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

<b>let </b><font color="Maroon" title="c1">X</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">Y</font> be   <a href="clvect_1.html#NM5" title="CLVECT_1:NM.5">ComplexNormSpace</a>;<br/>



<b>assume </b><a NAME="E1:55"/><i><font color="Green" title="E43">A1</font></i>: 
<font color="Maroon" title="c2">Y</font> is <a href="clopban1.html#V4" title="CLOPBAN1:attr.4">complete</a>
 ;<br/>

<b>let </b><font color="Maroon" title="c3">vseq</font> be   <a href="normsp_1.html#NM2" title="NORMSP_1:NM.2">sequence</a> of <span class="p1">(<span class="default"><a href="csspace4.html#K9" title="CSSPACE4:func.9">C_NormSpace_of_BoundedFunctions</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span>;<br/>

<b>assume </b><a NAME="E2:55"/><i><font color="Green" title="E44">A2</font></i>: 
<font color="Maroon" title="c3">vseq</font> is <a href="csspace3.html#V1" title="CSSPACE3:attr.1">CCauchy</a>
 ;<br/>

<a NAME="E3:55"/><i><font color="Green" title="E45">A3</font></i>: 
<span class="p1"><a href="clvect_1.html#K10" title="CLVECT_1:func.10">||.</a><span class="default"><font color="Maroon" title="c3">vseq</font></span><a href="clvect_1.html#K10" title="CLVECT_1:func.10">.||</a></span> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">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" title="c4">e1</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/><b>assume </b><a NAME="E1:55_1_1"/><i><font color="Green" title="E46">A4</font></i>: 
<font color="Maroon" title="c4">e1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 ;<br/><b>reconsider </b><font color="Maroon" title="c5">e</font> = <font color="Maroon" title="c4">e1</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/><b>consider </b><font color="Maroon" title="c6">k</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E4:55_1_1"/><i><font color="Green" title="E47">A5</font></i>: 
for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c6">k</font> &amp; <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c6">k</font> holds <br/><span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">vseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">vseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b2">m</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">e</font>
 <b>by </b><i><a class="txt" href="csspace4.html#E2:55"><i><font color="Green" title="E44">A2</font></i></a>, <a class="txt" href="csspace4.html#E1:55_1_1"><i><font color="Green" title="E46">A4</font></i></a>, <a class="ref" href="csspace3.html#T10" title="CSSPACE3:th.10">CSSPACE3:10</a></i>;<br/><div><i><font color="Green" title="E48">A6</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" title="c7">m</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:55_1_1_1"/><i><font color="Green" title="E49">A7</font></i>: 
<font color="Maroon" title="c7">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c6">k</font>
 ;<br/><a NAME="E2:55_1_1_1"/><i><font color="Green" title="E50">A8</font></i>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">vseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c7">m</font></span>)</span></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">vseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">vseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">vseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span>
 <b>by </b><i><a class="ref" href="clvect_1.html#T111" title="CLVECT_1:th.111">CLVECT_1:111</a></i>;<br/><a NAME="E3:55_1_1_1"/><i><font color="Green" title="E51">A9</font></i>: 
<span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">vseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="clvect_1.html#K10" title="CLVECT_1:func.10">||.</a><span class="default"><font color="Maroon" title="c3">vseq</font></span><a href="clvect_1.html#K10" title="CLVECT_1:func.10">.||</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">k</font>
 <b>by </b><i><a class="ref" href="clvect_1.html#D16" title="CLVECT_1:def.16">CLVECT_1:def 16</a></i>;<br/><a NAME="E4:55_1_1_1"/><i><font color="Green" title="E52">A10</font></i>: 
<span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">vseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c7">m</font></span>)</span></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="clvect_1.html#K10" title="CLVECT_1:func.10">||.</a><span class="default"><font color="Maroon" title="c3">vseq</font></span><a href="clvect_1.html#K10" title="CLVECT_1:func.10">.||</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">m</font>
 <b>by </b><i><a class="ref" href="clvect_1.html#D16" title="CLVECT_1:def.16">CLVECT_1:def 16</a></i>;<br/><a NAME="E5:55_1_1_1"/>
<span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">vseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">vseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">e</font>
 <b>by </b><i><a class="txt" href="csspace4.html#E4:55_1_1"><i><font color="Green" title="E47">A5</font></i></a>, <a class="txt" href="csspace4.html#E1:55_1_1_1"><i><font color="Green" title="E49">A7</font></i></a></i>;<br/><b>hence </b><a NAME="E6:55_1_1_1"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="clvect_1.html#K10" title="CLVECT_1:func.10">||.</a><span class="default"><font color="Maroon" title="c3">vseq</font></span><a href="clvect_1.html#K10" title="CLVECT_1:func.10">.||</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3"><a href="clvect_1.html#K10" title="CLVECT_1:func.10">||.</a><span class="default"><font color="Maroon" title="c3">vseq</font></span><a href="clvect_1.html#K10" title="CLVECT_1:func.10">.||</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">e1</font>
 <b>by </b><i><a class="txt" href="csspace4.html#E2:55_1_1_1"><i><font color="Green" title="E50">A8</font></i></a>, <a class="txt" href="csspace4.html#E3:55_1_1_1"><i><font color="Green" title="E51">A9</font></i></a>, <a class="txt" href="csspace4.html#E4:55_1_1_1"><i><font color="Green" title="E52">A10</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div><b>take </b><font color="Maroon" title="c7">k</font> = <font color="Maroon" title="c6">k</font>;<br/><b>thus </b><a NAME="E6:55_1_1"/>
for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">k</font> holds <br/> <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="clvect_1.html#K10" title="CLVECT_1:func.10">||.</a><span class="default"><font color="Maroon" title="c3">vseq</font></span><a href="clvect_1.html#K10" title="CLVECT_1:func.10">.||</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3"><a href="clvect_1.html#K10" title="CLVECT_1:func.10">||.</a><span class="default"><font color="Maroon" title="c3">vseq</font></span><a href="clvect_1.html#K10" title="CLVECT_1:func.10">.||</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">k</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">e1</font>
 <b>by </b><i><a class="txt" href="csspace4.html#E5:55_1_1"><i><font color="Green" title="E48">A6</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" title="CLVECT_1:func.10">||.</a><span class="default"><font color="Maroon" title="c3">vseq</font></span><a href="clvect_1.html#K10" title="CLVECT_1:func.10">.||</a></span> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 <b>by </b><i><a class="ref" href="seq_4.html#T58" title="SEQ_4:th.58">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" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">xseq</font> being  <a href="normsp_1.html#NM2" title="NORMSP_1:NM.2">sequence</a> of <font color="Maroon" title="c2">Y</font> st <br/>( ( for <font color="Olive" title="b2">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Olive" title="b1">xseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="csspace4.html#K6" title="CSSPACE4:func.6">modetrans</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">vseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b2">n</font></span>)</span>,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 ) &amp; <font color="Olive" title="b1">xseq</font> is <a href="clvect_1.html#V6" title="CLVECT_1:attr.6">convergent</a> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="clvect_1.html#K11" title="CLVECT_1:func.11">lim</a> <font color="Olive" title="b1">xseq</font> );<br/>
<a NAME="E4:55"/><i><font color="Green" title="E46">A11</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>  ex <font color="Olive" title="b2">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">Y</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">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" title="c4">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>;<br/>

