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

<b>let </b><font color="Maroon">X</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">Y</font> be   <a href="clvect_1.html#NM5">ComplexNormSpace</a>;<br/>



<b>assume </b><a NAME="E1:55"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">Y</font> is <a href="clopban1.html#V4">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="csspace4.html#K9" target="_self">C_NormSpace_of_BoundedFunctions</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span>;<br/>

<b>assume </b><a NAME="E2:55"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">vseq</font> is <a href="csspace3.html#V1">CCauchy</a>
 ;<br/>

<a NAME="E3:55"/><i><font color="Green">E34</font></i>: 
<span class="p1"><a href="clvect_1.html#K10">||.</a><span class="default"><font color="Maroon">vseq</font></span><a href="clvect_1.html#K10">.||</a></span> is <a href="seq_2.html#V4">convergent</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_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="E1:55_1_1"/><i><font color="Green">E47</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="E4:55_1_1"/><i><font color="Green">E52</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="clvect_1.html#K5">||.</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="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font>
 <b>by </b><i>, , <a class="ref" href="csspace3.html#T10">CSSPACE3:10</a></i>;<br/><div><i><font color="Green">E53</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="55_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="E1:55_1_1_1"/><i><font color="Green">E54</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="E2:55_1_1_1"/><i><font color="Green">E78</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="real_1.html#K5">-</a> <span class="p2"><a href="clvect_1.html#K5">||.</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="clvect_1.html#K5">.||</a></span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><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="clvect_1.html#K5">.||</a></span>
 <b>by </b><i><a class="ref" href="clvect_1.html#T111">CLVECT_1:111</a></i>;<br/><a NAME="E3:55_1_1_1"/><i><font color="Green">E79</font></i>: 
<span class="p1"><a href="clvect_1.html#K5">||.</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="clvect_1.html#K5">.||</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="clvect_1.html#K10">||.</a><span class="default"><font color="Maroon">vseq</font></span><a href="clvect_1.html#K10">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="clvect_1.html#D16">CLVECT_1:def 16</a></i>;<br/><a NAME="E4:55_1_1_1"/><i><font color="Green">E80</font></i>: 
<span class="p1"><a href="clvect_1.html#K5">||.</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="clvect_1.html#K5">.||</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="clvect_1.html#K10">||.</a><span class="default"><font color="Maroon">vseq</font></span><a href="clvect_1.html#K10">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="clvect_1.html#D16">CLVECT_1:def 16</a></i>;<br/><a NAME="E5:55_1_1_1"/>
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">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="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="txt" href="csspace4.html#E1:55_1_1"><i><font color="Green">E47</font></i></a>, </i>;<br/><b>hence </b><a NAME="E6:55_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="clvect_1.html#K10">||.</a><span class="default"><font color="Maroon">vseq</font></span><a href="clvect_1.html#K10">.||</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="clvect_1.html#K10">||.</a><span class="default"><font color="Maroon">vseq</font></span><a href="clvect_1.html#K10">.||</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="txt" href="csspace4.html#E4:55_1_1"><i><font color="Green">E52</font></i></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="E6:55_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="clvect_1.html#K10">||.</a><span class="default"><font color="Maroon">vseq</font></span><a href="clvect_1.html#K10">.||</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="clvect_1.html#K10">||.</a><span class="default"><font color="Maroon">vseq</font></span><a href="clvect_1.html#K10">.||</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="txt" href="csspace4.html#E5"><i><font color="Green">Lemma48</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:55_1"/>
<span class="p1"><a href="clvect_1.html#K10">||.</a><span class="default"><font color="Maroon">vseq</font></span><a href="clvect_1.html#K10">.||</a></span> is <a href="seq_2.html#V4">convergent</a>
 <b>by </b><i><a class="ref" href="seq_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="csspace4.html#K6" 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="clvect_1.html#V6">convergent</a> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="clvect_1.html#K11">lim</a> <font color="Olive">xseq</font> );<br/>
<a NAME="E4:55"/><i><font color="Green">E82</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">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="55_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">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="csspace4.html#K6" 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="E2:55_2"/><i><font color="Green">E83</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="E3:55_2"/><i><font color="Green">E84</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="clvect_1.html#K5">||.</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="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><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="clvect_1.html#K5">.||</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_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="E1:55_2_1"/>
<font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font> is  <a href="csspace4.html#V1" target="_self">bounded</a> <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">X</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>
 
