Th43</a></i>
<br/>.= 
<font color="Maroon" title="c6">yn</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">yn</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lopban_3.html#T43" target="_self" title="LOPBAN_3:th.43">Th43</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_3.html#K1" title="LOPBAN_3:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</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_3.html#K1" title="LOPBAN_3:func.1">Partial_Sums</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span></span>)</span> <a href="lopban_3.html#K3" title="LOPBAN_3:func.3">^\</a> 1</span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> 0</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="lopban_3.html#E4:75"><i><font color="Green" title="E49">A3</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lopban_3.html#K1" title="LOPBAN_3:func.1">Partial_Sums</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="lopban_3.html#K1" title="LOPBAN_3:func.1">Partial_Sums</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span></span>)</span> <a href="lopban_3.html#K3" title="LOPBAN_3:func.3">^\</a> 1</span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</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="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="lopban_3.html#T43" target="_self" title="LOPBAN_3:th.43">Th43</a></i>
;<br/><a NAME="E4:75_4"/>
<span class="p1">(<span class="default"><a href="lopban_3.html#K1" title="LOPBAN_3:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="lopban_3.html#K1" title="LOPBAN_3:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</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="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="lopban_3.html#D1" target="_self" title="LOPBAN_3:def.1">Def1</a></i>;<br/><a NAME="E5:75_4"/><b>then </b>
<span class="p1">(<span class="default"><a href="lopban_3.html#K1" title="LOPBAN_3:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="lopban_3.html#K1" title="LOPBAN_3:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</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="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="lopban_3.html#K3" title="LOPBAN_3:func.3">^\</a> 1</span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="lopban_3.html#D6" target="_self" title="LOPBAN_3:def.6">Def6</a></i>;<br/><b>then </b><font color="Maroon" title="c6">yn</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lopban_3.html#K1" title="LOPBAN_3:func.1">Partial_Sums</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="lopban_3.html#K1" title="LOPBAN_3:func.1">Partial_Sums</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="lopban_3.html#K3" title="LOPBAN_3:func.3">^\</a> 1</span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</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="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> 0</span>)</span>
<b>by </b><i><a class="txt" href="lopban_3.html#E3:75_4"><i><font color="Green" title="E54">A22</font></i></a>, <a class="ref" href="lopban_3.html#D6" target="_self" title="LOPBAN_3: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_3.html#K1" title="LOPBAN_3:func.1">Partial_Sums</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="lopban_3.html#K1" title="LOPBAN_3:func.1">Partial_Sums</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="lopban_3.html#K3" title="LOPBAN_3:func.3">^\</a> 1</span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</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="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="lopban_3.html#T43" target="_self" title="LOPBAN_3:th.43">Th43</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="lopban_3.html#K3" title="LOPBAN_3:func.3">^\</a> 1</span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</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="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T28" title="RLVECT_1:th.28">RLVECT_1:28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</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"><span class="p0">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="lopban_3.html#K3" title="LOPBAN_3:func.3">^\</a> 1</span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> 0</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T43" title="RLVECT_1:th.43">RLVECT_1:43</a></i>
<br/>.= 
 <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</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"><span class="p0">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="lopban_3.html#K3" title="LOPBAN_3:func.3">^\</a> 1</span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> 0</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T27" title="RLVECT_1:th.27">RLVECT_1:27</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="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> 0</span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</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="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="lopban_3.html#K3" title="LOPBAN_3:func.3">^\</a> 1</span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T47" title="RLVECT_1:th.47">RLVECT_1:47</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="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> 0</span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</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="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lopban_3.html#D6" target="_self" title="LOPBAN_3:def.6">Def6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</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="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lopban_3.html#D13" target="_self" title="LOPBAN_3:def.13">Def13</a></i>