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font> = <span class="p1">(<span class="default"><a href="csspace4.html#K6" title="CSSPACE4:func.6">modetrans</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">vseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> $1</span>)</span>,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font>;<br/>
<b>consider </b><font color="Maroon" title="c5">xseq</font> being   <a href="normsp_1.html#NM2" title="NORMSP_1:NM.2">sequence</a> of <font color="Maroon" title="c2">Y</font><b> such that </b><br/><a NAME="E2:55_2"/><i><font color="Green" title="E47">A12</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c5">xseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">n</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S4" title="FUNCT_2:sch.4">FUNCT_2:sch 4</a>();<br/></i>
<a NAME="E3:55_2"/><i><font color="Green" title="E48">A13</font></i>: 
for <font color="Olive" title="b1">m</font>, <font color="Olive" title="b2">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">xseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">xseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b2">k</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">vseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">vseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b2">k</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</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" title="c6">m</font>, <font color="Maroon" title="c7">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<a NAME="E1:55_2_1"/>
<font color="Maroon" title="c3">vseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">m</font> is  <a href="csspace4.html#V1" title="CSSPACE4:attr.1">bounded</a> <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font>
 
<b>by </b><i><a class="ref" href="csspace4.html#D5" target="_self" title="CSSPACE4:def.5">Def5</a></i>;<br/>
<a NAME="E2:55_2_1"/><b>then </b><i><font color="Green" title="E49">A14</font></i>: 
 <a href="csspace4.html#K6" title="CSSPACE4:func.6">modetrans</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">vseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">vseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">m</font>
 
<b>by </b><i><a class="ref" href="csspace4.html#T17" target="_self" title="CSSPACE4:th.17">Th17</a></i>;<br/>
<a NAME="E3:55_2_1"/>
<font color="Maroon" title="c3">vseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c7">k</font> is  <a href="csspace4.html#V1" title="CSSPACE4:attr.1">bounded</a> <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font>
 
<b>by </b><i><a class="ref" href="csspace4.html#D5" target="_self" title="CSSPACE4:def.5">Def5</a></i>;<br/>
<a NAME="E4:55_2_1"/><b>then </b><i><font color="Green" title="E50">A15</font></i>: 
 <a href="csspace4.html#K6" title="CSSPACE4:func.6">modetrans</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">vseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c7">k</font></span>)</span>,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">vseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c7">k</font>
 
<b>by </b><i><a class="ref" href="csspace4.html#T17" target="_self" title="CSSPACE4:th.17">Th17</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">h1</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c3">vseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">m</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">vseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c7">k</font></span>)</span> as  <a href="csspace4.html#V1" title="CSSPACE4:attr.1">bounded</a> <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font> <b>by </b><i><a class="ref" href="csspace4.html#D5" target="_self" title="CSSPACE4:def.5">Def5</a></i>;<br/>
<a NAME="E6:55_2_1"/><i><font color="Green" title="E51">A16</font></i>: 
<font color="Maroon" title="c5">xseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="csspace4.html#K6" title="CSSPACE4:func.6">modetrans</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">vseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font>
 
