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

<b>let </b><font color="Maroon">X</font> be   <a href="clopban2.html#NM3">Complex_Banach_Algebra</a>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>;<br/>



<b>assume </b><a NAME="E1:71"/><i><font color="Green">E33</font></i>: 
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> 1
 ;<br/>

<b>set </b><font color="Maroon">seq</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> ;<br/>
<a NAME="E2:71"/><i><font color="Green">E34</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a>  is <a href="clopban3.html#V1" target="_self">summable</a>
 
<b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">y</font> =  <a href="clopban3.html#K2" target="_self">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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font> ;<br/>
<a NAME="E4:71"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban3.html#K1" target="_self">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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban3.html#K1" target="_self">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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span></span>)</span> <a href="clopban3.html#K3" target="_self">^\</a> 1</span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> <a href="rlvect_1.html#K6">-</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_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">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban3.html#K1" target="_self">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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban3.html#K1" target="_self">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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span></span>)</span> <a href="clopban3.html#K3" target="_self">^\</a> 1</span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span>;<br/>
<i><font color="Green">E37</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban3.html#K1" target="_self">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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">X</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="txt" href="clopban3.html#E3:71_1"><i><font color="Green">E40</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font>
<b>by </b><i><a class="txt" href="clopban3.html#E2:71_1"><i><font color="Green">E38</font></i></a></i>
;<br/>
<i><font color="Green">E38</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban3.html#K1" target="_self">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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span></span>)</span> <a href="clopban3.html#K3" target="_self">^\</a> 1</span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span> <a href="rlvect_1.html#K6">-</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban3.html#K1" target="_self">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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban3.html#K1" target="_self">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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span> <a href="rlvect_1.html#K4">+</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span>
<b>by </b><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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span> <a href="rlvect_1.html#K4">+</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span>
<b>by </b><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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> 1</span>)</span> <a href="rlvect_1.html#K4">+</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span> <a href="rlvect_1.html#K6">-</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="clopban3.html#E2:71_1"><i><font color="Green">E38</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> 1</span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T28">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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="txt" href="clopban3.html#E3:71_1"><i><font color="Green">E40</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">X</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="txt" href="clopban3.html#E3:71_1"><i><font color="Green">E40</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font>
<b>by </b><i><a class="txt" href="clopban3.html#E2:71_1"><i><font color="Green">E38</font></i></a></i>
;<br/>
<a NAME="E3:71_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i><a class="txt" href="clopban3.html#E5:71"><i><font color="Green">E48</font></i></a>, <a class="ref" href="clopban3.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E4:71_1"/><i><font color="Green">E45</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:71_1_3"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>]
 ;<br/>