;<br/><b>then </b><i><font color="Green" title="E55">A23</font></i>: <span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">yn</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">x</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lopban_3.html#T43" target="_self" title="LOPBAN_3:th.43">Th43</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="lopban_3.html#T43" target="_self" title="LOPBAN_3:th.43">Th43</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="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="lopban_3.html#K3" title="LOPBAN_3:func.3">^\</a> 1</span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font>
<b>by </b><i><a class="ref" href="lopban_3.html#D6" target="_self" title="LOPBAN_3:def.6">Def6</a></i>
;<br/><a NAME="E8:75_4"/><i><font color="Green" title="E56">A24</font></i>: 
<span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><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"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="lopban_3.html#K3" title="LOPBAN_3:func.3">^\</a> 1</span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">yn</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><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"><span class="p0">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="lopban_3.html#K3" title="LOPBAN_3:func.3">^\</a> 1</span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">yn</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">x</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span>
 <b>by </b><i><a class="ref" href="normsp_1.html#D2" title="NORMSP_1:def.2">NORMSP_1:def 2</a></i>;<br/><a NAME="E9:75_4"/><i><font color="Green" title="E57">A25</font></i>: 
<span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><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"><span class="p0">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="lopban_3.html#K3" title="LOPBAN_3:func.3">^\</a> 1</span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">yn</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">x</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="lopban_3.html#K3" title="LOPBAN_3:func.3">^\</a> 1</span>)</span></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">yn</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">x</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span>
 <b>by </b><i><a class="ref" href="normsp_1.html#D10" title="NORMSP_1:def.10">NORMSP_1:def 10</a></i>;<br/><a NAME="E10:75_4"/>
<span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">yn</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">x</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">yn</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c4">y</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span>
 <b>by </b><i><a class="ref" href="lopban_3.html#T43" target="_self" title="LOPBAN_3:th.43">Th43</a></i>;<br/><a NAME="E11:75_4"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="lopban_3.html#K3" title="LOPBAN_3:func.3">^\</a> 1</span>)</span></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">yn</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">x</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="lopban_3.html#K3" title="LOPBAN_3:func.3">^\</a> 1</span>)</span></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">yn</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c4">y</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/><a NAME="E12:75_4"/><b>then </b>
<span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><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"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="lopban_3.html#K3" title="LOPBAN_3:func.3">^\</a> 1</span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">yn</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="lopban_3.html#K3" title="LOPBAN_3:func.3">^\</a> 1</span>)</span></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">yn</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c4">y</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="lopban_3.html#E8:75_4"><i><font color="Green" title="E56">A24</font></i></a>, <a class="txt" href="lopban_3.html#E9:75_4"><i><font color="Green" title="E57">A25</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E13:75_4"/><b>then </b>
<span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="lopban_3.html#K3" title="LOPBAN_3:func.3">^\</a> 1</span>)</span></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="lopban_3.html#K1" title="LOPBAN_3:func.1">Partial_Sums</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K5" title="NORMSP_1:func.5">-</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="lopban_3.html#E2:75_4"><i><font color="Green" title="E53">A21</font></i></a>, <a class="txt" href="lopban_3.html#E7:75_4"><i><font color="Green" title="E55">A23</font></i></a>, <a class="ref" href="normsp_1.html#D7" title="NORMSP_1:def.7">NORMSP_1:def 7</a></i>;<br/><a NAME="E14:75_4"/><b>then </b>