<b>by </b><i><a class="txt" href="csspace4.html#E2:55_2"><i><font color="Green" title="E47">A12</font></i></a></i>;<br/>
<a NAME="E7:55_2_1"/>
<font color="Maroon" title="c5">xseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="csspace4.html#K6" title="CSSPACE4:func.6">modetrans</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">vseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c7">k</font></span>)</span>,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font>
 
<b>by </b><i><a class="txt" href="csspace4.html#E2:55_2"><i><font color="Green" title="E47">A12</font></i></a></i>;<br/>
<a NAME="E8:55_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">xseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">m</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">xseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c7">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">h1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font>
 
<b>by </b><i><a class="txt" href="csspace4.html#E2:55_2_1"><i><font color="Green" title="E49">A14</font></i></a>, <a class="txt" href="csspace4.html#E4:55_2_1"><i><font color="Green" title="E50">A15</font></i></a>, <a class="txt" href="csspace4.html#E6:55_2_1"><i><font color="Green" title="E51">A16</font></i></a>, <a class="ref" href="csspace4.html#T28" target="_self" title="CSSPACE4:th.28">Th28</a></i>;<br/>
<b>hence </b><a NAME="E9:55_2_1"/>
<span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">xseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">m</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">xseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c7">k</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">vseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">m</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">vseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c7">k</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span>
 <b>by </b><i><a class="ref" href="csspace4.html#T20" target="_self" title="CSSPACE4:th.20">Th20</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:55_2"/><i><font color="Green" title="E49">A17</font></i>: 
<font color="Maroon" title="c5">xseq</font> is <a href="clvect_1.html#V6" title="CLVECT_1:attr.6">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" title="c6">e</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>assume </b><a NAME="E1:55_2_2_1"/><i><font color="Green" title="E50">A18</font></i>: 
<font color="Maroon" title="c6">e</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 ;<br/><b>thus </b><a NAME="E2:55_2_2_1"/>
 ex <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>for <font color="Olive" title="b2">n</font>, <font color="Olive" title="b3">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b3">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">k</font> holds <br/><span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">xseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b2">n</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">xseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b3">m</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c6">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" title="c7">k</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E2:55_2_2_1_1"/><i><font color="Green" title="E51">A19</font></i>: 
for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">k</font> &amp; <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">k</font> holds <br/><span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">vseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">vseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b2">m</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c6">e</font>
 <b>by </b><i><a class="txt" href="csspace4.html#E2:55"><i><font color="Green" title="E44">A2</font></i></a>, <a class="txt" href="csspace4.html#E1:55_2_2_1"><i><font color="Green" title="E50">A18</font></i></a>, <a class="ref" href="csspace3.html#T10" title="CSSPACE3:th.10">CSSPACE3:10</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c7">k</font>
;<br/>

<b>thus </b><a NAME="E3:55_2_2_1_1"/>
for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">k</font> &amp; <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">k</font> holds <br/><span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">xseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">xseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b2">m</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c6">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" title="c8">n</font>, <font color="Maroon" title="c9">m</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:55_2_2_1_1_1"/><i><font color="Green" title="E52">A20</font></i>: 
( <font color="Maroon" title="c8">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">k</font> &amp; <font color="Maroon" title="c9">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">k</font> )
 ;<br/>

<a NAME="E2:55_2_2_1_1_1"/><i><font color="Green" title="E53">A21</font></i>: 
<span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">xseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">n</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">xseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c9">m</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">vseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">n</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">vseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c9">m</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span>
 
<b>by </b><i><a class="txt" href="csspace4.html#E3:55_2"><i><font color="Green" title="E48">A13</font></i></a></i>;<br/>
<a NAME="E3:55_2_2_1_1_1"/>
<span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">vseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">n</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">vseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c9">m</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c6">e</font>
 
<b>by </b><i><a class="txt" href="csspace4.html#E2:55_2_2_1_1"><i><font color="Green" title="E51">A19</font></i></a>, <a class="txt" href="csspace4.html#E1:55_2_2_1_1_1"><i><font color="Green" title="E52">A20</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:55_2_2_1_1_1"/>
<span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">xseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">n</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">xseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c9">m</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c6">e</font>
 <b>by </b><i><a class="txt" href="csspace4.html#E2:55_2_2_1_1_1"><i><font color="Green" title="E53">A21</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">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" title="c5">xseq</font> is <a href="csspace3.html#V1" title="CSSPACE3:attr.1">CCauchy</a>
 
<b>by </b><i><a class="ref" href="csspace3.html#T10" title="CSSPACE3:th.10">CSSPACE3:10</a></i>;<br/>
<b>hence </b><a NAME="E3:55_2_2"/>
<font color="Maroon" title="c5">xseq</font> is <a href="clvect_1.html#V6" title="CLVECT_1:attr.6">convergent</a>
 <b>by </b><i><a class="txt" href="csspace4.html#E1:55"><i><font color="Green" title="E43">A1</font></i></a>, <a class="ref" href="clopban1.html#D14" title="CLOPBAN1:def.14">CLOPBAN1:def 14</a></i>;<br/>


</div><b>end;</b></div>
<b>take </b><font color="Maroon" title="c6">y</font> =  <a href="clvect_1.html#K11" title="CLVECT_1:func.11">lim</a> <font color="Maroon" title="c5">xseq</font>;<br/>