<i><font color="Green">E47</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban3.html#K1" target="_self">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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban3.html#K1" target="_self">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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="rlvect_1.html#K4">+</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban3.html#K1" target="_self">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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="clopban3.html#E2:71_1"><i><font color="Green">E38</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban3.html#K1" target="_self">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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span></span>)</span> <a href="clopban3.html#K3" target="_self">^\</a> 1</span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="rlvect_1.html#K6">-</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i>, <a class="txt" href="clopban3.html#E3:71_1"><i><font color="Green">E40</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="clopban3.html#K1" target="_self">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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span></span>)</span> <a href="clopban3.html#K3" target="_self">^\</a> 1</span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban3.html#K1" target="_self">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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban3.html#K1" target="_self">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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span>
<b>by </b><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="clopban3.html#K1" target="_self">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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span></span>)</span> <a href="clopban3.html#K3" target="_self">^\</a> 1</span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="rlvect_1.html#K4">+</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span>)</span>
<b>by </b><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="clopban3.html#K1" target="_self">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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span></span>)</span> <a href="clopban3.html#K3" target="_self">^\</a> 1</span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="rlvect_1.html#K6">-</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
;<br/>
<b>hence </b><a NAME="E4:71_1_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban3.html#K1" target="_self">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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban3.html#K1" target="_self">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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span></span>)</span> <a href="clopban3.html#K3" target="_self">^\</a> 1</span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E5:71_1"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, <a class="ref" href="clopban3.html#D2" target="_self">Def2</a>);<br/></i>
<b>hence </b><a NAME="E6:71_1"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban3.html#K1" target="_self">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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban3.html#K1" target="_self">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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span></span>)</span> <a href="clopban3.html#K3" target="_self">^\</a> 1</span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> <a href="rlvect_1.html#K6">-</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E5:71"/><i><font color="Green">E48</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="group_1.html#K1">*</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> <a href="hidden.html#R1">=</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_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">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="group_1.html#K1">*</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/>
<i><font color="Green">E105</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="group_1.html#K1">*</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">X</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font>
<b>by </b><i/>
;<br/>
<i><font color="Green">E123</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">X</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font>
<b>by </b><i/>
;<br/>
<a NAME="E3:71_2"/><i><font color="Green">E124</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i><a class="ref" href="clopban3.html#T1" target="_self">Th1</a>, </i>;<br/>
<a NAME="E4:71_2"/><i><font color="Green">E125</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:71_2_3"/><i><font color="Green">E126</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>]
 ;<br/>

<b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="group_1.html#K1">*</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="group_1.html#K1">*</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="group_1.html#K1">*</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i>, </i>
;<br/>


</div><b>end;</b></div>
<a NAME="E5:71_2"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="ref" href="clopban3.html#D2" target="_self">Def2</a>, );<br/></i>
<b>hence </b><a NAME="E6:71_2"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="group_1.html#K1">*</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> <a href="hidden.html#R1">=</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E6:71"/><i><font color="Green">E127</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban3.html#K1" target="_self">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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban3.html#K1" target="_self">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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span></span>)</span> <a href="clopban3.html#K3" target="_self">^\</a> 1</span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> <a href="rlvect_1.html#K6">-</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_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">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban3.html#K1" target="_self">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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban3.html#K1" target="_self">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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span></span>)</span> <a href="clopban3.html#K3" target="_self">^\</a> 1</span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span>;<br/>
<i><font color="Green">E128</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban3.html#K1" target="_self">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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="group_1.html#K1">*</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">X</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font>
<b>by </b><i/>
;<br/>
<i><font color="Green">E129</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban3.html#K1" target="_self">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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span></span>)</span> <a href="clopban3.html#K3" target="_self">^\</a> 1</span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span> <a href="rlvect_1.html#K6">-</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban3.html#K1" target="_self">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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban3.html#K1" target="_self">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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span> <a href="rlvect_1.html#K4">+</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span>
<b>by </b><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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span> <a href="rlvect_1.html#K4">+</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span>
<b>by </b><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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> 1</span>)</span> <a href="rlvect_1.html#K4">+</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span> <a href="rlvect_1.html#K6">-</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span>)</span>
<b>by </b><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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> 1</span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T28">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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">X</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font>
<b>by </b><i/>
;<br/>
<a NAME="E3:71_3"/><i><font color="Green">E130</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i>, <a class="ref" href="clopban3.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E4:71_3"/><i><font color="Green">E131</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_3_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:71_3_3"/><i><font color="Green">E132</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>]
 ;<br/>