<b>by </b><i/>;<br/>
<a NAME="E2:55_2_1"/><b>then </b><i><font color="Green">E85</font></i>: 
 <a href="csspace4.html#K6" 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/>;<br/>
<a NAME="E3:55_2_1"/>
<font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font> is  <a href="csspace4.html#V1" target="_self">bounded</a> <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">X</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>
 
<b>by </b><i/>;<br/>
<a NAME="E4:55_2_1"/><b>then </b><i><font color="Green">E86</font></i>: 
 <a href="csspace4.html#K6" 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/>;<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="csspace4.html#V1" target="_self">bounded</a> <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">X</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font> <b>by </b><i/>;<br/>
<a NAME="E6:55_2_1"/><i><font color="Green">E87</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="csspace4.html#K6" 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/>
<a NAME="E7:55_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="csspace4.html#K6" 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/>;<br/>
<a NAME="E8:55_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>, , , </i>;<br/>
<b>hence </b><a NAME="E9:55_2_1"/>
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">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="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><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="clvect_1.html#K5">.||</a></span>
 <b>by </b><i><a class="txt" href="csspace4.html#E2:55"><i><font color="Green">E33</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:55_2"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">xseq</font> is <a href="clvect_1.html#V6">convergent</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_2_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_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="E1:55_2_2_1"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">e</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><b>thus </b><a NAME="E2:55_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="clvect_1.html#K5">||.</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="clvect_1.html#K5">.||</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="55_2_2_1_1"><b>proof </b></a><div class="add">

<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="E2:55_2_2_1_1"/><i><font color="Green">E90</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="clvect_1.html#K5">||.</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="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font>
 <b>by </b><i>, , <a class="ref" href="csspace3.html#T10">CSSPACE3:10</a></i>;<br/>
<b>take </b>
<font color="Maroon">k</font>
;<br/>

<b>thus </b><a NAME="E3:55_2_2_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="clvect_1.html#K5">||.</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="clvect_1.html#K5">.||</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="55_2_2_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="E1:55_2_2_1_1_1"/><i><font color="Green">E91</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="E2:55_2_2_1_1_1"/><i><font color="Green">E92</font></i>: 
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><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="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><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="clvect_1.html#K5">.||</a></span>
 
<b>by </b><i><a class="txt" href="csspace4.html#E5"><i><font color="Green">Lemma48</font></i></a></i>;<br/>
<a NAME="E3:55_2_2_1_1_1"/>
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">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="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E4:55_2_2_1_1_1"/>
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">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="clvect_1.html#K5">.||</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><b>end;</b></div></div><b>end;</b></div>
<a NAME="E2:55_2_2"/><b>then </b>
<font color="Maroon">xseq</font> is <a href="csspace3.html#V1">CCauchy</a>
 
<b>by </b><i><a class="ref" href="csspace3.html#T10">CSSPACE3:10</a></i>;<br/>
<b>hence </b><a NAME="E3:55_2_2"/>
<font color="Maroon">xseq</font> is <a href="clvect_1.html#V6">convergent</a>
 <b>by </b><i>, <a class="ref" href="clopban1.html#D14">CLOPBAN1:def 14</a></i>;<br/>


</div><b>end;</b></div>
<b>take </b><font color="Maroon">y</font> =  <a href="clvect_1.html#K11">lim</a> <font color="Maroon">xseq</font>;<br/>

<b>thus </b><a NAME="E5:55_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">y</font>]
 <b>by </b><i>, </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 <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="E6:55"/><i><font color="Green">E93</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">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>();<br/></i>
<b>reconsider </b><font color="Maroon">tseq</font> = <font color="Maroon">f</font> as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">X</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font> ;<br/>
<a NAME="E8:55"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">tseq</font> is <a href="csspace4.html#V1" target="_self">bounded</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_3"><b>proof </b></a><div class="add">