<b>thus </b><a NAME="E5:55_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c6">y</font>]
 <b>by </b><i><a class="txt" href="csspace4.html#E2:55_2"><i><font color="Green" title="E47">A12</font></i></a>, <a class="txt" href="csspace4.html#E4:55_2"><i><font color="Green" title="E49">A17</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c4">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font><b> such that </b><br/><a NAME="E6:55"/><i><font color="Green" title="E47">A22</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="csspace4.html#E4:55"><i><font color="Green" title="E46">A11</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon" title="c5">tseq</font> = <font color="Maroon" title="c4">f</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font> ;<br/>
<a NAME="E8:55"/><i><font color="Green" title="E48">A23</font></i>: 
<font color="Maroon" title="c5">tseq</font> is <a href="csspace4.html#V1" title="CSSPACE4:attr.1">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" title="E49">A24</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" title="c6">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>;<br/><b>consider </b><font color="Maroon" title="c7">xseq</font> being   <a href="normsp_1.html#NM2" title="NORMSP_1:NM.2">sequence</a> of <font color="Maroon" title="c2">Y</font><b> such that </b><br/><a NAME="E2:55_3_1"/><i><font color="Green" title="E50">A25</font></i>: 
( ( for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c7">xseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="csspace4.html#K6" title="CSSPACE4:func.6">modetrans</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">vseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font> ) &amp; <font color="Maroon" title="c7">xseq</font> is <a href="clvect_1.html#V6" title="CLVECT_1:attr.6">convergent</a> &amp; <font color="Maroon" title="c5">tseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="clvect_1.html#K11" title="CLVECT_1:func.11">lim</a> <font color="Maroon" title="c7">xseq</font> )
 <b>by </b><i><a class="txt" href="csspace4.html#E6:55"><i><font color="Green" title="E47">A22</font></i></a></i>;<br/><a NAME="E3:55_3_1"/><i><font color="Green" title="E51">A26</font></i>: 
for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">xseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b1">m</font></span>)</span></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">vseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b1">m</font></span>)</span></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</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" title="c8">m</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<a NAME="E1:55_3_1_1"/>
<font color="Maroon" title="c3">vseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">m</font> is  <a href="csspace4.html#V1" title="CSSPACE4:attr.1">bounded</a> <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font>
 
<b>by </b><i><a class="ref" href="csspace4.html#D5" target="_self" title="CSSPACE4:def.5">Def5</a></i>;<br/>
<a NAME="E2:55_3_1_1"/><b>then </b><i><font color="Green" title="E52">A27</font></i>: 
 <a href="csspace4.html#K6" title="CSSPACE4:func.6">modetrans</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">vseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">m</font></span>)</span>,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">vseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">m</font>
 
<b>by </b><i><a class="ref" href="csspace4.html#T17" target="_self" title="CSSPACE4:th.17">Th17</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">h1</font> = <font color="Maroon" title="c3">vseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">m</font> as  <a href="csspace4.html#V1" title="CSSPACE4:attr.1">bounded</a> <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font> <b>by </b><i><a class="ref" href="csspace4.html#D5" target="_self" title="CSSPACE4:def.5">Def5</a></i>;<br/>
<a NAME="E4:55_3_1_1"/>
<font color="Maroon" title="c7">xseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="csspace4.html#K6" title="CSSPACE4:func.6">modetrans</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">vseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">m</font></span>)</span>,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font>
 
<b>by </b><i><a class="txt" href="csspace4.html#E2:55_3_1"><i><font color="Green" title="E50">A25</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:55_3_1_1"/>
<span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">xseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">m</font></span>)</span></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">vseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">m</font></span>)</span></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span>
 <b>by </b><i><a class="txt" href="csspace4.html#E2:55_3_1_1"><i><font color="Green" title="E52">A27</font></i></a>, <a class="ref" href="csspace4.html#T20" target="_self" title="CSSPACE4:th.20">Th20</a></i>;<br/>


</div><b>end;</b></div><a NAME="E4:55_3_1"/><i><font color="Green" title="E52">A28</font></i>: 
<span class="p1"><a href="clvect_1.html#K10" title="CLVECT_1:func.10">||.</a><span class="default"><font color="Maroon" title="c7">xseq</font></span><a href="clvect_1.html#K10" title="CLVECT_1:func.10">.||</a></span> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 <b>by </b><i><a class="txt" href="csspace4.html#E2:55_3_1"><i><font color="Green" title="E50">A25</font></i></a>, <a class="ref" href="clopban1.html#T45" title="CLOPBAN1:th.45">CLOPBAN1:45</a></i>;<br/><a NAME="E5:55_3_1"/><i><font color="Green" title="E53">A29</font></i>: 
<span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">tseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1"><a href="clvect_1.html#K10" title="CLVECT_1:func.10">||.</a><span class="default"><font color="Maroon" title="c7">xseq</font></span><a href="clvect_1.html#K10" title="CLVECT_1:func.10">.||</a></span>
 <b>by </b><i><a class="txt" href="csspace4.html#E2:55_3_1"><i><font color="Green" title="E50">A25</font></i></a>, <a class="ref" href="clopban1.html#T45" title="CLOPBAN1:th.45">CLOPBAN1:45</a></i>;<br/><a NAME="E6:55_3_1"/>
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <span class="p1"><a href="clvect_1.html#K10" title="CLVECT_1:func.10">||.</a><span class="default"><font color="Maroon" title="c7">xseq</font></span><a href="clvect_1.html#K10" title="CLVECT_1:func.10">.||</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="clvect_1.html#K10" title="CLVECT_1:func.10">||.</a><span class="default"><font color="Maroon" title="c3">vseq</font></span><a href="clvect_1.html#K10" title="CLVECT_1:func.10">.||</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">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" title="c8">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<a NAME="E1:55_3_1_2"/><i><font color="Green" title="E54">A30</font></i>: 
<span class="p1"><a href="clvect_1.html#K10" title="CLVECT_1:func.10">||.</a><span class="default"><font color="Maroon" title="c7">xseq</font></span><a href="clvect_1.html#K10" title="CLVECT_1:func.10">.||</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">xseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span>
 