<i><font color="Green">E133</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban3.html#K1" target="_self">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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban3.html#K1" target="_self">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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="rlvect_1.html#K4">+</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban3.html#K1" target="_self">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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="group_1.html#K1">*</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><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="clopban3.html#K1" target="_self">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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span></span>)</span> <a href="clopban3.html#K3" target="_self">^\</a> 1</span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="rlvect_1.html#K6">-</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="clopban3.html#E5:71"><i><font color="Green">E48</font></i></a>, <a class="ref" href="clopban3.html#D4" target="_self">Def4</a></i>
;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban3.html#K1" target="_self">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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span></span>)</span> <a href="clopban3.html#K3" target="_self">^\</a> 1</span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban3.html#K1" target="_self">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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban3.html#K1" target="_self">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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span>
<b>by </b><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="clopban3.html#K1" target="_self">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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span></span>)</span> <a href="clopban3.html#K3" target="_self">^\</a> 1</span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="rlvect_1.html#K4">+</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span>)</span>
<b>by </b><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="clopban3.html#K1" target="_self">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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span></span>)</span> <a href="clopban3.html#K3" target="_self">^\</a> 1</span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="rlvect_1.html#K6">-</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
;<br/>
<b>hence </b><a NAME="E4:71_3_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban3.html#K1" target="_self">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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban3.html#K1" target="_self">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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span></span>)</span> <a href="clopban3.html#K3" target="_self">^\</a> 1</span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span>
 <b>by </b><i><a class="ref" href="clopban3.html#T2" target="_self">Th2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:71_3"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<b>hence </b><a NAME="E6:71_3"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban3.html#K1" target="_self">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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban3.html#K1" target="_self">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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span></span>)</span> <a href="clopban3.html#K3" target="_self">^\</a> 1</span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> <a href="rlvect_1.html#K6">-</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span>
 ;<br/>