<div><i><font color="Green">E106</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="55_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</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="E2:55_3_1"/><i><font color="Green">E107</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="csspace4.html#K6" 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="clvect_1.html#V6">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="clvect_1.html#K11">lim</a> <font color="Maroon">xseq</font> )
 <b>by </b><i/>;<br/><a NAME="E3:55_3_1"/><i><font color="Green">E108</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="clvect_1.html#K5">||.</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="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">m</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_3_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="E1:55_3_1_1"/>
<font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font> is  <a href="csspace4.html#V1" target="_self">bounded</a> <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">X</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>
 
<b>by </b><i/>;<br/>
<a NAME="E2:55_3_1_1"/><b>then </b><i><font color="Green">E109</font></i>: 
 <a href="csspace4.html#K6" 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/>;<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="csspace4.html#V1" target="_self">bounded</a> <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">X</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font> <b>by </b><i/>;<br/>
<a NAME="E4:55_3_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="csspace4.html#K6" 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="E5:55_3_1_1"/>
<span class="p1"><a href="clvect_1.html#K5">||.</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="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span>
 <b>by </b><i>, <a class="txt" href="csspace4.html#E2:55"><i><font color="Green">E33</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E4:55_3_1"/><i><font color="Green">E110</font></i>: 
<span class="p1"><a href="clvect_1.html#K10">||.</a><span class="default"><font color="Maroon">xseq</font></span><a href="clvect_1.html#K10">.||</a></span> is <a href="seq_2.html#V4">convergent</a>
 <b>by </b><i>, <a class="ref" href="clopban1.html#T45">CLOPBAN1:45</a></i>;<br/><a NAME="E5:55_3_1"/><i><font color="Green">E111</font></i>: 
<span class="p1"><a href="clvect_1.html#K5">||.</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="clvect_1.html#K5">.||</a></span> <a href="hidden.html#R1">=</a>  <a href="seq_2.html#K2">lim</a> <span class="p1"><a href="clvect_1.html#K10">||.</a><span class="default"><font color="Maroon">xseq</font></span><a href="clvect_1.html#K10">.||</a></span>
 <b>by </b><i>, <a class="ref" href="clopban1.html#T45">CLOPBAN1:45</a></i>;<br/><a NAME="E6:55_3_1"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <span class="p1"><a href="clvect_1.html#K10">||.</a><span class="default"><font color="Maroon">xseq</font></span><a href="clvect_1.html#K10">.||</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"><a href="clvect_1.html#K10">||.</a><span class="default"><font color="Maroon">vseq</font></span><a href="clvect_1.html#K10">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_3_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="E1:55_3_1_2"/><i><font color="Green">E112</font></i>: 
<span class="p1"><a href="clvect_1.html#K10">||.</a><span class="default"><font color="Maroon">xseq</font></span><a href="clvect_1.html#K10">.||</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="clvect_1.html#K5">||.</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="clvect_1.html#K5">.||</a></span>
 
<b>by </b><i><a class="ref" href="clvect_1.html#D16">CLVECT_1:def 16</a></i>;<br/>
<a NAME="E2:55_3_1_2"/>
<span class="p1"><a href="clvect_1.html#K5">||.</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="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:55_3_1_2"/>
<span class="p1"><a href="clvect_1.html#K10">||.</a><span class="default"><font color="Maroon">xseq</font></span><a href="clvect_1.html#K10">.||</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"><a href="clvect_1.html#K10">||.</a><span class="default"><font color="Maroon">vseq</font></span><a href="clvect_1.html#K10">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="clvect_1.html#D16">CLVECT_1:def 16</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E7:55_3_1"/>
<span class="p1"><a href="clvect_1.html#K5">||.</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="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="seq_2.html#K2">lim</a> <span class="p1"><a href="clvect_1.html#K10">||.</a><span class="default"><font color="Maroon">vseq</font></span><a href="clvect_1.html#K10">.||</a></span>
 <b>by </b><i>, , , <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="clvect_1.html#K10">||.</a><span class="default"><font color="Maroon">vseq</font></span><a href="clvect_1.html#K10">.||</a></span>; <i><font color="Red">:: according to </font></i><a class="ref" href="csspace4.html#D4" target="_self">CSSPACE4:def 4</a><br/>

