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

<b>let </b><font color="Maroon">X</font> be   <a href="normsp_1.html#NM1">RealNormSpace</a>, <font color="Maroon">Y</font> be   <a href="normsp_1.html#NM1">RealNormSpace</a>;<br/>

<b>assume </b><a NAME="E1:84"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">Y</font> is <a href="lopban_1.html#V4" target="_self">complete</a>
 ;<br/>

<b>let </b><font color="Maroon">vseq</font> be   <a href="normsp_1.html#NM2">sequence</a> of <span class="p1">(<span class="default"><a href="lopban_1.html#K16" target="_self">R_NormSpace_of_BoundedLinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span>;<br/>

<b>assume </b><a NAME="E2:84"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">vseq</font> is <a href="rsspace3.html#V1">CCauchy</a>
 ;<br/>

<a NAME="E3:84"/><i><font color="Green">E52</font></i>: 
<span class="p1"><a href="normsp_1.html#K7">||.</a><span class="default"><font color="Maroon">vseq</font></span><a href="normsp_1.html#K7">.||</a></span> is <a href="seq_2.html#V4">convergent</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">e1</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>assume </b><a NAME="E1:84_1_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">e1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><b>reconsider </b><font color="Maroon">e</font> = <font color="Maroon">e1</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/><b>consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:84_1_1"/><i><font color="Green">E54</font></i>: 
for <font color="Olive">n</font>, <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k</font> &amp; <font color="Olive">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k</font> holds <br/><span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">m</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font>
 <b>by </b><i>, <a class="ref" href="lopban_1.html#T6" target="_self">Th6</a>, <a class="ref" href="rsspace3.html#T10">RSSPACE3:10</a></i>;<br/><div><i><font color="Green">E86</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="84_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:84_1_1_1"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k</font>
 ;<br/><a NAME="E2:84_1_1_1"/><i><font color="Green">E90</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K5">-</a> <span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
 <b>by </b><i><a class="ref" href="normsp_1.html#T13">NORMSP_1:13</a></i>;<br/><a NAME="E3:84_1_1_1"/><i><font color="Green">E91</font></i>: 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="normsp_1.html#K7">||.</a><span class="default"><font color="Maroon">vseq</font></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="normsp_1.html#D10">NORMSP_1:def 10</a></i>;<br/><a NAME="E4:84_1_1_1"/><i><font color="Green">E92</font></i>: 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="normsp_1.html#K7">||.</a><span class="default"><font color="Maroon">vseq</font></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="normsp_1.html#D10">NORMSP_1:def 10</a></i>;<br/><a NAME="E5:84_1_1_1"/>
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="ref" href="lopban_1.html#T7" target="_self">Th7</a>, <a class="ref" href="lopban_1.html#T9" target="_self">Th9</a></i>;<br/><b>hence </b><a NAME="E6:84_1_1_1"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="normsp_1.html#K7">||.</a><span class="default"><font color="Maroon">vseq</font></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3"><a href="normsp_1.html#K7">||.</a><span class="default"><font color="Maroon">vseq</font></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e1</font>
 <b>by </b><i>, <a class="ref" href="lopban_1.html#T10" target="_self">Th10</a>, <a class="ref" href="lopban_1.html#T11" target="_self">Th11</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div><b>take </b><font color="Maroon">k</font> = <font color="Maroon">k</font>;<br/><b>thus </b><a NAME="E6:84_1_1"/>
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k</font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="normsp_1.html#K7">||.</a><span class="default"><font color="Maroon">vseq</font></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Olive">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3"><a href="normsp_1.html#K7">||.</a><span class="default"><font color="Maroon">vseq</font></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e1</font>
 <b>by </b><i><a class="ref" href="lopban_1.html#T8" target="_self">Th8</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:84_1"/>
<span class="p1"><a href="normsp_1.html#K7">||.</a><span class="default"><font color="Maroon">vseq</font></span><a href="normsp_1.html#K7">.||</a></span> is <a href="seq_2.html#V4">convergent</a>
 <b>by </b><i><a class="ref" href="seq_4.html#T58">SEQ_4:58</a></i>;<br/>


</div><b>end;</b></div>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">xseq</font> being  <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">Y</font> st <br/>( ( 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="Olive">xseq</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="lopban_1.html#K13" target="_self">modetrans</a> <span class="p2">(<span class="default"><font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font></span>)</span>,<font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> ) &amp; <font color="Olive">xseq</font> is <a href="normsp_1.html#V4">convergent</a> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="normsp_1.html#K8">lim</a> <font color="Olive">xseq</font> );<br/>
<a NAME="E4:84"/><i><font color="Green">E93</font></i>: 
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>  ex <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Y</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>;<br/>

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font> = <span class="p1">(<span class="default"><a href="lopban_1.html#K13" target="_self">modetrans</a> <span class="p2">(<span class="default"><font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>,<font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>;<br/>
<b>consider </b><font color="Maroon">xseq</font> being   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E2:84_2"/><i><font color="Green">E94</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  <font color="Maroon">xseq</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">n</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S4">FUNCT_2:sch 4</a>();<br/></i>
<a NAME="E3:84_2"/><i><font color="Green">E95</font></i>: 
for <font color="Olive">m</font>, <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">xseq</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">m</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">xseq</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">k</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">m</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">k</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">x</font></span><a href="normsp_1.html#K1">.||</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2_1"><b>proof </b></a><div class="add">

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

<a NAME="E1:84_2_1"/>
<font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font> is  <a href="lopban_1.html#V3" target="_self">bounded</a> <a href="lopban_1.html#NM1">LinearOperator</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font>
 
<b>by </b><i/>;<br/>
<a NAME="E2:84_2_1"/><b>then </b><i><font color="Green">E131</font></i>: 
 <a href="lopban_1.html#K13" target="_self">modetrans</a> <span class="p1">(<span class="default"><font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span>,<font color="Maroon">X</font>,<font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="txt" href="lopban_1.html#E2:84"><i><font color="Green">E42</font></i></a></i>;<br/>
<a NAME="E3:84_2_1"/>
<font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font> is  <a href="lopban_1.html#V3" target="_self">bounded</a> <a href="lopban_1.html#NM1">LinearOperator</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font>
 
<b>by </b><i/>;<br/>
<a NAME="E4:84_2_1"/><b>then </b><i><font color="Green">E132</font></i>: 
 <a href="lopban_1.html#K13" target="_self">modetrans</a> <span class="p1">(<span class="default"><font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span>,<font color="Maroon">X</font>,<font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font>
 