</div><b>end;</b></div>
<div><i><font color="Green">E134</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="71_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>reconsider </b><font color="Maroon">yn</font> = <span class="p1">(<span class="default"><a href="clopban3.html#K1" target="_self">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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font> ;<br/><i><font color="Green">E136</font></i>: <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T26">RLVECT_1:26</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">yn</font> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">yn</font> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T28">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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">yn</font> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">yn</font> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T43">RLVECT_1:43</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">yn</font> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">yn</font> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">yn</font> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">yn</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">y</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i/>
;<br/><i><font color="Green">E137</font></i>: <font color="Maroon">yn</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">yn</font> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">yn</font> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">yn</font> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">yn</font> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban3.html#K1" target="_self">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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K6">-</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="clopban3.html#K1" target="_self">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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span></span>)</span> <a href="clopban3.html#K3" target="_self">^\</a> 1</span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K6">-</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban3.html#K1" target="_self">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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban3.html#K1" target="_self">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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span></span>)</span> <a href="clopban3.html#K3" target="_self">^\</a> 1</span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span>
<b>by </b><i/>
;<br/><a NAME="E4:71_4"/>
<span class="p1">(<span class="default"><a href="clopban3.html#K1" target="_self">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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban3.html#K1" target="_self">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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K4">+</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E5:71_4"/><b>then </b>
<span class="p1">(<span class="default"><a href="clopban3.html#K1" target="_self">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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban3.html#K1" target="_self">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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K4">+</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="clopban3.html#K3" target="_self">^\</a> 1</span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i/>;<br/><b>then </b><i><font color="Green">E138</font></i>: <font color="Maroon">yn</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">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="clopban3.html#K1" target="_self">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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban3.html#K1" target="_self">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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K4">+</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="clopban3.html#K3" target="_self">^\</a> 1</span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span>
<b>by </b><i>, </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="clopban3.html#K1" target="_self">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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban3.html#K1" target="_self">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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="clopban3.html#K3" target="_self">^\</a> 1</span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="clopban3.html#K3" target="_self">^\</a> 1</span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T28">RLVECT_1:28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="clopban3.html#K3" target="_self">^\</a> 1</span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K6">-</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T43">RLVECT_1:43</a></i>
<br/>.= 
 <a href="rlvect_1.html#K5">-</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="clopban3.html#K3" target="_self">^\</a> 1</span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K6">-</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T27">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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span> <a href="rlvect_1.html#K6">-</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="clopban3.html#K3" target="_self">^\</a> 1</span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T47">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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span> <a href="rlvect_1.html#K6">-</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i/>
;<br/><i><font color="Green">E139</font></i>: <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">yn</font> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i>, </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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="clopban3.html#K3" target="_self">^\</a> 1</span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font>
<b>by </b><i/>
;<br/><a NAME="E8:71_4"/><i><font color="Green">E140</font></i>: 
<span class="p1"><a href="clvect_1.html#K5">||.</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="clopban3.html#K3" target="_self">^\</a> 1</span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">yn</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">y</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="clvect_1.html#K5">||.</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="clopban3.html#K3" target="_self">^\</a> 1</span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">yn</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">y</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span>
 <b>by </b><i><a class="ref" href="clvect_1.html#D10">CLVECT_1:def 10</a></i>;<br/><a NAME="E9:71_4"/><i><font color="Green">E141</font></i>: 
<span class="p1"><a href="clvect_1.html#K5">||.</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="clopban3.html#K3" target="_self">^\</a> 1</span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">yn</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">y</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="clvect_1.html#K10">||.</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="clopban3.html#K3" target="_self">^\</a> 1</span>)</span></span><a href="clvect_1.html#K10">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">yn</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">y</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span>