<a NAME="E2:55_3"/>
<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="55_3_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_3_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="E1:55_3_2_1"/>
<span class="p1"><a href="clvect_1.html#K5">||.</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="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="clvect_1.html#T106">CLVECT_1:106</a></i>;<br/><b>hence </b><a NAME="E2:55_3_2_1"/>
<span class="p1"><a href="clvect_1.html#K10">||.</a><span class="default"><font color="Maroon">vseq</font></span><a href="clvect_1.html#K10">.||</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="clvect_1.html#D16">CLVECT_1:def 16</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:55_3_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="E3:55_3"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">K</font> &amp; ( for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> holds  <span class="p1"><a href="clvect_1.html#K5">||.</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="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">K</font> ) )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E9:55"/><i><font color="Green">E113</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="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> holds  <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace4.html#K6" 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="clvect_1.html#K5">.||</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="55_4"><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="E1:55_4"/><i><font color="Green">E114</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="E3:55_4"/><i><font color="Green">E115</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="clvect_1.html#K5">||.</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="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font>
 <b>by </b><i>, , <a class="ref" href="csspace3.html#T10">CSSPACE3:10</a></i>;<br/>
<div><i><font color="Green">E116</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="55_4_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="E1:55_4_1"/><i><font color="Green">E117</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="55_4_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</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="E2:55_4_1_1"/><i><font color="Green">E118</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="csspace4.html#K6" 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="clvect_1.html#V6">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="clvect_1.html#K11">lim</a> <font color="Maroon">xseq</font> )
 <b>by </b><i/>;<br/><a NAME="E3:55_4_1_1"/><i><font color="Green">E122</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="clvect_1.html#K5">||.</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="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><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="clvect_1.html#K5">.||</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_4_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="E1:55_4_1_1_1"/>
<font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font> is  <a href="csspace4.html#V1" target="_self">bounded</a> <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">X</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>
 
<b>by </b><i/>;<br/>
<a NAME="E2:55_4_1_1_1"/><b>then </b><i><font color="Green">E123</font></i>: 
 <a href="csspace4.html#K6" 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/>;<br/>
<a NAME="E3:55_4_1_1_1"/>
<font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font> is  <a href="csspace4.html#V1" target="_self">bounded</a> <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">X</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>
 
<b>by </b><i/>;<br/>
<a NAME="E4:55_4_1_1_1"/><b>then </b><i><font color="Green">E124</font></i>: 
 <a href="csspace4.html#K6" 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/>;<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="csspace4.html#V1" target="_self">bounded</a> <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">X</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font> <b>by </b><i/>;<br/>
<a NAME="E6:55_4_1_1_1"/><i><font color="Green">E125</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="csspace4.html#K6" 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/>
<a NAME="E7:55_4_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="csspace4.html#K6" 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/>;<br/>
<a NAME="E8:55_4_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>, , <a class="txt" href="csspace4.html#E6"><i><font color="Green">Lemma58</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E9:55_4_1_1_1"/>
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">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="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><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="clvect_1.html#K5">.||</a></span>
 <b>by </b><i><a class="txt" href="csspace4.html#E2:55"><i><font color="Green">E33</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E4:55_4_1_1"/><i><font color="Green">E126</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="clvect_1.html#K5">||.</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="clvect_1.html#K5">.||</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="55_4_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="E1:55_4_1_1_2"/><i><font color="Green">E127</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="E2:55_4_1_1_2"/><i><font color="Green">E128</font></i>: 
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><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="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><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="clvect_1.html#K5">.||</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E3:55_4_1_1_2"/>
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">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="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font>
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E4:55_4_1_1_2"/>
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">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="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="txt" href="csspace4.html#E6"><i><font color="Green">Lemma58</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>


</div><b>end;</b></div><a NAME="E5:55_4_1_1"/>
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">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="clvect_1.html#K5">.||</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="55_4_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="clvect_1.html#K5">||.</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="clvect_1.html#K5">.||</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="E2:55_4_1_1_3"/><i><font color="Green">E129</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="E3:55_4_1_1_3"/><i><font color="Green">E130</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="clvect_1.html#K5">||.</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="clvect_1.html#K5">.||</a></span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_4_1_1_3_1"><b>proof </b></a><div class="add">