<b>by </b><i><a class="ref" href="clvect_1.html#D16" title="CLVECT_1:def.16">CLVECT_1:def 16</a></i>;<br/>
<a NAME="E2:55_3_1_2"/>
<span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">xseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">vseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span>
 
<b>by </b><i><a class="txt" href="csspace4.html#E3:55_3_1"><i><font color="Green" title="E51">A26</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:55_3_1_2"/>
<span class="p1"><a href="clvect_1.html#K10" title="CLVECT_1:func.10">||.</a><span class="default"><font color="Maroon" title="c7">xseq</font></span><a href="clvect_1.html#K10" title="CLVECT_1:func.10">.||</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="clvect_1.html#K10" title="CLVECT_1:func.10">||.</a><span class="default"><font color="Maroon" title="c3">vseq</font></span><a href="clvect_1.html#K10" title="CLVECT_1:func.10">.||</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">n</font>
 <b>by </b><i><a class="txt" href="csspace4.html#E1:55_3_1_2"><i><font color="Green" title="E54">A30</font></i></a>, <a class="ref" href="clvect_1.html#D16" title="CLVECT_1:def.16">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" title="CLVECT_1:func.5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">tseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1"><a href="clvect_1.html#K10" title="CLVECT_1:func.10">||.</a><span class="default"><font color="Maroon" title="c3">vseq</font></span><a href="clvect_1.html#K10" title="CLVECT_1:func.10">.||</a></span>
 <b>by </b><i><a class="txt" href="csspace4.html#E3:55"><i><font color="Green" title="E45">A3</font></i></a>, <a class="txt" href="csspace4.html#E4:55_3_1"><i><font color="Green" title="E52">A28</font></i></a>, <a class="txt" href="csspace4.html#E5:55_3_1"><i><font color="Green" title="E53">A29</font></i></a>, <a class="ref" href="seq_2.html#T32" title="SEQ_2:th.32">SEQ_2:32</a></i>;<br/></div><b>end;</b></div>
<b>take </b><font color="Maroon" title="c6">K</font> =  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1"><a href="clvect_1.html#K10" title="CLVECT_1:func.10">||.</a><span class="default"><font color="Maroon" title="c3">vseq</font></span><a href="clvect_1.html#K10" title="CLVECT_1:func.10">.||</a></span>; <i><font color="Red">:: according to </font></i><a class="ref" href="csspace4.html#D4" target="_self" title="CSSPACE4:def.4">CSSPACE4:def 4</a><br/>

<a NAME="E2:55_3"/>
<font color="Maroon" title="c6">K</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&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" title="c7">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><a NAME="E1:55_3_2_1"/>
<span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">vseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c7">n</font></span>)</span></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="clvect_1.html#T106" title="CLVECT_1:th.106">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" title="CLVECT_1:func.10">||.</a><span class="default"><font color="Maroon" title="c3">vseq</font></span><a href="clvect_1.html#K10" title="CLVECT_1:func.10">.||</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="clvect_1.html#D16" title="CLVECT_1:def.16">CLVECT_1:def 16</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:55_3_2"/>
<font color="Maroon" title="c6">K</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0
 <b>by </b><i><a class="txt" href="csspace4.html#E3:55"><i><font color="Green" title="E45">A3</font></i></a>, <a class="ref" href="seq_2.html#T31" title="SEQ_2:th.31">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" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">K</font> &amp; ( for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> holds  <span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">tseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">K</font> ) )
 <b>by </b><i><a class="txt" href="csspace4.html#E1:55_3"><i><font color="Green" title="E49">A24</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:55"/><i><font color="Green" title="E49">A31</font></i>: 
for <font color="Olive" title="b1">e</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">e</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 holds <br/> ex <font color="Olive" title="b2">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>for <font color="Olive" title="b3">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b3">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b2">k</font> holds <br/>for <font color="Olive" title="b4">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> holds  <span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</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" title="CSSPACE4:func.6">modetrans</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">vseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b3">n</font></span>)</span>,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">x</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">tseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">x</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">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" title="c6">e</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>

<b>assume </b><a NAME="E1:55_4"/><i><font color="Green" title="E50">A32</font></i>: 
<font color="Maroon" title="c6">e</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 ;<br/>