<span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="lopban_3.html#K3" title="LOPBAN_3:func.3">^\</a> 1</span>)</span></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="lopban_3.html#K1" title="LOPBAN_3:func.1">Partial_Sums</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K5" title="NORMSP_1:func.5">-</a> <font color="Maroon" title="c4">y</font></span>)</span></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="normsp_1.html#D10" title="NORMSP_1:def.10">NORMSP_1:def 10</a></i>;<br/><a NAME="E15:75_4"/><b>then </b>
<span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="lopban_3.html#K3" title="LOPBAN_3:func.3">^\</a> 1</span>)</span></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="seq_1.html#K14" title="SEQ_1:func.14">(#)</a> <span class="p3"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="lopban_3.html#K1" title="LOPBAN_3:func.1">Partial_Sums</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K5" title="NORMSP_1:func.5">-</a> <font color="Maroon" title="c4">y</font></span>)</span></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="seq_1.html#T13" title="SEQ_1:th.13">SEQ_1:13</a></i>;<br/><b>hence </b><a NAME="E16:75_4"/>
<span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="lopban_3.html#K3" title="LOPBAN_3:func.3">^\</a> 1</span>)</span></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <span class="p2">(<span class="default"><span class="p3"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="seq_1.html#K14" title="SEQ_1:func.14">(#)</a> <span class="p3"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="lopban_3.html#K1" title="LOPBAN_3:func.1">Partial_Sums</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K5" title="NORMSP_1:func.5">-</a> <font color="Maroon" title="c4">y</font></span>)</span></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">n</font>
 <b>by </b><i><a class="ref" href="seq_1.html#T11" title="SEQ_1:th.11">SEQ_1:11</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green" title="E53">A26</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="75_5"><b>now </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> ;<br/><b>reconsider </b><font color="Maroon" title="c6">yn</font> = <span class="p1">(<span class="default"><a href="lopban_3.html#K1" title="LOPBAN_3:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font> ;<br/><i><font color="Green" title="E54">A27</font></i>: <span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c4">y</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c4">y</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T26" title="RLVECT_1:th.26">RLVECT_1:26</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c6">yn</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c6">yn</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c4">y</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T28" title="RLVECT_1:th.28">RLVECT_1:28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c6">yn</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c6">yn</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c4">y</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T43" title="RLVECT_1:th.43">RLVECT_1:43</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c6">yn</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c6">yn</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c4">y</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c6">yn</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">yn</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c4">y</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lopban_3.html#T43" target="_self" title="LOPBAN_3:th.43">Th43</a></i>
;<br/><i><font color="Green" title="E55">A28</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c6">yn</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c6">yn</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c6">yn</font></span>)</span>
<b>by </b><i><a class="ref" href="lopban_3.html#T43" target="_self" title="LOPBAN_3:th.43">Th43</a></i>
<br/>.= 
<font color="Maroon" title="c6">yn</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c6">yn</font></span>)</span>
<b>by </b><i><a class="ref" href="lopban_3.html#T43" target="_self" title="LOPBAN_3:th.43">Th43</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_3.html#K1" title="LOPBAN_3:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</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_3.html#K1" title="LOPBAN_3:func.1">Partial_Sums</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span></span>)</span> <a href="lopban_3.html#K3" title="LOPBAN_3:func.3">^\</a> 1</span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> 0</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="lopban_3.html#E6:75"><i><font color="Green" title="E51">A14</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lopban_3.html#K1" title="LOPBAN_3:func.1">Partial_Sums</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="lopban_3.html#K1" title="LOPBAN_3:func.1">Partial_Sums</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span></span>)</span> <a href="lopban_3.html#K3" title="LOPBAN_3:func.3">^\</a> 1</span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</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="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="lopban_3.html#T43" target="_self" title="LOPBAN_3:th.43">Th43</a></i>