<b>by </b><i><a class="txt" href="lopban_1.html#E2:84"><i><font color="Green">E42</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">h1</font> = <span class="p1">(<span class="default"><font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> as  <a href="lopban_1.html#V3" target="_self">bounded</a> <a href="lopban_1.html#NM1">LinearOperator</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font> <b>by </b><i/>;<br/>
<a NAME="E6:84_2_1"/><i><font color="Green">E148</font></i>: 
<font color="Maroon">xseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="lopban_1.html#K13" target="_self">modetrans</a> <span class="p2">(<span class="default"><font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span>,<font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="lopban_1.html#T7" target="_self">Th7</a></i>;<br/>
<a NAME="E7:84_2_1"/>
<font color="Maroon">xseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="lopban_1.html#K13" target="_self">modetrans</a> <span class="p2">(<span class="default"><font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span>,<font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="lopban_1.html#T7" target="_self">Th7</a></i>;<br/>
<a NAME="E8:84_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">xseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">xseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="lopban_1.html#T9" target="_self">Th9</a>, , <a class="ref" href="lopban_1.html#T10" target="_self">Th10</a>, </i>;<br/>
<b>hence </b><a NAME="E9:84_2_1"/>
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">xseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">xseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">x</font></span><a href="normsp_1.html#K1">.||</a></span>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E4:84_2"/><i><font color="Green">E149</font></i>: 
<font color="Maroon">xseq</font> is <a href="normsp_1.html#V4">convergent</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">e</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>assume </b><a NAME="E1:84_2_2_1"/><i><font color="Green">E150</font></i>: 
<font color="Maroon">e</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><b>thus </b><a NAME="E2:84_2_2_1"/>
 ex <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>for <font color="Olive">n</font>, <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">k</font> holds <br/><span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">xseq</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">xseq</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">m</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2_2_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2_2_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_2_2_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:84_2_2_1_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font> or <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_2_2_1_1_1_1_1"><b>case </b></a><a NAME="E1:84_2_2_1_1_1_1_1"/><i><font color="Green">E151</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font>
 ;<br/><div class="add"><b>take </b><font color="Maroon">k</font> = 0;<br/><b>thus </b><a NAME="E2:84_2_2_1_1_1_1_1"/>
for <font color="Olive">n</font>, <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k</font> &amp; <font color="Olive">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k</font> holds <br/><span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">xseq</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">xseq</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">m</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2_2_1_1_1_1_1_1"><b>proof </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> , <font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:84_2_2_1_1_1_1_1_1"/>
( <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k</font> )
 ;<br/>

<i><font color="Green">E152</font></i>: <font color="Maroon">xseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><a href="lopban_1.html#K13" target="_self">modetrans</a> <span class="p2">(<span class="default"><font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="lopban_1.html#T7" target="_self">Th7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lopban_1.html#K13" target="_self">modetrans</a> <span class="p2">(<span class="default"><font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">0 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="lopban_1.html#T10" target="_self">Th10</a>, <a class="ref" href="rlvect_1.html#T23">RLVECT_1:23</a></i>
<br/>.= 
0 <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_1.html#K13" target="_self">modetrans</a> <span class="p3">(<span class="default"><font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">Y</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T23">RLVECT_1:23</a></i>
;<br/>
<font color="Maroon">xseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font> = 
<span class="p1">(<span class="default"><a href="lopban_1.html#K13" target="_self">modetrans</a> <span class="p2">(<span class="default"><font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span>,<font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="lopban_1.html#T7" target="_self">Th7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lopban_1.html#K13" target="_self">modetrans</a> <span class="p2">(<span class="default"><font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span>,<font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">0 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="lopban_1.html#T10" target="_self">Th10</a>, <a class="ref" href="rlvect_1.html#T23">RLVECT_1:23</a></i>
<br/>.= 
0 <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_1.html#K13" target="_self">modetrans</a> <span class="p3">(<span class="default"><font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span>,<font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">Y</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T23">RLVECT_1:23</a></i>
;<br/>
<b>then </b><span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">xseq</font> <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"><font color="Maroon">xseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> = 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">Y</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
<b>by </b><i><a class="ref" href="lopban_1.html#T11" target="_self">Th11</a>, <a class="ref" href="rlvect_1.html#T26">RLVECT_1:26</a></i>
<br/>.= 
0
<b>by </b><i><a class="ref" href="normsp_1.html#D2">NORMSP_1:def 2</a></i>
;<br/>
<b>hence </b><a NAME="E5:84_2_2_1_1_1_1_1_1"/>
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">xseq</font> <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"><font color="Maroon">xseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_2_2_1_1_1_1_2"><b>case </b></a><a NAME="E1:84_2_2_1_1_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font>
 ;<br/><div class="add"><a NAME="E2:84_2_2_1_1_1_1_2"/><b>then </b><i><font color="Green">E153</font></i>: 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">x</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="ref" href="normsp_1.html#D2">NORMSP_1:def 2</a></i>;<br/><a NAME="E3:84_2_2_1_1_1_1_2"/><b>then </b><i><font color="Green">E154</font></i>: 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">x</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="normsp_1.html#T8">NORMSP_1:8</a></i>;<br/><a NAME="E4:84_2_2_1_1_1_1_2"/><b>then </b>
<font color="Maroon">e</font> <a href="real_1.html#K6">/</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">x</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T141">XREAL_1:141</a></i>;<br/><b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E6:84_2_2_1_1_1_1_2"/><i><font color="Green">E155</font></i>: 
for <font color="Olive">n</font>, <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k</font> &amp; <font color="Olive">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k</font> holds <br/><span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">m</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font> <a href="real_1.html#K6">/</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">x</font></span><a href="normsp_1.html#K1">.||</a></span>
 <b>by </b><i>, <a class="ref" href="rsspace3.html#T10">RSSPACE3:10</a></i>;<br/><b>take </b><font color="Maroon">k</font> = <font color="Maroon">k</font>;<br/><b>thus </b><a NAME="E7:84_2_2_1_1_1_1_2"/>
for <font color="Olive">n</font>, <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k</font> &amp; <font color="Olive">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k</font> holds <br/><span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">xseq</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">xseq</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">m</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2_2_1_1_1_1_2_1"><b>proof </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> , <font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:84_2_2_1_1_1_1_2_1"/><i><font color="Green">E156</font></i>: 
( <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k</font> )
 ;<br/>