 <b>by </b><i><a class="ref" href="clvect_1.html#D16">CLVECT_1:def 16</a></i>;<br/><a NAME="E10:71_4"/>
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">yn</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">y</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">yn</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">y</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">x</font></span><a href="clvect_1.html#K5">.||</a></span>
 <b>by </b><i/>;<br/><a NAME="E11:71_4"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2"><a href="clvect_1.html#K10">||.</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="clopban3.html#K3" target="_self">^\</a> 1</span>)</span></span><a href="clvect_1.html#K10">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">yn</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">y</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="clvect_1.html#K10">||.</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="clopban3.html#K3" target="_self">^\</a> 1</span>)</span></span><a href="clvect_1.html#K10">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">yn</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">y</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p2"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">x</font></span><a href="clvect_1.html#K5">.||</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E12:71_4"/><b>then </b>
<span class="p1"><a href="clvect_1.html#K5">||.</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="clopban3.html#K3" target="_self">^\</a> 1</span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">yn</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">y</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="clvect_1.html#K10">||.</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="clopban3.html#K3" target="_self">^\</a> 1</span>)</span></span><a href="clvect_1.html#K10">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">yn</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">y</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p2"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">x</font></span><a href="clvect_1.html#K5">.||</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="clopban3.html#T2" target="_self">Th2</a>, <a class="ref" href="clopban3.html#T3" target="_self">Th3</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E13:71_4"/><b>then </b>
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="clvect_1.html#K10">||.</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="clopban3.html#K3" target="_self">^\</a> 1</span>)</span></span><a href="clvect_1.html#K10">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="clopban3.html#K1" target="_self">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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span></span>)</span> <a href="clvect_1.html#K8">-</a> <font color="Maroon">y</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p2"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">x</font></span><a href="clvect_1.html#K5">.||</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="clopban3.html#D2" target="_self">Def2</a>, <a class="ref" href="clopban3.html#D4" target="_self">Def4</a>, <a class="ref" href="clvect_1.html#D13">CLVECT_1:def 13</a></i>;<br/><a NAME="E14:71_4"/><b>then </b>
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="clvect_1.html#K10">||.</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="clopban3.html#K3" target="_self">^\</a> 1</span>)</span></span><a href="clvect_1.html#K10">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="clvect_1.html#K10">||.</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="clopban3.html#K1" target="_self">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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span></span>)</span> <a href="clvect_1.html#K8">-</a> <font color="Maroon">y</font></span>)</span></span><a href="clvect_1.html#K10">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">x</font></span><a href="clvect_1.html#K5">.||</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="clvect_1.html#D16">CLVECT_1:def 16</a></i>;<br/><a NAME="E15:71_4"/><b>then </b>
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="clvect_1.html#K10">||.</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="clopban3.html#K3" target="_self">^\</a> 1</span>)</span></span><a href="clvect_1.html#K10">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">x</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="seq_1.html#K14">(#)</a> <span class="p3"><a href="clvect_1.html#K10">||.</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="clopban3.html#K1" target="_self">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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span></span>)</span> <a href="clvect_1.html#K8">-</a> <font color="Maroon">y</font></span>)</span></span><a href="clvect_1.html#K10">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="seq_1.html#T13">SEQ_1:13</a></i>;<br/><b>hence </b><a NAME="E16:71_4"/>
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="clvect_1.html#K10">||.</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="clopban3.html#K3" target="_self">^\</a> 1</span>)</span></span><a href="clvect_1.html#K10">.||</a></span> <a href="seq_1.html#K9">+</a> <span class="p2">(<span class="default"><span class="p3"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">x</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="seq_1.html#K14">(#)</a> <span class="p3"><a href="clvect_1.html#K10">||.</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="clopban3.html#K1" target="_self">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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span></span>)</span> <a href="clvect_1.html#K8">-</a> <font color="Maroon">y</font></span>)</span></span><a href="clvect_1.html#K10">.||</a></span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="seq_1.html#T11">SEQ_1:11</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E142</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="71_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>reconsider </b><font color="Maroon">yn</font> = <span class="p1">(<span class="default"><a href="clopban3.html#K1" target="_self">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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font> ;<br/><i><font color="Green">E143</font></i>: <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T26">RLVECT_1:26</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">yn</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">yn</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T28">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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">yn</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">yn</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T43">RLVECT_1:43</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">yn</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">yn</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">yn</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">yn</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">y</font></span>)</span></span>)</span>
<b>by </b><i/>