<a NAME="E1:55_4_1_1_3_1"/><i><font color="Green">E131</font></i>: 
<font color="Maroon">xseq</font> <a href="clvect_1.html#K8">-</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="clvect_1.html#V6">convergent</a>
 
<b>by </b><i>, <a class="ref" href="clvect_1.html#T117">CLVECT_1:117</a></i>;<br/>
<a NAME="E2:55_4_1_1_3_1"/><i><font color="Green">E132</font></i>: 
 <a href="clvect_1.html#K11">lim</a> <span class="p1">(<span class="default"><font color="Maroon">xseq</font> <a href="clvect_1.html#K8">-</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="ref" href="clvect_1.html#T123">CLVECT_1:123</a></i>;<br/>
<a NAME="E3:55_4_1_1_3_1"/>
<font color="Maroon">rseq</font> <a href="funct_2.html#R2">=</a> <span class="p1"><a href="clvect_1.html#K10">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">xseq</font> <a href="clvect_1.html#K8">-</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="clvect_1.html#K10">.||</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_4_1_1_3_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_4_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="E1:55_4_1_1_3_1_1_1"/><i><font color="Green">E135</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="clvect_1.html#K5">||.</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="clvect_1.html#K5">.||</a></span>
<b>by </b><i/>
<br/>.= 
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">xseq</font> <a href="clvect_1.html#K8">-</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="clvect_1.html#K5">.||</a></span>
<b>by </b><i><a class="ref" href="clvect_1.html#D13">CLVECT_1:def 13</a></i>
<br/>.= 
<span class="p1"><a href="clvect_1.html#K10">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">xseq</font> <a href="clvect_1.html#K8">-</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="clvect_1.html#K10">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="clvect_1.html#D16">CLVECT_1:def 16</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:55_4_1_1_3_1_1"/>
<font color="Maroon">rseq</font> <a href="funct_2.html#R2">=</a> <span class="p1"><a href="clvect_1.html#K10">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">xseq</font> <a href="clvect_1.html#K8">-</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="clvect_1.html#K10">.||</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="E4:55_4_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="clvect_1.html#K5">||.</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="clvect_1.html#K5">.||</a></span> )
 <b>by </b><i><a class="txt" href="csspace4.html#E7"><i><font color="Green">Lemma59</font></i></a>, <a class="txt" href="csspace4.html#E8"><i><font color="Green">Lemma60</font></i></a>, <a class="ref" href="clopban1.html#T22">CLOPBAN1:22</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:55_4_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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_4_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="E1:55_4_1_1_3_2"/><i><font color="Green">E136</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="clvect_1.html#K5">||.</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="clvect_1.html#K5">.||</a></span>
<b>by </b><i/>
<br/>.= 
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">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="clvect_1.html#K5">.||</a></span>
<b>by </b><i><a class="ref" href="clvect_1.html#T109">CLVECT_1:109</a></i>
;<br/>
<b>hence </b><a NAME="E3:55_4_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>
 <b>by </b><i>, <a class="txt" href="csspace4.html#E7"><i><font color="Green">Lemma59</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:55_4_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>
 
<b>by </b><i><a class="txt" href="csspace4.html#E6"><i><font color="Green">Lemma58</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E6:55_4_1_1_3"/>
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">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="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="txt" href="csspace4.html#E6"><i><font color="Green">Lemma58</font></i></a>, <a class="ref" href="clvect_1.html#T109">CLVECT_1:109</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E6:55_4_1_1"/>
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace4.html#K6" 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="clvect_1.html#K5">.||</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><b>hence </b><a NAME="E3:55_4_1"/>
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> holds  <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace4.html#K6" 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="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">e</font>
 ;<br/></div><b>end;</b></div>
<b>take </b>
<font color="Maroon">k</font>
;<br/>