<a NAME="E2:84_2_2_1_1_1_1_2_1"/><i><font color="Green">E157</font></i>: 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">xseq</font> <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"><font color="Maroon">xseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">vseq</font> <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"><font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">x</font></span><a href="normsp_1.html#K1">.||</a></span>
 
<b>by </b><i><a class="ref" href="lopban_1.html#T8" target="_self">Th8</a></i>;<br/>
<a NAME="E3:84_2_2_1_1_1_1_2_1"/>
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">vseq</font> <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"><font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font> <a href="real_1.html#K6">/</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">x</font></span><a href="normsp_1.html#K1">.||</a></span>
 
<b>by </b><i><a class="ref" href="lopban_1.html#T12" target="_self">Th12</a>, <a class="txt" href="lopban_1.html#E17"><i><font color="Green">Lemma69</font></i></a></i>;<br/>
<a NAME="E4:84_2_2_1_1_1_1_2_1"/><b>then </b><i><font color="Green">E158</font></i>: 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">vseq</font> <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"><font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">x</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K6">/</a> <span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">x</font></span><a href="normsp_1.html#K1">.||</a></span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">x</font></span><a href="normsp_1.html#K1">.||</a></span>
 
<b>by </b><i><a class="ref" href="lopban_1.html#T11" target="_self">Th11</a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K6">/</a> <span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">x</font></span><a href="normsp_1.html#K1">.||</a></span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">x</font></span><a href="normsp_1.html#K1">.||</a></span> = 
<span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">x</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K2">"</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">x</font></span><a href="normsp_1.html#K1">.||</a></span>
<b>by </b><i><a class="ref" href="xcmplx_0.html#D9">XCMPLX_0:def 9</a></i>
<br/>.= 
<font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">x</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">x</font></span><a href="normsp_1.html#K1">.||</a></span></span>)</span>

<br/>.= 
<font color="Maroon">e</font> <a href="real_1.html#K4">*</a> 1
<b>by </b><i><a class="ref" href="lopban_1.html#T10" target="_self">Th10</a>, <a class="ref" href="xcmplx_0.html#D7">XCMPLX_0:def 7</a></i>
<br/>.= 
<font color="Maroon">e</font>

;<br/>
<b>hence </b><a NAME="E6:84_2_2_1_1_1_1_2_1"/>
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">xseq</font> <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"><font color="Maroon">xseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="ref" href="lopban_1.html#T13" target="_self">Th13</a>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:84_2_2_1_1"/>
 ex <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>for <font color="Olive">n</font>, <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">k</font> holds <br/><span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">xseq</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">xseq</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">m</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font>
 ;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E2:84_2_2"/><b>then </b>
<font color="Maroon">xseq</font> is <a href="rsspace3.html#V1">CCauchy</a>
 
<b>by </b><i><a class="ref" href="rsspace3.html#T10">RSSPACE3:10</a></i>;<br/>
<b>hence </b><a NAME="E3:84_2_2"/>
<font color="Maroon">xseq</font> is <a href="normsp_1.html#V4">convergent</a>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>take </b><font color="Maroon">y</font> =  <a href="normsp_1.html#K8">lim</a> <font color="Maroon">xseq</font>;<br/>

<b>thus </b><a NAME="E5:84_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">y</font>]
 <b>by </b><i><a class="ref" href="lopban_1.html#T7" target="_self">Th7</a>, <a class="ref" href="lopban_1.html#T9" target="_self">Th9</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E6:84"/><i><font color="Green">E159</font></i>: 
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>(<a class="ref" href="lopban_1.html#T6" target="_self">Th6</a>);<br/></i>
<b>reconsider </b><font color="Maroon">tseq</font> = <font color="Maroon">f</font> as   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font> ;<br/>
<a NAME="E8:84"/>
<font color="Maroon">tseq</font> is   <a href="lopban_1.html#NM1">LinearOperator</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_3"><b>proof </b></a><div class="add">