;<br/><i><font color="Green">E144</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">yn</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">X</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">yn</font></span>)</span> <a href="rlvect_1.html#K6">-</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">yn</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">yn</font> <a href="rlvect_1.html#K6">-</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">yn</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban3.html#K1" target="_self">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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K6">-</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="clopban3.html#K1" target="_self">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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span></span>)</span> <a href="clopban3.html#K3" target="_self">^\</a> 1</span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K6">-</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="clopban3.html#T1" target="_self">Th1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban3.html#K1" target="_self">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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban3.html#K1" target="_self">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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span></span>)</span> <a href="clopban3.html#K3" target="_self">^\</a> 1</span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span>
<b>by </b><i/>
;<br/><a NAME="E4:71_5"/>
<span class="p1">(<span class="default"><a href="clopban3.html#K1" target="_self">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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban3.html#K1" target="_self">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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K4">+</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E5:71_5"/><b>then </b>
<span class="p1">(<span class="default"><a href="clopban3.html#K1" target="_self">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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban3.html#K1" target="_self">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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K4">+</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="clopban3.html#K3" target="_self">^\</a> 1</span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i/>;<br/><b>then </b><i><font color="Green">E145</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">yn</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban3.html#K1" target="_self">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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban3.html#K1" target="_self">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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K4">+</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="clopban3.html#K3" target="_self">^\</a> 1</span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span>
<b>by </b><i>, </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="clopban3.html#K1" target="_self">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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban3.html#K1" target="_self">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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="clopban3.html#K3" target="_self">^\</a> 1</span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="clopban3.html#K3" target="_self">^\</a> 1</span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T28">RLVECT_1:28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="clopban3.html#K3" target="_self">^\</a> 1</span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K6">-</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T43">RLVECT_1:43</a></i>
<br/>.= 
 <a href="rlvect_1.html#K5">-</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="clopban3.html#K3" target="_self">^\</a> 1</span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K6">-</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T27">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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span> <a href="rlvect_1.html#K6">-</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="clopban3.html#K3" target="_self">^\</a> 1</span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T47">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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span> <a href="rlvect_1.html#K6">-</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i/>
;<br/><i><font color="Green">E146</font></i>: <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">yn</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="clopban3.html#D4" target="_self">Def4</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="clopban3.html#K3" target="_self">^\</a> 1</span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font>
<b>by </b><i/>
;<br/><a NAME="E8:71_5"/><i><font color="Green">E147</font></i>: 
<span class="p1"><a href="clvect_1.html#K5">||.</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="clopban3.html#K3" target="_self">^\</a> 1</span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">yn</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="clvect_1.html#K5">||.</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="clopban3.html#K3" target="_self">^\</a> 1</span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">yn</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">y</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span>
 <b>by </b><i><a class="ref" href="clvect_1.html#D10">CLVECT_1:def 10</a></i>;<br/><a NAME="E9:71_5"/><i><font color="Green">E148</font></i>: 
<span class="p1"><a href="clvect_1.html#K5">||.</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="clopban3.html#K3" target="_self">^\</a> 1</span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">yn</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">y</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="clvect_1.html#K10">||.</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="clopban3.html#K3" target="_self">^\</a> 1</span>)</span></span><a href="clvect_1.html#K10">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">yn</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">y</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span>
 <b>by </b><i><a class="ref" href="clvect_1.html#D16">CLVECT_1:def 16</a></i>;<br/><a NAME="E10:71_5"/>
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">yn</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">y</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">x</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">yn</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">y</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span>
 <b>by </b><i/>;<br/><a NAME="E11:71_5"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2"><a href="clvect_1.html#K10">||.</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="clopban3.html#K3" target="_self">^\</a> 1</span>)</span></span><a href="clvect_1.html#K10">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">yn</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">y</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="clvect_1.html#K10">||.</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="clopban3.html#K3" target="_self">^\</a> 1</span>)</span></span><a href="clvect_1.html#K10">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">x</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p2"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">yn</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">y</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E12:71_5"/><b>then </b>