;<br/><a NAME="E4:75_5"/>
<span class="p1">(<span class="default"><a href="lopban_3.html#K1" title="LOPBAN_3:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="lopban_3.html#K1" title="LOPBAN_3:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</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="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="lopban_3.html#D1" target="_self" title="LOPBAN_3:def.1">Def1</a></i>;<br/><a NAME="E5:75_5"/><b>then </b>
<span class="p1">(<span class="default"><a href="lopban_3.html#K1" title="LOPBAN_3:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="lopban_3.html#K1" title="LOPBAN_3:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</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="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="lopban_3.html#K3" title="LOPBAN_3:func.3">^\</a> 1</span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="lopban_3.html#D6" target="_self" title="LOPBAN_3:def.6">Def6</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c6">yn</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lopban_3.html#K1" title="LOPBAN_3:func.1">Partial_Sums</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="lopban_3.html#K1" title="LOPBAN_3:func.1">Partial_Sums</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="lopban_3.html#K3" title="LOPBAN_3:func.3">^\</a> 1</span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</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="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> 0</span>)</span>
<b>by </b><i><a class="txt" href="lopban_3.html#E3:75_5"><i><font color="Green" title="E55">A28</font></i></a>, <a class="ref" href="lopban_3.html#D6" target="_self" title="LOPBAN_3: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_3.html#K1" title="LOPBAN_3:func.1">Partial_Sums</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="lopban_3.html#K1" title="LOPBAN_3:func.1">Partial_Sums</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="lopban_3.html#K3" title="LOPBAN_3:func.3">^\</a> 1</span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</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="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="lopban_3.html#T43" target="_self" title="LOPBAN_3:th.43">Th43</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="lopban_3.html#K3" title="LOPBAN_3:func.3">^\</a> 1</span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</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="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T28" title="RLVECT_1:th.28">RLVECT_1:28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</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"><span class="p0">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="lopban_3.html#K3" title="LOPBAN_3:func.3">^\</a> 1</span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> 0</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T43" title="RLVECT_1:th.43">RLVECT_1:43</a></i>
<br/>.= 
 <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</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"><span class="p0">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="lopban_3.html#K3" title="LOPBAN_3:func.3">^\</a> 1</span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> 0</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T27" title="RLVECT_1:th.27">RLVECT_1:27</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="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> 0</span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</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="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="lopban_3.html#K3" title="LOPBAN_3:func.3">^\</a> 1</span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T47" title="RLVECT_1:th.47">RLVECT_1:47</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="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> 0</span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</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="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lopban_3.html#D6" target="_self" title="LOPBAN_3:def.6">Def6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</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="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lopban_3.html#D13" target="_self" title="LOPBAN_3:def.13">Def13</a></i>
;<br/><b>then </b><i><font color="Green" title="E56">A29</font></i>: <span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c6">yn</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lopban_3.html#T43" target="_self" title="LOPBAN_3:th.43">Th43</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="lopban_3.html#T43" target="_self" title="LOPBAN_3:th.43">Th43</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="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="lopban_3.html#K3" title="LOPBAN_3:func.3">^\</a> 1</span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font>
<b>by </b><i><a class="ref" href="lopban_3.html#D6" target="_self" title="LOPBAN_3:def.6">Def6</a></i>