<div><i><font color="Green">E160</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="84_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font>, <font color="Maroon">y</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font>;<br/><b>consider </b><font color="Maroon">xseq</font> being   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E2:84_3_1"/><i><font color="Green">E161</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  <font color="Maroon">xseq</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="lopban_1.html#K13" target="_self">modetrans</a> <span class="p2">(<span class="default"><font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font></span>)</span>,<font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> ) &amp; <font color="Maroon">xseq</font> is <a href="normsp_1.html#V4">convergent</a> &amp; <font color="Maroon">tseq</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="normsp_1.html#K8">lim</a> <font color="Maroon">xseq</font> )
 <b>by </b><i><a class="ref" href="lopban_1.html#T7" target="_self">Th7</a></i>;<br/><b>consider </b><font color="Maroon">yseq</font> being   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E4:84_3_1"/><i><font color="Green">E162</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  <font color="Maroon">yseq</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="lopban_1.html#K13" target="_self">modetrans</a> <span class="p2">(<span class="default"><font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font></span>)</span>,<font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> ) &amp; <font color="Maroon">yseq</font> is <a href="normsp_1.html#V4">convergent</a> &amp; <font color="Maroon">tseq</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="normsp_1.html#K8">lim</a> <font color="Maroon">yseq</font> )
 <b>by </b><i><a class="ref" href="lopban_1.html#T7" target="_self">Th7</a></i>;<br/><b>consider </b><font color="Maroon">zseq</font> being   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E6:84_3_1"/><i><font color="Green">E163</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  <font color="Maroon">zseq</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="lopban_1.html#K13" target="_self">modetrans</a> <span class="p2">(<span class="default"><font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font></span>)</span>,<font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</font></span>)</span> ) &amp; <font color="Maroon">zseq</font> is <a href="normsp_1.html#V4">convergent</a> &amp; <font color="Maroon">tseq</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="normsp_1.html#K8">lim</a> <font color="Maroon">zseq</font> )
 <b>by </b><i><a class="ref" href="lopban_1.html#T7" target="_self">Th7</a></i>;<br/><a NAME="E7:84_3_1"/>
<font color="Maroon">zseq</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">xseq</font> <a href="normsp_1.html#K3">+</a> <font color="Maroon">yseq</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_3_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_3_1_1_1"><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>thus </b><font color="Maroon">zseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><a href="lopban_1.html#K13" target="_self">modetrans</a> <span class="p2">(<span class="default"><font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i><a class="ref" href="lopban_1.html#T10" target="_self">Th10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_1.html#K13" target="_self">modetrans</a> <span class="p3">(<span class="default"><font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_1.html#K13" target="_self">modetrans</a> <span class="p3">(<span class="default"><font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">xseq</font> <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"><a href="lopban_1.html#K13" target="_self">modetrans</a> <span class="p3">(<span class="default"><font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i><a class="ref" href="lopban_1.html#T9" target="_self">Th9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">xseq</font> <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"><font color="Maroon">yseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i/>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:84_3_1_1"/>
<font color="Maroon">zseq</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">xseq</font> <a href="normsp_1.html#K3">+</a> <font color="Maroon">yseq</font>
 <b>by </b><i><a class="ref" href="normsp_1.html#D5">NORMSP_1:def 5</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E8:84_3_1"/>
<font color="Maroon">tseq</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">tseq</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">tseq</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span>
 <b>by </b><i><a class="ref" href="lopban_1.html#T9" target="_self">Th9</a>, , <a class="ref" href="lopban_1.html#T10" target="_self">Th10</a>, <a class="ref" href="normsp_1.html#T42">NORMSP_1:42</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_3_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>consider </b><font color="Maroon">xseq</font> being   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E2:84_3_2"/><i><font color="Green">E164</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  <font color="Maroon">xseq</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="lopban_1.html#K13" target="_self">modetrans</a> <span class="p2">(<span class="default"><font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font></span>)</span>,<font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> ) &amp; <font color="Maroon">xseq</font> is <a href="normsp_1.html#V4">convergent</a> &amp; <font color="Maroon">tseq</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="normsp_1.html#K8">lim</a> <font color="Maroon">xseq</font> )
 <b>by </b><i><a class="ref" href="lopban_1.html#T7" target="_self">Th7</a></i>;<br/><b>consider </b><font color="Maroon">zseq</font> being   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E4:84_3_2"/><i><font color="Green">E165</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  <font color="Maroon">zseq</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="lopban_1.html#K13" target="_self">modetrans</a> <span class="p2">(<span class="default"><font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font></span>)</span>,<font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x</font></span>)</span> ) &amp; <font color="Maroon">zseq</font> is <a href="normsp_1.html#V4">convergent</a> &amp; <font color="Maroon">tseq</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="normsp_1.html#K8">lim</a> <font color="Maroon">zseq</font> )
 <b>by </b><i><a class="ref" href="lopban_1.html#T7" target="_self">Th7</a></i>;<br/><a NAME="E5:84_3_2"/>
<font color="Maroon">zseq</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">a</font> <a href="normsp_1.html#K6">*</a> <font color="Maroon">xseq</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_3_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_3_2_1_1"><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>thus </b><font color="Maroon">zseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><a href="lopban_1.html#K13" target="_self">modetrans</a> <span class="p2">(<span class="default"><font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_1.html#K13" target="_self">modetrans</a> <span class="p3">(<span class="default"><font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">xseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="lopban_1.html#T9" target="_self">Th9</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:84_3_2_1"/>
<font color="Maroon">zseq</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">a</font> <a href="normsp_1.html#K6">*</a> <font color="Maroon">xseq</font>
 <b>by </b><i><a class="ref" href="normsp_1.html#D8">NORMSP_1:def 8</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E6:84_3_2"/>
<font color="Maroon">tseq</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">tseq</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>
 <b>by </b><i><a class="ref" href="lopban_1.html#T9" target="_self">Th9</a>, , <a class="ref" href="normsp_1.html#T45">NORMSP_1:45</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:84_3"/>
<font color="Maroon">tseq</font> is   <a href="lopban_1.html#NM1">LinearOperator</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="lopban_1.html#T8" target="_self">Th8</a>, , </i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">tseq</font> = <font color="Maroon">tseq</font> as   <a href="lopban_1.html#NM1">LinearOperator</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font> ;<br/>
<a NAME="E10:84"/><i><font color="Green">E166</font></i>: 
<font color="Maroon">tseq</font> is <a href="lopban_1.html#V3" target="_self">bounded</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_4"><b>proof </b></a><div class="add">

<div><i><font color="Green">E167</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="84_4_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font>;<br/><b>consider </b><font color="Maroon">xseq</font> being   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E2:84_4_1"/><i><font color="Green">E168</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  <font color="Maroon">xseq</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="lopban_1.html#K13" target="_self">modetrans</a> <span class="p2">(<span class="default"><font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font></span>)</span>,<font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> ) &amp; <font color="Maroon">xseq</font> is <a href="normsp_1.html#V4">convergent</a> &amp; <font color="Maroon">tseq</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="normsp_1.html#K8">lim</a> <font color="Maroon">xseq</font> )
 <b>by </b><i><a class="ref" href="lopban_1.html#T7" target="_self">Th7</a></i>;<br/><a NAME="E3:84_4_1"/><i><font color="Green">E169</font></i>: 
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">xseq</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">m</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">m</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">x</font></span><a href="normsp_1.html#K1">.||</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_4_1_1"><b>proof </b></a><div class="add">

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

<a NAME="E1:84_4_1_1"/>
<font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font> is  <a href="lopban_1.html#V3" target="_self">bounded</a> <a href="lopban_1.html#NM1">LinearOperator</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font>
 
<b>by </b><i/>;<br/>
<a NAME="E2:84_4_1_1"/><b>then </b><i><font color="Green">E170</font></i>: 
 <a href="lopban_1.html#K13" target="_self">modetrans</a> <span class="p1">(<span class="default"><font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span>,<font color="Maroon">X</font>,<font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="txt" href="lopban_1.html#E2:84"><i><font color="Green">E42</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">h1</font> = <font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font> as  <a href="lopban_1.html#V3" target="_self">bounded</a> <a href="lopban_1.html#NM1">LinearOperator</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font> <b>by </b><i/>;<br/>
<a NAME="E4:84_4_1_1"/>
<font color="Maroon">xseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="lopban_1.html#K13" target="_self">modetrans</a> <span class="p2">(<span class="default"><font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span>,<font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:84_4_1_1"/>
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">xseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">x</font></span><a href="normsp_1.html#K1">.||</a></span>
 <b>by </b><i><a class="ref" href="lopban_1.html#T11" target="_self">Th11</a>, </i>;<br/>