<span class="p1"><a href="clvect_1.html#K5">||.</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="clopban3.html#K3" target="_self">^\</a> 1</span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">yn</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="clvect_1.html#K10">||.</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="clopban3.html#K3" target="_self">^\</a> 1</span>)</span></span><a href="clvect_1.html#K10">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">x</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p2"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">yn</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">y</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="clopban3.html#T3" target="_self">Th3</a>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E13:71_5"/><b>then </b>
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="clvect_1.html#K10">||.</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="clopban3.html#K3" target="_self">^\</a> 1</span>)</span></span><a href="clvect_1.html#K10">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">x</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p2"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="clopban3.html#K1" target="_self">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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span></span>)</span> <a href="clvect_1.html#K8">-</a> <font color="Maroon">y</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span></span>)</span>
 <b>by </b><i>, <a class="ref" href="clopban3.html#T2" target="_self">Th2</a>, <a class="ref" href="clvect_1.html#D13">CLVECT_1:def 13</a></i>;<br/><a NAME="E14:71_5"/><b>then </b>
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="clvect_1.html#K10">||.</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="clopban3.html#K3" target="_self">^\</a> 1</span>)</span></span><a href="clvect_1.html#K10">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">x</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="clvect_1.html#K10">||.</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="clopban3.html#K1" target="_self">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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span></span>)</span> <a href="clvect_1.html#K8">-</a> <font color="Maroon">y</font></span>)</span></span><a href="clvect_1.html#K10">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="clvect_1.html#D16">CLVECT_1:def 16</a></i>;<br/><a NAME="E15:71_5"/><b>then </b>
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="clvect_1.html#K10">||.</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="clopban3.html#K3" target="_self">^\</a> 1</span>)</span></span><a href="clvect_1.html#K10">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">x</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="seq_1.html#K14">(#)</a> <span class="p3"><a href="clvect_1.html#K10">||.</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="clopban3.html#K1" target="_self">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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span></span>)</span> <a href="clvect_1.html#K8">-</a> <font color="Maroon">y</font></span>)</span></span><a href="clvect_1.html#K10">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="seq_1.html#T13">SEQ_1:13</a></i>;<br/><b>hence </b><a NAME="E16:71_5"/>
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="clvect_1.html#K10">||.</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="clopban3.html#K3" target="_self">^\</a> 1</span>)</span></span><a href="clvect_1.html#K10">.||</a></span> <a href="seq_1.html#K9">+</a> <span class="p2">(<span class="default"><span class="p3"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">x</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="seq_1.html#K14">(#)</a> <span class="p3"><a href="clvect_1.html#K10">||.</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="clopban3.html#K1" target="_self">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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span></span>)</span> <a href="clvect_1.html#K8">-</a> <font color="Maroon">y</font></span>)</span></span><a href="clvect_1.html#K10">.||</a></span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="seq_1.html#T11">SEQ_1:11</a></i>;<br/></div><b>end;</b></div>
<a NAME="E9:71"/><i><font color="Green">E149</font></i>: 
 <a href="clopban3.html#K1" target="_self">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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> is <a href="clvect_1.html#V6">convergent</a>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E10:71"/><i><font color="Green">E150</font></i>: 
( <span class="p1"><a href="clvect_1.html#K10">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban3.html#K1" target="_self">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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span></span>)</span> <a href="clvect_1.html#K8">-</a> <font color="Maroon">y</font></span>)</span></span><a href="clvect_1.html#K10">.||</a></span> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <span class="p1"><a href="clvect_1.html#K10">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban3.html#K1" target="_self">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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span></span>)</span> <a href="clvect_1.html#K8">-</a> <font color="Maroon">y</font></span>)</span></span><a href="clvect_1.html#K10">.||</a></span> <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i>, <a class="ref" href="clvect_1.html#T120">CLVECT_1:120</a></i>;<br/>
<a NAME="E11:71"/><i><font color="Green">E151</font></i>: 
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">x</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="seq_1.html#K14">(#)</a> <span class="p1"><a href="clvect_1.html#K10">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban3.html#K1" target="_self">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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span></span>)</span> <a href="clvect_1.html#K8">-</a> <font color="Maroon">y</font></span>)</span></span><a href="clvect_1.html#K10">.||</a></span> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i>, <a class="ref" href="seq_2.html#T21">SEQ_2:21</a></i>;<br/>
<i><font color="Green">E152</font></i>:  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><span class="p2"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">x</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="seq_1.html#K14">(#)</a> <span class="p2"><a href="clvect_1.html#K10">||.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban3.html#K1" target="_self">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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span></span>)</span> <a href="clvect_1.html#K8">-</a> <font color="Maroon">y</font></span>)</span></span><a href="clvect_1.html#K10">.||</a></span></span>)</span> = 
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">x</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="real_1.html#K4">*</a> 0
<b>by </b><i>, <a class="ref" href="seq_2.html#T22">SEQ_2:22</a></i>
<br/>.= 
0