<b>consider </b><font color="Maroon" title="c7">k</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:55_4"/><i><font color="Green" title="E51">A33</font></i>: 
for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">k</font> &amp; <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">k</font> holds <br/><span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">vseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">vseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b2">m</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c6">e</font>
 <b>by </b><i><a class="txt" href="csspace4.html#E2:55"><i><font color="Green" title="E44">A2</font></i></a>, <a class="txt" href="csspace4.html#E1:55_4"><i><font color="Green" title="E50">A32</font></i></a>, <a class="ref" href="csspace3.html#T10" title="CSSPACE3:th.10">CSSPACE3:10</a></i>;<br/>
<div><i><font color="Green" title="E52">A34</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" title="c8">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:55_4_1"/><i><font color="Green" title="E53">A35</font></i>: 
<font color="Maroon" title="c8">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">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" title="c9">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>;<br/><b>consider </b><font color="Maroon" title="c10">xseq</font> being   <a href="normsp_1.html#NM2" title="NORMSP_1:NM.2">sequence</a> of <font color="Maroon" title="c2">Y</font><b> such that </b><br/><a NAME="E2:55_4_1_1"/><i><font color="Green" title="E54">A36</font></i>: 
( ( for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c10">xseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="csspace4.html#K6" title="CSSPACE4:func.6">modetrans</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">vseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">x</font> ) &amp; <font color="Maroon" title="c10">xseq</font> is <a href="clvect_1.html#V6" title="CLVECT_1:attr.6">convergent</a> &amp; <font color="Maroon" title="c5">tseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="clvect_1.html#K11" title="CLVECT_1:func.11">lim</a> <font color="Maroon" title="c10">xseq</font> )
 <b>by </b><i><a class="txt" href="csspace4.html#E6:55"><i><font color="Green" title="E47">A22</font></i></a></i>;<br/><a NAME="E3:55_4_1_1"/><i><font color="Green" title="E55">A37</font></i>: 
for <font color="Olive" title="b1">m</font>, <font color="Olive" title="b2">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">xseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">xseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b2">k</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">vseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">vseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b2">k</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</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" title="c11">m</font>, <font color="Maroon" title="c12">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<a NAME="E1:55_4_1_1_1"/>
<font color="Maroon" title="c3">vseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c11">m</font> is  <a href="csspace4.html#V1" title="CSSPACE4:attr.1">bounded</a> <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font>
 
<b>by </b><i><a class="ref" href="csspace4.html#D5" target="_self" title="CSSPACE4:def.5">Def5</a></i>;<br/>
<a NAME="E2:55_4_1_1_1"/><b>then </b><i><font color="Green" title="E56">A38</font></i>: 
 <a href="csspace4.html#K6" title="CSSPACE4:func.6">modetrans</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">vseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c11">m</font></span>)</span>,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">vseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c11">m</font>
 
<b>by </b><i><a class="ref" href="csspace4.html#T17" target="_self" title="CSSPACE4:th.17">Th17</a></i>;<br/>
<a NAME="E3:55_4_1_1_1"/>
<font color="Maroon" title="c3">vseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c12">k</font> is  <a href="csspace4.html#V1" title="CSSPACE4:attr.1">bounded</a> <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font>
 
<b>by </b><i><a class="ref" href="csspace4.html#D5" target="_self" title="CSSPACE4:def.5">Def5</a></i>;<br/>
<a NAME="E4:55_4_1_1_1"/><b>then </b><i><font color="Green" title="E57">A39</font></i>: 
 <a href="csspace4.html#K6" title="CSSPACE4:func.6">modetrans</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">vseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c12">k</font></span>)</span>,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">vseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c12">k</font>
 
<b>by </b><i><a class="ref" href="csspace4.html#T17" target="_self" title="CSSPACE4:th.17">Th17</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">h1</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c3">vseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c11">m</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">vseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c12">k</font></span>)</span> as  <a href="csspace4.html#V1" title="CSSPACE4:attr.1">bounded</a> <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font> <b>by </b><i><a class="ref" href="csspace4.html#D5" target="_self" title="CSSPACE4:def.5">Def5</a></i>;<br/>
<a NAME="E6:55_4_1_1_1"/><i><font color="Green" title="E58">A40</font></i>: 
<font color="Maroon" title="c10">xseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c11">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="csspace4.html#K6" title="CSSPACE4:func.6">modetrans</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">vseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c11">m</font></span>)</span>,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">x</font>
 
<b>by </b><i><a class="txt" href="csspace4.html#E2:55_4_1_1"><i><font color="Green" title="E54">A36</font></i></a></i>;<br/>
<a NAME="E7:55_4_1_1_1"/>
<font color="Maroon" title="c10">xseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c12">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="csspace4.html#K6" title="CSSPACE4:func.6">modetrans</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">vseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c12">k</font></span>)</span>,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">x</font>
 
<b>by </b><i><a class="txt" href="csspace4.html#E2:55_4_1_1"><i><font color="Green" title="E54">A36</font></i></a></i>;<br/>
<a NAME="E8:55_4_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c10">xseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c11">m</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">xseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c12">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">h1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">x</font>
 
<b>by </b><i><a class="txt" href="csspace4.html#E2:55_4_1_1_1"><i><font color="Green" title="E56">A38</font></i></a>, <a class="txt" href="csspace4.html#E4:55_4_1_1_1"><i><font color="Green" title="E57">A39</font></i></a>, <a class="txt" href="csspace4.html#E6:55_4_1_1_1"><i><font color="Green" title="E58">A40</font></i></a>, <a class="ref" href="csspace4.html#T28" target="_self" title="CSSPACE4:th.28">Th28</a></i>;<br/>
<b>hence </b><a NAME="E9:55_4_1_1_1"/>
<span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">xseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c11">m</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">xseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">vseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c11">m</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">vseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span>
 <b>by </b><i><a class="ref" href="csspace4.html#T20" target="_self" title="CSSPACE4:th.20">Th20</a></i>;<br/>