;<br/><a NAME="E8:75_5"/><i><font color="Green" title="E57">A30</font></i>: 
<span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><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"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="lopban_3.html#K3" title="LOPBAN_3:func.3">^\</a> 1</span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">yn</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c4">y</font></span>)</span></span>)</span></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><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"><span class="p0">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="lopban_3.html#K3" title="LOPBAN_3:func.3">^\</a> 1</span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">yn</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c4">y</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span>
 <b>by </b><i><a class="ref" href="normsp_1.html#D2" title="NORMSP_1:def.2">NORMSP_1:def 2</a></i>;<br/><a NAME="E9:75_5"/><i><font color="Green" title="E58">A31</font></i>: 
<span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><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"><span class="p0">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="lopban_3.html#K3" title="LOPBAN_3:func.3">^\</a> 1</span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">yn</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c4">y</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="lopban_3.html#K3" title="LOPBAN_3:func.3">^\</a> 1</span>)</span></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">yn</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c4">y</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span>
 <b>by </b><i><a class="ref" href="normsp_1.html#D10" title="NORMSP_1:def.10">NORMSP_1:def 10</a></i>;<br/><a NAME="E10:75_5"/>
<span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">yn</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c4">y</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">yn</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c4">y</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span>
 <b>by </b><i><a class="ref" href="lopban_3.html#T43" target="_self" title="LOPBAN_3:th.43">Th43</a></i>;<br/><a NAME="E11:75_5"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="lopban_3.html#K3" title="LOPBAN_3:func.3">^\</a> 1</span>)</span></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">yn</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c4">y</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="lopban_3.html#K3" title="LOPBAN_3:func.3">^\</a> 1</span>)</span></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">yn</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c4">y</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/><a NAME="E12:75_5"/><b>then </b>
<span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><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"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="lopban_3.html#K3" title="LOPBAN_3:func.3">^\</a> 1</span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">yn</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c4">y</font></span>)</span></span>)</span></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="lopban_3.html#K3" title="LOPBAN_3:func.3">^\</a> 1</span>)</span></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">yn</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c4">y</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="lopban_3.html#E8:75_5"><i><font color="Green" title="E57">A30</font></i></a>, <a class="txt" href="lopban_3.html#E9:75_5"><i><font color="Green" title="E58">A31</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E13:75_5"/><b>then </b>
<span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c4">y</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="lopban_3.html#K3" title="LOPBAN_3:func.3">^\</a> 1</span>)</span></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="lopban_3.html#K1" title="LOPBAN_3:func.1">Partial_Sums</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K5" title="NORMSP_1:func.5">-</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="lopban_3.html#E2:75_5"><i><font color="Green" title="E54">A27</font></i></a>, <a class="txt" href="lopban_3.html#E7:75_5"><i><font color="Green" title="E56">A29</font></i></a>, <a class="ref" href="normsp_1.html#D7" title="NORMSP_1:def.7">NORMSP_1:def 7</a></i>;<br/><a NAME="E14:75_5"/><b>then </b>
<span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c4">y</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="lopban_3.html#K3" title="LOPBAN_3:func.3">^\</a> 1</span>)</span></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="lopban_3.html#K1" title="LOPBAN_3:func.1">Partial_Sums</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K5" title="NORMSP_1:func.5">-</a> <font color="Maroon" title="c4">y</font></span>)</span></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="normsp_1.html#D10" title="NORMSP_1:def.10">NORMSP_1:def 10</a></i>;<br/><a NAME="E15:75_5"/><b>then </b>