</div><b>end;</b></div><a NAME="E4:84_4_1"/><i><font color="Green">E172</font></i>: 
<span class="p1"><a href="normsp_1.html#K7">||.</a><span class="default"><font color="Maroon">xseq</font></span><a href="normsp_1.html#K7">.||</a></span> is <a href="seq_2.html#V4">convergent</a>
 <b>by </b><i>, </i>;<br/><a NAME="E5:84_4_1"/><i><font color="Green">E173</font></i>: 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">tseq</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="hidden.html#R1">=</a>  <a href="seq_2.html#K2">lim</a> <span class="p1"><a href="normsp_1.html#K7">||.</a><span class="default"><font color="Maroon">xseq</font></span><a href="normsp_1.html#K7">.||</a></span>
 <b>by </b><i>, </i>;<br/><a NAME="E6:84_4_1"/><i><font color="Green">E174</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"><a href="normsp_1.html#K7">||.</a><span class="default"><font color="Maroon">xseq</font></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">x</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="seq_1.html#K14">(#)</a> <span class="p2"><a href="normsp_1.html#K7">||.</a><span class="default"><font color="Maroon">vseq</font></span><a href="normsp_1.html#K7">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_4_1_2"><b>proof </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/>

<a NAME="E1:84_4_1_2"/><i><font color="Green">E175</font></i>: 
<span class="p1"><a href="normsp_1.html#K7">||.</a><span class="default"><font color="Maroon">xseq</font></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">xseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
 
<b>by </b><i><a class="ref" href="normsp_1.html#D10">NORMSP_1:def 10</a></i>;<br/>
<a NAME="E2:84_4_1_2"/><i><font color="Green">E176</font></i>: 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">xseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">x</font></span><a href="normsp_1.html#K1">.||</a></span>
 
<b>by </b><i><a class="ref" href="lopban_1.html#T10" target="_self">Th10</a></i>;<br/>
<a NAME="E3:84_4_1_2"/>
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="normsp_1.html#K7">||.</a><span class="default"><font color="Maroon">vseq</font></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="normsp_1.html#D10">NORMSP_1:def 10</a></i>;<br/>
<b>hence </b><a NAME="E4:84_4_1_2"/>
<span class="p1"><a href="normsp_1.html#K7">||.</a><span class="default"><font color="Maroon">xseq</font></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">x</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="seq_1.html#K14">(#)</a> <span class="p2"><a href="normsp_1.html#K7">||.</a><span class="default"><font color="Maroon">vseq</font></span><a href="normsp_1.html#K7">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="lopban_1.html#T13" target="_self">Th13</a>, , <a class="ref" href="seq_1.html#T13">SEQ_1:13</a></i>;<br/>


</div><b>end;</b></div><a NAME="E7:84_4_1"/><i><font color="Green">E177</font></i>: 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">x</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="seq_1.html#K14">(#)</a> <span class="p1"><a href="normsp_1.html#K7">||.</a><span class="default"><font color="Maroon">vseq</font></span><a href="normsp_1.html#K7">.||</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/><a NAME="E8:84_4_1"/>
 <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">x</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="seq_1.html#K14">(#)</a> <span class="p2"><a href="normsp_1.html#K7">||.</a><span class="default"><font color="Maroon">vseq</font></span><a href="normsp_1.html#K7">.||</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p2"><a href="normsp_1.html#K7">||.</a><span class="default"><font color="Maroon">vseq</font></span><a href="normsp_1.html#K7">.||</a></span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">x</font></span><a href="normsp_1.html#K1">.||</a></span>
 <b>by </b><i>, <a class="ref" href="seq_2.html#T22">SEQ_2:22</a></i>;<br/><b>hence </b><a NAME="E9:84_4_1"/>
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">tseq</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p2"><a href="normsp_1.html#K7">||.</a><span class="default"><font color="Maroon">vseq</font></span><a href="normsp_1.html#K7">.||</a></span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">x</font></span><a href="normsp_1.html#K1">.||</a></span>
 <b>by </b><i><a class="ref" href="lopban_1.html#T11" target="_self">Th11</a>, <a class="ref" href="lopban_1.html#T12" target="_self">Th12</a>, <a class="txt" href="lopban_1.html#E17"><i><font color="Green">Lemma69</font></i></a>, <a class="ref" href="lopban_1.html#T13" target="_self">Th13</a>, <a class="ref" href="seq_2.html#T32">SEQ_2:32</a></i>;<br/></div><b>end;</b></div>
<b>take </b><font color="Maroon">K</font> =  <a href="seq_2.html#K2">lim</a> <span class="p1"><a href="normsp_1.html#K7">||.</a><span class="default"><font color="Maroon">vseq</font></span><a href="normsp_1.html#K7">.||</a></span>; <i><font color="Red">:: according to </font></i><a class="ref" href="lopban_1.html#D9" target="_self">LOPBAN_1:def 9</a><br/>

<a NAME="E2:84_4"/>
<font color="Maroon">K</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_4_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_4_2_1"><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/><a NAME="E1:84_4_2_1"/>
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="normsp_1.html#T8">NORMSP_1:8</a></i>;<br/><b>hence </b><a NAME="E2:84_4_2_1"/>
<span class="p1"><a href="normsp_1.html#K7">||.</a><span class="default"><font color="Maroon">vseq</font></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="normsp_1.html#D10">NORMSP_1:def 10</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:84_4_2"/>
<font color="Maroon">K</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i>, <a class="ref" href="seq_2.html#T31">SEQ_2:31</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:84_4"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">K</font> &amp; ( for <font color="Olive">x</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font> holds  <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">tseq</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">K</font> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Olive">x</font></span><a href="normsp_1.html#K1">.||</a></span> ) )
 <b>by </b><i><a class="ref" href="lopban_1.html#T9" target="_self">Th9</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:84"/><i><font color="Green">E178</font></i>: 
for <font color="Olive">e</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">e</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 holds <br/> ex <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">k</font> holds <br/>for <font color="Olive">x</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font> holds  <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="lopban_1.html#K13" target="_self">modetrans</a> <span class="p5">(<span class="default"><font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font></span>)</span>,<font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">tseq</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">e</font> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Olive">x</font></span><a href="normsp_1.html#K1">.||</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">e</font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E1:84_5"/><i><font color="Green">E179</font></i>: 
<font color="Maroon">e</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<b>consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:84_5"/><i><font color="Green">E180</font></i>: 
for <font color="Olive">n</font>, <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k</font> &amp; <font color="Olive">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k</font> holds <br/><span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">m</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font>
 <b>by </b><i>, , <a class="ref" href="rsspace3.html#T10">RSSPACE3:10</a></i>;<br/>
<div><i><font color="Green">E181</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="84_5_1"><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>assume </b><a NAME="E1:84_5_1"/><i><font color="Green">E182</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_5_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font>;<br/><b>consider </b><font color="Maroon">xseq</font> being   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E2:84_5_1_1"/><i><font color="Green">E183</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  <font color="Maroon">xseq</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="lopban_1.html#K13" target="_self">modetrans</a> <span class="p2">(<span class="default"><font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font></span>)</span>,<font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> ) &amp; <font color="Maroon">xseq</font> is <a href="normsp_1.html#V4">convergent</a> &amp; <font color="Maroon">tseq</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="normsp_1.html#K8">lim</a> <font color="Maroon">xseq</font> )
 <b>by </b><i><a class="ref" href="lopban_1.html#T7" target="_self">Th7</a></i>;<br/><a NAME="E3:84_5_1_1"/><i><font color="Green">E201</font></i>: 
for <font color="Olive">m</font>, <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">xseq</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">m</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">xseq</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">k</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">m</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">k</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">x</font></span><a href="normsp_1.html#K1">.||</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_5_1_1_1"><b>proof </b></a><div class="add">