</div><b>end;</b></div><a NAME="E4:55_4_1_1"/><i><font color="Green" title="E56">A41</font></i>: 
for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">k</font> holds <br/><span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">xseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">n</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">xseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b1">m</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">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" title="c11">m</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:55_4_1_1_2"/><i><font color="Green" title="E57">A42</font></i>: 
<font color="Maroon" title="c11">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">k</font>
 ;<br/>

<a NAME="E2:55_4_1_1_2"/><i><font color="Green" title="E58">A43</font></i>: 
<span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">xseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">n</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">xseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c11">m</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">vseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">n</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">vseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c11">m</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span>
 
<b>by </b><i><a class="txt" href="csspace4.html#E3:55_4_1_1"><i><font color="Green" title="E55">A37</font></i></a></i>;<br/>
<a NAME="E3:55_4_1_1_2"/>
<span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">vseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">n</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">vseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c11">m</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c6">e</font>
 
<b>by </b><i><a class="txt" href="csspace4.html#E3:55_4"><i><font color="Green" title="E51">A33</font></i></a>, <a class="txt" href="csspace4.html#E1:55_4_1"><i><font color="Green" title="E53">A35</font></i></a>, <a class="txt" href="csspace4.html#E1:55_4_1_1_2"><i><font color="Green" title="E57">A42</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:55_4_1_1_2"/>
<span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">xseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">n</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">xseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c11">m</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">e</font>
 <b>by </b><i><a class="txt" href="csspace4.html#E2:55_4_1_1_2"><i><font color="Green" title="E58">A43</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">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" title="CLVECT_1:func.5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">xseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">n</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">tseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">x</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">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" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  = <span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">xseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> $1</span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">xseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span>;<br/>
<b>consider </b><font color="Maroon" title="c11">rseq</font> being   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a><b> such that </b><br/><a NAME="E2:55_4_1_1_3"/><i><font color="Green" title="E57">A44</font></i>: 
for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c11">rseq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">m</font>)
 <b>from </b><i><a class="ref" href="seq_1.html#S1" title="SEQ_1:sch.1">SEQ_1:sch 1</a>();<br/></i>
<a NAME="E3:55_4_1_1_3"/><i><font color="Green" title="E58">A45</font></i>: 
( <font color="Maroon" title="c11">rseq</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c11">rseq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">tseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">x</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">xseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</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" title="E59">A46</font></i>: 
<font color="Maroon" title="c10">xseq</font> <a href="clvect_1.html#K8" title="CLVECT_1:func.8">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">xseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">n</font></span>)</span> is <a href="clvect_1.html#V6" title="CLVECT_1:attr.6">convergent</a>
 
<b>by </b><i><a class="txt" href="csspace4.html#E2:55_4_1_1"><i><font color="Green" title="E54">A36</font></i></a>, <a class="ref" href="clvect_1.html#T117" title="CLVECT_1:th.117">CLVECT_1:117</a></i>;<br/>
<a NAME="E2:55_4_1_1_3_1"/><i><font color="Green" title="E60">A47</font></i>: 
 <a href="clvect_1.html#K11" title="CLVECT_1:func.11">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">xseq</font> <a href="clvect_1.html#K8" title="CLVECT_1:func.8">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">xseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">tseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">x</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">xseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="csspace4.html#E2:55_4_1_1"><i><font color="Green" title="E54">A36</font></i></a>, <a class="ref" href="clvect_1.html#T123" title="CLVECT_1:th.123">CLVECT_1:123</a></i>;<br/>
<a NAME="E3:55_4_1_1_3_1"/>
<font color="Maroon" title="c11">rseq</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1"><a href="clvect_1.html#K10" title="CLVECT_1:func.10">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">xseq</font> <a href="clvect_1.html#K8" title="CLVECT_1:func.8">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">xseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span><a href="clvect_1.html#K10" title="CLVECT_1:func.10">.||</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" title="c12">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:55_4_1_1_3_1_1_1"/><i><font color="Green" title="E61">A48</font></i>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 ;<br/><b>reconsider </b><font color="Maroon" title="c13">k</font> = <font color="Maroon" title="c12">x</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="csspace4.html#E1:55_4_1_1_3_1_1_1"><i><font color="Green" title="E61">A48</font></i></a></i>;<br/><b>thus </b><font color="Maroon" title="c11">rseq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c12">x</font> = 
<span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">xseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c13">k</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">xseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span>
<b>by </b><i><a class="txt" href="csspace4.html#E2:55_4_1_1_3"><i><font color="Green" title="E57">A44</font></i></a></i>
<br/>.= 
<span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">xseq</font> <a href="clvect_1.html#K8" title="CLVECT_1:func.8">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c10">xseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c13">k</font></span>)</span></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span>
<b>by </b><i><a class="ref" href="clvect_1.html#D13" title="CLVECT_1:def.13">CLVECT_1:def 13</a></i>
<br/>.= 
<span class="p1"><a href="clvect_1.html#K10" title="CLVECT_1:func.10">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">xseq</font> <a href="clvect_1.html#K8" title="CLVECT_1:func.8">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">xseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span><a href="clvect_1.html#K10" title="CLVECT_1:func.10">.||</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c12">x</font>
<b>by </b><i><a class="ref" href="clvect_1.html#D16" title="CLVECT_1:def.16">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" title="c11">rseq</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1"><a href="clvect_1.html#K10" title="CLVECT_1:func.10">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">xseq</font> <a href="clvect_1.html#K8" title="CLVECT_1:func.8">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">xseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span><a href="clvect_1.html#K10" title="CLVECT_1:func.10">.||</a></span>
 <b>by </b><i><a class="ref" href="seq_1.html#T8" title="SEQ_1:th.8">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" title="c11">rseq</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c11">rseq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">tseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">x</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">xseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span> )
 <b>by </b><i><a class="txt" href="csspace4.html#E1:55_4_1_1_3_1"><i><font color="Green" title="E59">A46</font></i></a>, <a class="txt" href="csspace4.html#E2:55_4_1_1_3_1"><i><font color="Green" title="E60">A47</font></i></a>, <a class="ref" href="clopban1.html#T22" title="CLOPBAN1:th.22">CLOPBAN1:22</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:55_4_1_1_3"/>
for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">k</font> holds <br/><font color="Maroon" title="c11">rseq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">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" title="c12">m</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:55_4_1_1_3_2"/><i><font color="Green" title="E59">A49</font></i>: 
<font color="Maroon" title="c12">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">k</font>
 ;<br/>