<span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c4">y</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="lopban_3.html#K3" title="LOPBAN_3:func.3">^\</a> 1</span>)</span></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="seq_1.html#K14" title="SEQ_1:func.14">(#)</a> <span class="p3"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="lopban_3.html#K1" title="LOPBAN_3:func.1">Partial_Sums</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K5" title="NORMSP_1:func.5">-</a> <font color="Maroon" title="c4">y</font></span>)</span></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="seq_1.html#T13" title="SEQ_1:th.13">SEQ_1:13</a></i>;<br/><b>hence </b><a NAME="E16:75_5"/>
<span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c4">y</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="lopban_3.html#K3" title="LOPBAN_3:func.3">^\</a> 1</span>)</span></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <span class="p2">(<span class="default"><span class="p3"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="seq_1.html#K14" title="SEQ_1:func.14">(#)</a> <span class="p3"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="lopban_3.html#K1" title="LOPBAN_3:func.1">Partial_Sums</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K5" title="NORMSP_1:func.5">-</a> <font color="Maroon" title="c4">y</font></span>)</span></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">n</font>
 <b>by </b><i><a class="ref" href="seq_1.html#T11" title="SEQ_1:th.11">SEQ_1:11</a></i>;<br/></div><b>end;</b></div>
<a NAME="E9:75"/><i><font color="Green" title="E54">A32</font></i>: 
 <a href="lopban_3.html#K1" title="LOPBAN_3:func.1">Partial_Sums</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> is <a href="normsp_1.html#V4" title="NORMSP_1:attr.4">convergent</a>
 
<b>by </b><i><a class="txt" href="lopban_3.html#E2:75"><i><font color="Green" title="E48">A2</font></i></a>, <a class="ref" href="lopban_3.html#D2" target="_self" title="LOPBAN_3:def.2">Def2</a></i>;<br/>
<a NAME="E10:75"/><i><font color="Green" title="E55">A33</font></i>: 
( <span class="p1"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lopban_3.html#K1" title="LOPBAN_3:func.1">Partial_Sums</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K5" title="NORMSP_1:func.5">-</a> <font color="Maroon" title="c4">y</font></span>)</span></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lopban_3.html#K1" title="LOPBAN_3:func.1">Partial_Sums</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K5" title="NORMSP_1:func.5">-</a> <font color="Maroon" title="c4">y</font></span>)</span></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 
<b>by </b><i><a class="txt" href="lopban_3.html#E9:75"><i><font color="Green" title="E54">A32</font></i></a>, <a class="ref" href="normsp_1.html#T41" title="NORMSP_1:th.41">NORMSP_1:41</a></i>;<br/>
<a NAME="E11:75"/><b>then </b><i><font color="Green" title="E56">A34</font></i>: 
<span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="seq_1.html#K14" title="SEQ_1:func.14">(#)</a> <span class="p1"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lopban_3.html#K1" title="LOPBAN_3:func.1">Partial_Sums</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K5" title="NORMSP_1:func.5">-</a> <font color="Maroon" title="c4">y</font></span>)</span></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 
<b>by </b><i><a class="ref" href="seq_2.html#T21" title="SEQ_2:th.21">SEQ_2:21</a></i>;<br/>
<i><font color="Green" title="E57">A35</font></i>:  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="seq_1.html#K14" title="SEQ_1:func.14">(#)</a> <span class="p2"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="lopban_3.html#K1" title="LOPBAN_3:func.1">Partial_Sums</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K5" title="NORMSP_1:func.5">-</a> <font color="Maroon" title="c4">y</font></span>)</span></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span></span>)</span> = 
<span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> 0
<b>by </b><i><a class="txt" href="lopban_3.html#E10:75"><i><font color="Green" title="E55">A33</font></i></a>, <a class="ref" href="seq_2.html#T22" title="SEQ_2:th.22">SEQ_2:22</a></i>
<br/>.= 
0

;<br/>
<a NAME="E13:75"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a>  is <a href="normsp_1.html#V4" title="NORMSP_1:attr.4">convergent</a> &amp;  <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">X</font> )
 
<b>by </b><i><a class="txt" href="lopban_3.html#E2:75"><i><font color="Green" title="E48">A2</font></i></a>, <a class="ref" href="lopban_3.html#T19" target="_self" title="LOPBAN_3:th.19">Th19</a></i>;<br/>
<a NAME="E14:75"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="lopban_3.html#K3" title="LOPBAN_3:func.3">^\</a> 1 is <a href="normsp_1.html#V4" title="NORMSP_1:attr.4">convergent</a> &amp;  <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</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="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="lopban_3.html#K3" title="LOPBAN_3:func.3">^\</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">X</font> )
 
<b>by </b><i><a class="ref" href="lopban_3.html#T14" target="_self" title="LOPBAN_3:th.14">Th14</a></i>;<br/>
<a NAME="E15:75"/><b>then </b><i><font color="Green" title="E58">A36</font></i>: 