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

<a NAME="E1:84_5_1_1_1"/>
<font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font> is  <a href="lopban_1.html#V3" target="_self">bounded</a> <a href="lopban_1.html#NM1">LinearOperator</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font>
 
<b>by </b><i/>;<br/>
<a NAME="E2:84_5_1_1_1"/><b>then </b><i><font color="Green">E202</font></i>: 
 <a href="lopban_1.html#K13" target="_self">modetrans</a> <span class="p1">(<span class="default"><font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span>,<font color="Maroon">X</font>,<font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="txt" href="lopban_1.html#E2:84"><i><font color="Green">E42</font></i></a></i>;<br/>
<a NAME="E3:84_5_1_1_1"/>
<font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font> is  <a href="lopban_1.html#V3" target="_self">bounded</a> <a href="lopban_1.html#NM1">LinearOperator</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font>
 
<b>by </b><i/>;<br/>
<a NAME="E4:84_5_1_1_1"/><b>then </b><i><font color="Green">E203</font></i>: 
 <a href="lopban_1.html#K13" target="_self">modetrans</a> <span class="p1">(<span class="default"><font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span>,<font color="Maroon">X</font>,<font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font>
 
<b>by </b><i><a class="txt" href="lopban_1.html#E2:84"><i><font color="Green">E42</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">h1</font> = <span class="p1">(<span class="default"><font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> as  <a href="lopban_1.html#V3" target="_self">bounded</a> <a href="lopban_1.html#NM1">LinearOperator</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font> <b>by </b><i/>;<br/>
<a NAME="E6:84_5_1_1_1"/><i><font color="Green">E204</font></i>: 
<font color="Maroon">xseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="lopban_1.html#K13" target="_self">modetrans</a> <span class="p2">(<span class="default"><font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span>,<font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="txt" href="lopban_1.html#E17"><i><font color="Green">Lemma69</font></i></a></i>;<br/>
<a NAME="E7:84_5_1_1_1"/>
<font color="Maroon">xseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="lopban_1.html#K13" target="_self">modetrans</a> <span class="p2">(<span class="default"><font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span>,<font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="txt" href="lopban_1.html#E17"><i><font color="Green">Lemma69</font></i></a></i>;<br/>
<a NAME="E8:84_5_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">xseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">xseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 
<b>by </b><i>, , , </i>;<br/>
<b>hence </b><a NAME="E9:84_5_1_1_1"/>
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">xseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">xseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">x</font></span><a href="normsp_1.html#K1">.||</a></span>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><a NAME="E4:84_5_1_1"/><i><font color="Green">E205</font></i>: 
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k</font> holds <br/><span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">xseq</font> <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"><font color="Maroon">xseq</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">m</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">x</font></span><a href="normsp_1.html#K1">.||</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_5_1_1_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:84_5_1_1_2"/><i><font color="Green">E206</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k</font>
 ;<br/>

<a NAME="E2:84_5_1_1_2"/><i><font color="Green">E207</font></i>: 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">xseq</font> <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"><font color="Maroon">xseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">vseq</font> <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"><font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">x</font></span><a href="normsp_1.html#K1">.||</a></span>
 
<b>by </b><i><a class="ref" href="lopban_1.html#T13" target="_self">Th13</a></i>;<br/>
<a NAME="E3:84_5_1_1_2"/><i><font color="Green">E208</font></i>: 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">vseq</font> <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"><font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font>
 
<b>by </b><i><a class="ref" href="lopban_1.html#T10" target="_self">Th10</a>, <a class="ref" href="lopban_1.html#T12" target="_self">Th12</a>, </i>;<br/>
<a NAME="E4:84_5_1_1_2"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">x</font></span><a href="normsp_1.html#K1">.||</a></span>
 
<b>by </b><i><a class="ref" href="normsp_1.html#T8">NORMSP_1:8</a></i>;<br/>
<a NAME="E5:84_5_1_1_2"/><b>then </b>
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">vseq</font> <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"><font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">x</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">x</font></span><a href="normsp_1.html#K1">.||</a></span>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<b>hence </b><a NAME="E6:84_5_1_1_2"/>
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">xseq</font> <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"><font color="Maroon">xseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">x</font></span><a href="normsp_1.html#K1">.||</a></span>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>


</div><b>end;</b></div><a NAME="E5:84_5_1_1"/>
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">xseq</font> <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"><font color="Maroon">tseq</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">x</font></span><a href="normsp_1.html#K1">.||</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_5_1_1_3"><b>proof </b></a><div class="add">

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  = <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">xseq</font> <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="p3">(<span class="default"><font color="Maroon">xseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>;<br/>
<b>consider </b><font color="Maroon">rseq</font> being   <a href="seq_1.html#NM1">Real_Sequence</a><b> such that </b><br/><a NAME="E2:84_5_1_1_3"/><i><font color="Green">E210</font></i>: 
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">rseq</font> <a href="seq_1.html#K2">.</a> <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">m</font>)
 <b>from </b><i><a class="ref" href="seq_1.html#S1">SEQ_1:sch 1</a>();<br/></i>
<a NAME="E3:84_5_1_1_3"/><i><font color="Green">E211</font></i>: 
( <font color="Maroon">rseq</font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <font color="Maroon">rseq</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">tseq</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">xseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_5_1_1_3_1"><b>proof </b></a><div class="add">

<a NAME="E1:84_5_1_1_3_1"/><i><font color="Green">E212</font></i>: 
<font color="Maroon">xseq</font> <a href="normsp_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">xseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> is <a href="normsp_1.html#V4">convergent</a>
 