<b>thus </b><a NAME="E5:55_4"/>
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="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> holds  <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace4.html#K6" 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="clvect_1.html#K5">.||</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>
<b>reconsider </b><font color="Maroon">tseq</font> = <font color="Maroon">tseq</font> as  <a href="csspace4.html#V1" target="_self">bounded</a> <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">X</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font> <b>by </b><i><a class="txt" href="csspace4.html#E5"><i><font color="Green">Lemma48</font></i></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="csspace4.html#K9" target="_self">C_NormSpace_of_BoundedFunctions</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> <b>by </b><i/>;<br/>
<a NAME="E12:55"/><i><font color="Green">E137</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="clvect_1.html#K5">||.</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="clvect_1.html#K5">.||</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="55_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="E1:55_5"/><i><font color="Green">E138</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="E3:55_5"/><i><font color="Green">E139</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="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> holds  <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace4.html#K6" 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="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">e</font>
 <b>by </b><i>, </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_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="E1:55_5_1"/><i><font color="Green">E140</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="csspace4.html#K6" 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="csspace4.html#V1" target="_self">bounded</a> <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">X</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font> <b>by </b><i/>;<br/><div><i><font color="Green">E141</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="55_5_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">t</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>;<br/><a NAME="E1:55_5_1_1"/>
<font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font> is  <a href="csspace4.html#V1" target="_self">bounded</a> <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">X</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>
 <b>by </b><i/>;<br/><a NAME="E2:55_5_1_1"/><b>then </b>
 <a href="csspace4.html#K6" 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/>;<br/><a NAME="E3:55_5_1_1"/><b>then </b>
<span class="p1"><a href="clvect_1.html#K5">||.</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="clvect_1.html#K5">.||</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace4.html#K6" 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="clvect_1.html#K5">.||</a></span>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:55_5_1_1"/>
<span class="p1"><a href="clvect_1.html#K5">||.</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="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">e</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><i><font color="Green">E188</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="55_5_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="E1:55_5_1_2"/><i><font color="Green">E189</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="csspace4.html#K7" target="_self">PreNorms</a> <font color="Maroon">h1</font>
 ;<br/><a NAME="E2:55_5_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="clvect_1.html#K5">||.</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="clvect_1.html#K5">.||</a></span> where <font color="Olive">t</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> : verum </span>}</span> 
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">t</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E4:55_5_1_2"/><i><font color="Green">E190</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span>
 ;<br/><b>thus </b><a NAME="E5:55_5_1_2"/>
<font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">e</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><a NAME="E5:55_5_1"/><i><font color="Green">E191</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="csspace4.html#K7" 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="csspace4.html#K7" 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="E6:55_5_1"/>
<span class="p1">(<span class="default"><a href="csspace4.html#K8" target="_self">ComplexBoundedFunctionsNorm</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="csspace4.html#K7" target="_self">PreNorms</a> <font color="Maroon">h1</font></span>)</span>
 <b>by </b><i><a class="txt" href="csspace4.html#E1:55"><i><font color="Green">E31</font></i></a></i>;<br/><b>hence </b><a NAME="E7:55_5_1"/>
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">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="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">e</font>
 <b>by </b><i>, , <a class="ref" href="clvect_1.html#D9">CLVECT_1:def 9</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:55_5"/>
 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="clvect_1.html#K5">||.</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="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">e</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E13:55"/>
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="clvect_1.html#K5">||.</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="clvect_1.html#K5">.||</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="55_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="E1:55_6"/><i><font color="Green">E192</font></i>: 
<font color="Maroon">e</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<a NAME="E2:55_6"/><i><font color="Green">E193</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="seq_2.html#T3">SEQ_2:3</a></i>;<br/>
<a NAME="E3:55_6"/><i><font color="Green">E194</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="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="E5:55_6"/><i><font color="Green">E195</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="clvect_1.html#K5">||.</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="clvect_1.html#K5">.||</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>, </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_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="E1:55_6_1"/><i><font color="Green">E196</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="E2:55_6_1"/>
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">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="clvect_1.html#K5">.||</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>, </i>;<br/><b>hence </b><a NAME="E3:55_6_1"/>
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">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="clvect_1.html#K5">.||</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>
<b>hence </b><a NAME="E7:55_6"/>
 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="clvect_1.html#K5">||.</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="clvect_1.html#K5">.||</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="E14:55"/>
<font color="Maroon">vseq</font> is <a href="clvect_1.html#V6">convergent</a>
 <b>by </b><i><a class="ref" href="clvect_1.html#D15">CLVECT_1:def 15</a></i>;<br/>


</div>