;<br/>
<a NAME="E13:71"/><i><font color="Green">E153</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a>  is <a href="clvect_1.html#V6">convergent</a> &amp;  <a href="clvect_1.html#K11">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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E14:71"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="clopban3.html#K3" target="_self">^\</a> 1 is <a href="clvect_1.html#V6">convergent</a> &amp;  <a href="clvect_1.html#K11">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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="clopban3.html#K3" target="_self">^\</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font> )
 
<b>by </b><i><a class="ref" href="clopban3.html#T3" target="_self">Th3</a>, </i>;<br/>
<a NAME="E15:71"/><b>then </b><i><font color="Green">E154</font></i>: 
( <span class="p1"><a href="clvect_1.html#K10">||.</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="clopban3.html#K3" target="_self">^\</a> 1</span>)</span></span><a href="clvect_1.html#K10">.||</a></span> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <span class="p1"><a href="clvect_1.html#K10">||.</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="clopban3.html#K3" target="_self">^\</a> 1</span>)</span></span><a href="clvect_1.html#K10">.||</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> )
 
<b>by </b><i><a class="ref" href="clopban1.html#T45">CLOPBAN1:45</a></i>;<br/>
<a NAME="E16:71"/><b>then </b><i><font color="Green">E155</font></i>: 
<span class="p1"><a href="clvect_1.html#K10">||.</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="clopban3.html#K3" target="_self">^\</a> 1</span>)</span></span><a href="clvect_1.html#K10">.||</a></span> <a href="seq_1.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">x</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="seq_1.html#K14">(#)</a> <span class="p2"><a href="clvect_1.html#K10">||.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban3.html#K1" target="_self">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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span></span>)</span> <a href="clvect_1.html#K8">-</a> <font color="Maroon">y</font></span>)</span></span><a href="clvect_1.html#K10">.||</a></span></span>)</span> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i><a class="ref" href="clopban3.html#D4" target="_self">Def4</a>, <a class="ref" href="seq_2.html#T19">SEQ_2:19</a></i>;<br/>
<i><font color="Green">E156</font></i>:  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><span class="p2"><a href="clvect_1.html#K10">||.</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="clopban3.html#K3" target="_self">^\</a> 1</span>)</span></span><a href="clvect_1.html#K10">.||</a></span> <a href="seq_1.html#K9">+</a> <span class="p2">(<span class="default"><span class="p3"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">x</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="seq_1.html#K14">(#)</a> <span class="p3"><a href="clvect_1.html#K10">||.</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="clopban3.html#K1" target="_self">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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span></span>)</span> <a href="clvect_1.html#K8">-</a> <font color="Maroon">y</font></span>)</span></span><a href="clvect_1.html#K10">.||</a></span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p2"><a href="clvect_1.html#K10">||.</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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span> <a href="clopban3.html#K3" target="_self">^\</a> 1</span>)</span></span><a href="clvect_1.html#K10">.||</a></span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p2">(<span class="default"><span class="p3"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">x</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="seq_1.html#K14">(#)</a> <span class="p3"><a href="clvect_1.html#K10">||.</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="clopban3.html#K1" target="_self">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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span></span>)</span> <a href="clvect_1.html#K8">-</a> <font color="Maroon">y</font></span>)</span></span><a href="clvect_1.html#K10">.||</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="clopban3.html#D4" target="_self">Def4</a>, , <a class="ref" href="seq_2.html#T20">SEQ_2:20</a></i>
<br/>.= 
0
<b>by </b><i><a class="ref" href="clopban3.html#T2" target="_self">Th2</a>, , <a class="ref" href="clvect_1.html#T103">CLVECT_1:103</a></i>
;<br/>
<a NAME="E18:71"/><i><font color="Green">E157</font></i>: 
 <a href="group_1.html#K2">1.</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font>
 
<b>by </b><i>, , , </i>;<br/>
<a NAME="E19:71"/><i><font color="Green">E158</font></i>: 
 <a href="group_1.html#K2">1.</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y</font>
 
<b>by </b><i><a class="ref" href="clopban3.html#D2" target="_self">Def2</a>, , , </i>;<br/>
<b>hence </b><a NAME="E20:71"/>
<font color="Maroon">x</font> is <a href="lopban_3.html#V4">invertible</a>
 <b>by </b><i><a class="ref" href="clopban3.html#T4" target="_self">Th4</a>, <a class="ref" href="lopban_3.html#D8">LOPBAN_3:def 8</a></i>;<br/>

<b>hence </b><a NAME="E21:71"/>
<font color="Maroon">x</font> <a href="clopban3.html#K7" target="_self">"</a>  <a href="hidden.html#R1">=</a>  <a href="clopban3.html#K2" target="_self">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">1.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="clopban3.html#K8" target="_self">GeoSeq</a> </span>)</span>
 <b>by </b><i><a class="ref" href="clopban3.html#T4" target="_self">Th4</a>, , </i>;<br/>


</div>