<b>by </b><i><a class="txt" href="lopban_1.html#E17"><i><font color="Green">Lemma69</font></i></a>, <a class="ref" href="normsp_1.html#T36">NORMSP_1:36</a></i>;<br/>
<a NAME="E2:84_5_1_1_3_1"/><i><font color="Green">E213</font></i>: 
 <a href="normsp_1.html#K8">lim</a> <span class="p1">(<span class="default"><font color="Maroon">xseq</font> <a href="normsp_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">xseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">tseq</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">xseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="lopban_1.html#E17"><i><font color="Green">Lemma69</font></i></a>, <a class="ref" href="normsp_1.html#T44">NORMSP_1:44</a></i>;<br/>
<a NAME="E3:84_5_1_1_3_1"/>
<font color="Maroon">rseq</font> <a href="funct_2.html#R2">=</a> <span class="p1"><a href="normsp_1.html#K7">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">xseq</font> <a href="normsp_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">xseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span><a href="normsp_1.html#K7">.||</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_5_1_1_3_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_5_1_1_3_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:84_5_1_1_3_1_1_1"/><i><font color="Green">E214</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 ;<br/><b>reconsider </b><font color="Maroon">k</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/><b>thus </b><font color="Maroon">rseq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> = 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">xseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">xseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
<b>by </b><i/>
<br/>.= 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">xseq</font> <a href="normsp_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">xseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
<b>by </b><i><a class="ref" href="normsp_1.html#D7">NORMSP_1:def 7</a></i>
<br/>.= 
<span class="p1"><a href="normsp_1.html#K7">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">xseq</font> <a href="normsp_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">xseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="normsp_1.html#D10">NORMSP_1:def 10</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:84_5_1_1_3_1_1"/>
<font color="Maroon">rseq</font> <a href="funct_2.html#R2">=</a> <span class="p1"><a href="normsp_1.html#K7">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">xseq</font> <a href="normsp_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">xseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span><a href="normsp_1.html#K7">.||</a></span>
 <b>by </b><i><a class="ref" href="seq_1.html#T8">SEQ_1:8</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:84_5_1_1_3_1"/>
( <font color="Maroon">rseq</font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <font color="Maroon">rseq</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">tseq</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">xseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> )
 <b>by </b><i>, <a class="ref" href="lopban_1.html#T14" target="_self">Th14</a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:84_5_1_1_3"/>
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k</font> holds <br/><font color="Maroon">rseq</font> <a href="seq_1.html#K2">.</a> <font color="Olive">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">x</font></span><a href="normsp_1.html#K1">.||</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_5_1_1_3_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:84_5_1_1_3_2"/><i><font color="Green">E215</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k</font>
 ;<br/>

<font color="Maroon">rseq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> = 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">xseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">xseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
<b>by </b><i/>
<br/>.= 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">xseq</font> <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"><font color="Maroon">xseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
<b>by </b><i><a class="ref" href="normsp_1.html#T11">NORMSP_1:11</a></i>
;<br/>
<b>hence </b><a NAME="E3:84_5_1_1_3_2"/>
<font color="Maroon">rseq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">x</font></span><a href="normsp_1.html#K1">.||</a></span>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:84_5_1_1_3"/><b>then </b>
 <a href="seq_2.html#K2">lim</a> <font color="Maroon">rseq</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">x</font></span><a href="normsp_1.html#K1">.||</a></span>
 
<b>by </b><i>, <a class="ref" href="lopban_1.html#T4" target="_self">Th4</a></i>;<br/>
<b>hence </b><a NAME="E6:84_5_1_1_3"/>
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">xseq</font> <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"><font color="Maroon">tseq</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">x</font></span><a href="normsp_1.html#K1">.||</a></span>
 <b>by </b><i>, <a class="ref" href="normsp_1.html#T11">NORMSP_1:11</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E6:84_5_1_1"/>
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="lopban_1.html#K13" target="_self">modetrans</a> <span class="p5">(<span class="default"><font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">tseq</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">x</font></span><a href="normsp_1.html#K1">.||</a></span>
 <b>by </b><i><a class="txt" href="lopban_1.html#E17"><i><font color="Green">Lemma69</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:84_5_1"/>
for <font color="Olive">x</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font> holds  <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="lopban_1.html#K13" target="_self">modetrans</a> <span class="p5">(<span class="default"><font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">tseq</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Olive">x</font></span><a href="normsp_1.html#K1">.||</a></span>
 ;<br/></div><b>end;</b></div>
<b>take </b>
<font color="Maroon">k</font>
;<br/>