( <span class="p1"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><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="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="lopban_3.html#K3" title="LOPBAN_3:func.3">^\</a> 1</span>)</span></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><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="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="lopban_3.html#K3" title="LOPBAN_3:func.3">^\</a> 1</span>)</span></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">X</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> )
 
<b>by </b><i><a class="ref" href="lopban_1.html#T47" title="LOPBAN_1:th.47">LOPBAN_1:47</a></i>;<br/>
<a NAME="E16:75"/><b>then </b><i><font color="Green" title="E59">A37</font></i>: 
<span class="p1"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><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="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="lopban_3.html#K3" title="LOPBAN_3:func.3">^\</a> 1</span>)</span></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="seq_1.html#K14" title="SEQ_1:func.14">(#)</a> <span class="p2"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="lopban_3.html#K1" title="LOPBAN_3:func.1">Partial_Sums</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K5" title="NORMSP_1:func.5">-</a> <font color="Maroon" title="c4">y</font></span>)</span></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span></span>)</span> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 
<b>by </b><i><a class="txt" href="lopban_3.html#E11:75"><i><font color="Green" title="E56">A34</font></i></a>, <a class="ref" href="seq_2.html#T19" title="SEQ_2:th.19">SEQ_2:19</a></i>;<br/>
<i><font color="Green" title="E60">A38</font></i>:  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="lopban_3.html#K3" title="LOPBAN_3:func.3">^\</a> 1</span>)</span></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <span class="p2">(<span class="default"><span class="p3"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="seq_1.html#K14" title="SEQ_1:func.14">(#)</a> <span class="p3"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="lopban_3.html#K1" title="LOPBAN_3:func.1">Partial_Sums</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K5" title="NORMSP_1:func.5">-</a> <font color="Maroon" title="c4">y</font></span>)</span></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span> <a href="lopban_3.html#K3" title="LOPBAN_3:func.3">^\</a> 1</span>)</span></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><span class="p3"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="seq_1.html#K14" title="SEQ_1:func.14">(#)</a> <span class="p3"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="lopban_3.html#K1" title="LOPBAN_3:func.1">Partial_Sums</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K5" title="NORMSP_1:func.5">-</a> <font color="Maroon" title="c4">y</font></span>)</span></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="lopban_3.html#E11:75"><i><font color="Green" title="E56">A34</font></i></a>, <a class="txt" href="lopban_3.html#E15:75"><i><font color="Green" title="E58">A36</font></i></a>, <a class="ref" href="seq_2.html#T20" title="SEQ_2:th.20">SEQ_2:20</a></i>
<br/>.= 
0
<b>by </b><i><a class="txt" href="lopban_3.html#E12:75"><i><font color="Green" title="E57">A35</font></i></a>, <a class="txt" href="lopban_3.html#E15:75"><i><font color="Green" title="E58">A36</font></i></a>, <a class="ref" href="normsp_1.html#T5" title="NORMSP_1:th.5">NORMSP_1:5</a></i>
;<br/>
<a NAME="E18:75"/><b>then </b><i><font color="Green" title="E61">A39</font></i>: 
 <a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">x</font>
 
<b>by </b><i><a class="txt" href="lopban_3.html#E7:75"><i><font color="Green" title="E52">A20</font></i></a>, <a class="txt" href="lopban_3.html#E16:75"><i><font color="Green" title="E59">A37</font></i></a>, <a class="txt" href="lopban_3.html#E66"><i><font color="Green" title="E46">Lm6</font></i></a></i>;<br/>
<a NAME="E19:75"/><i><font color="Green" title="E62">A40</font></i>: 
 <a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c4">y</font>
 
<b>by </b><i><a class="txt" href="lopban_3.html#E8:75"><i><font color="Green" title="E53">A26</font></i></a>, <a class="txt" href="lopban_3.html#E16:75"><i><font color="Green" title="E59">A37</font></i></a>, <a class="txt" href="lopban_3.html#E17:75"><i><font color="Green" title="E60">A38</font></i></a>, <a class="txt" href="lopban_3.html#E66"><i><font color="Green" title="E46">Lm6</font></i></a></i>;<br/>
<b>hence </b><a NAME="E20:75"/>
<font color="Maroon" title="c2">x</font> is <a href="lopban_3.html#V4" title="LOPBAN_3:attr.4">invertible</a>
 <b>by </b><i><a class="txt" href="lopban_3.html#E18:75"><i><font color="Green" title="E61">A39</font></i></a>, <a class="ref" href="lopban_3.html#D8" target="_self" title="LOPBAN_3:def.8">Def8</a></i>;<br/>

<b>hence </b><a NAME="E21:75"/>
<font color="Maroon" title="c2">x</font> <a href="lopban_3.html#K9" title="LOPBAN_3:func.9">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lopban_3.html#K2" title="LOPBAN_3:func.2">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">GeoSeq</a> </span>)</span>
 <b>by </b><i><a class="txt" href="lopban_3.html#E18:75"><i><font color="Green" title="E61">A39</font></i></a>, <a class="txt" href="lopban_3.html#E19:75"><i><font color="Green" title="E62">A40</font></i></a>, <a class="ref" href="lopban_3.html#D12" target="_self" title="LOPBAN_3:def.12">Def12</a></i>;<br/>


</div>