<b>thus </b><a NAME="E5:84_5"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k</font> holds <br/>for <font color="Olive">x</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font> holds  <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="lopban_1.html#K13" target="_self">modetrans</a> <span class="p5">(<span class="default"><font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font></span>)</span>,<font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">tseq</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Olive">x</font></span><a href="normsp_1.html#K1">.||</a></span>
 <b>by </b><i><a class="ref" href="lopban_1.html#T11" target="_self">Th11</a></i>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">tseq</font> = <font color="Maroon">tseq</font> as  <a href="lopban_1.html#V3" target="_self">bounded</a> <a href="lopban_1.html#NM1">LinearOperator</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font> <b>by </b><i><a class="ref" href="lopban_1.html#T8" target="_self">Th8</a></i>;<br/>
<b>reconsider </b><font color="Maroon">tv</font> = <font color="Maroon">tseq</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="lopban_1.html#K16" target="_self">R_NormSpace_of_BoundedLinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> <b>by </b><i/>;<br/>
<a NAME="E14:84"/><i><font color="Green">E217</font></i>: 
for <font color="Olive">e</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">e</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 holds <br/> ex <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">k</font> holds <br/><span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">tv</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">e</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">e</font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E1:84_6"/><i><font color="Green">E218</font></i>: 
<font color="Maroon">e</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<b>consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:84_6"/><i><font color="Green">E219</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>   st <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k</font> holds <br/>for <font color="Olive">x</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font> holds  <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="lopban_1.html#K13" target="_self">modetrans</a> <span class="p5">(<span class="default"><font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font></span>)</span>,<font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">tseq</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Olive">x</font></span><a href="normsp_1.html#K1">.||</a></span>
 <b>by </b><i><a class="ref" href="lopban_1.html#T9" target="_self">Th9</a>, <a class="ref" href="lopban_1.html#T10" target="_self">Th10</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_6_1"><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>assume </b><a NAME="E1:84_6_1"/><i><font color="Green">E220</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k</font>
 ;<br/><b>set </b><font color="Maroon">f1</font> =  <a href="lopban_1.html#K13" target="_self">modetrans</a> <span class="p1">(<span class="default"><font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">X</font>,<font color="Maroon">Y</font>;<br/><b>set </b><font color="Maroon">g1</font> = <font color="Maroon">tseq</font>;<br/><b>reconsider </b><font color="Maroon">h1</font> = <span class="p1">(<span class="default"><font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">tv</font> as  <a href="lopban_1.html#V3" target="_self">bounded</a> <a href="lopban_1.html#NM1">LinearOperator</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font> <b>by </b><i/>;<br/><div><i><font color="Green">E221</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="84_6_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">t</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font>;<br/><b>assume </b><a NAME="E1:84_6_1_1"/><i><font color="Green">E222</font></i>: 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">t</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> 1
 ;<br/><a NAME="E2:84_6_1_1"/>
<font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font> is  <a href="lopban_1.html#V3" target="_self">bounded</a> <a href="lopban_1.html#NM1">LinearOperator</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font>
 <b>by </b><i/>;<br/><a NAME="E3:84_6_1_1"/><b>then </b>
 <a href="lopban_1.html#K13" target="_self">modetrans</a> <span class="p1">(<span class="default"><font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">X</font>,<font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="txt" href="lopban_1.html#E2:84"><i><font color="Green">E42</font></i></a></i>;<br/><a NAME="E4:84_6_1_1"/><b>then </b><i><font color="Green">E223</font></i>: 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="lopban_1.html#K13" target="_self">modetrans</a> <span class="p5">(<span class="default"><font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">tseq</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
 <b>by </b><i/>;<br/><a NAME="E5:84_6_1_1"/><i><font color="Green">E224</font></i>: 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="lopban_1.html#K13" target="_self">modetrans</a> <span class="p5">(<span class="default"><font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">tseq</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">t</font></span><a href="normsp_1.html#K1">.||</a></span>
 <b>by </b><i><a class="ref" href="lopban_1.html#T11" target="_self">Th11</a>, <a class="ref" href="lopban_1.html#T12" target="_self">Th12</a></i>;<br/><a NAME="E6:84_6_1_1"/>
<font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">t</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">e</font> <a href="real_1.html#K4">*</a> 1
 <b>by </b><i><a class="ref" href="lopban_1.html#T10" target="_self">Th10</a>, <a class="ref" href="lopban_1.html#T13" target="_self">Th13</a>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><b>hence </b><a NAME="E7:84_6_1_1"/>
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">e</font>
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E225</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="84_6_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">r</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>assume </b><a NAME="E1:84_6_1_2"/><i><font color="Green">E226</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="lopban_1.html#K14" target="_self">PreNorms</a> <font color="Maroon">h1</font>
 ;<br/><a NAME="E2:84_6_1_2"/>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">h1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">t</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> where <font color="Olive">t</font> is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font> : <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Olive">t</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> 1 </span>}</span> 
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">t</font> being   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E4:84_6_1_2"/><i><font color="Green">E227</font></i>: 
( <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> &amp; <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">t</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 ;<br/><b>thus </b><a NAME="E5:84_6_1_2"/>
<font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="txt" href="lopban_1.html#E17"><i><font color="Green">Lemma69</font></i></a>, </i>;<br/></div><b>end;</b></div><a NAME="E5:84_6_1"/><i><font color="Green">E228</font></i>: 
( ( for <font color="Olive">s</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">s</font> <a href="hidden.html#R2">in</a>  <a href="lopban_1.html#K14" target="_self">PreNorms</a> <font color="Maroon">h1</font> holds <br/><font color="Olive">s</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">e</font> ) implies  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="lopban_1.html#K14" target="_self">PreNorms</a> <font color="Maroon">h1</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">e</font> )
 <b>by </b><i><a class="ref" href="seq_4.html#T62">SEQ_4:62</a></i>;<br/><a NAME="E6:84_6_1"/>
<span class="p1">(<span class="default"><a href="lopban_1.html#K15" target="_self">BoundedLinearOperatorsNorm</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">tv</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="lopban_1.html#K14" target="_self">PreNorms</a> <font color="Maroon">h1</font></span>)</span>
 <b>by </b><i><a class="ref" href="lopban_1.html#D1" target="_self">Def1</a></i>;<br/><b>hence </b><a NAME="E7:84_6_1"/>
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">tv</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="ref" href="lopban_1.html#T13" target="_self">Th13</a>, , <a class="ref" href="normsp_1.html#D1">NORMSP_1:def 1</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:84_6"/>
 ex <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">k</font> holds <br/><span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">tv</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">e</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E15:84"/>
for <font color="Olive">e</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">e</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 holds <br/> ex <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">m</font> holds <br/><span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">tv</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">e</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">e</font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E1:84_7"/><i><font color="Green">E229</font></i>: 
<font color="Maroon">e</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<a NAME="E2:84_7"/><i><font color="Green">E230</font></i>: 
<font color="Maroon">e</font> <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="lopban_1.html#T10" target="_self">Th10</a>, <a class="ref" href="seq_2.html#T3">SEQ_2:3</a></i>;<br/>
<a NAME="E3:84_7"/><i><font color="Green">E231</font></i>: 
<font color="Maroon">e</font> <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font>
 
<b>by </b><i><a class="ref" href="lopban_1.html#T10" target="_self">Th10</a>, <a class="ref" href="xreal_1.html#T218">XREAL_1:218</a></i>;<br/>
<b>consider </b><font color="Maroon">m</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E5:84_7"/><i><font color="Green">E232</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>   st <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">m</font> holds <br/><span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">tv</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">e</font> <a href="real_1.html#K6">/</a> 2
 <b>by </b><i>, <a class="ref" href="lopban_1.html#T11" target="_self">Th11</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_7_1"><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>assume </b><a NAME="E1:84_7_1"/><i><font color="Green">E233</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">m</font>
 ;<br/><a NAME="E2:84_7_1"/>
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">tv</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">e</font> <a href="real_1.html#K6">/</a> 2
 <b>by </b><i><a class="txt" href="lopban_1.html#E17"><i><font color="Green">Lemma69</font></i></a>, <a class="ref" href="lopban_1.html#T13" target="_self">Th13</a></i>;<br/><b>hence </b><a NAME="E3:84_7_1"/>
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">tv</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="ref" href="lopban_1.html#T12" target="_self">Th12</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:84_7"/>
 ex <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">m</font> holds <br/><span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">tv</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E16:84"/>
<font color="Maroon">vseq</font> is <a href="normsp_1.html#V4">convergent</a>
 <b>by </b><i><a class="ref" href="normsp_1.html#D9">NORMSP_1:def 9</a></i>;<br/>


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