<font color="Maroon" title="c11">rseq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c12">m</font> = 
<span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">xseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c12">m</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">xseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span>
<b>by </b><i><a class="txt" href="csspace4.html#E2:55_4_1_1_3"><i><font color="Green" title="E57">A44</font></i></a></i>
<br/>.= 
<span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">xseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">n</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">xseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c12">m</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span>
<b>by </b><i><a class="ref" href="clvect_1.html#T109" title="CLVECT_1:th.109">CLVECT_1:109</a></i>
;<br/>
<b>hence </b><a NAME="E3:55_4_1_1_3_2"/>
<font color="Maroon" title="c11">rseq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c12">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">e</font>
 <b>by </b><i><a class="txt" href="csspace4.html#E4:55_4_1_1"><i><font color="Green" title="E56">A41</font></i></a>, <a class="txt" href="csspace4.html#E1:55_4_1_1_3_2"><i><font color="Green" title="E59">A49</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" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c11">rseq</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">e</font>
 
<b>by </b><i><a class="txt" href="csspace4.html#E3:55_4_1_1_3"><i><font color="Green" title="E58">A45</font></i></a>, <a class="txt" href="csspace4.html#E51"><i><font color="Green" title="E41">Lm13</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" title="CLVECT_1:func.5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">xseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">n</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">tseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">x</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">e</font>
 <b>by </b><i><a class="txt" href="csspace4.html#E3:55_4_1_1_3"><i><font color="Green" title="E58">A45</font></i></a>, <a class="ref" href="clvect_1.html#T109" title="CLVECT_1:th.109">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" title="CLVECT_1:func.5">||.</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" title="CSSPACE4:func.6">modetrans</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">vseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">n</font></span>)</span>,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">x</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">tseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">x</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">e</font>
 <b>by </b><i><a class="txt" href="csspace4.html#E2:55_4_1_1"><i><font color="Green" title="E54">A36</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:55_4_1"/>
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> holds  <span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</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" title="CSSPACE4:func.6">modetrans</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">vseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">n</font></span>)</span>,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">tseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">e</font>
 ;<br/></div><b>end;</b></div>
<b>take </b>
<font color="Maroon" title="c7">k</font>
;<br/>

<b>thus </b><a NAME="E5:55_4"/>
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">k</font> holds <br/>for <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> holds  <span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</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" title="CSSPACE4:func.6">modetrans</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">vseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">x</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">tseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">x</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">e</font>
 <b>by </b><i><a class="txt" href="csspace4.html#E4:55_4"><i><font color="Green" title="E52">A34</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon" title="c6">tseq</font> = <font color="Maroon" title="c5">tseq</font> as  <a href="csspace4.html#V1" title="CSSPACE4:attr.1">bounded</a> <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font> <b>by </b><i><a class="txt" href="csspace4.html#E8:55"><i><font color="Green" title="E48">A23</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">tv</font> = <font color="Maroon" title="c6">tseq</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="csspace4.html#K9" title="CSSPACE4:func.9">C_NormSpace_of_BoundedFunctions</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span> <b>by </b><i><a class="ref" href="csspace4.html#D5" target="_self" title="CSSPACE4:def.5">Def5</a></i>;<br/>
<a NAME="E12:55"/><i><font color="Green" title="E50">A50</font></i>: 
for <font color="Olive" title="b1">e</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">e</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 holds <br/> ex <font color="Olive" title="b2">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>for <font color="Olive" title="b3">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b3">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b2">k</font> holds <br/><span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">vseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b3">n</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c7">tv</font></span>)</span></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">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" title="c8">e</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>

<b>assume </b><a NAME="E1:55_5"/><i><font color="Green" title="E51">A51</font></i>: 
<font color="Maroon" title="c8">e</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 ;<br/>

<b>consider </b><font color="Maroon" title="c9">k</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:55_5"/><i><font color="Green" title="E52">A52</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c9">k</font> holds <br/>for <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> holds  <span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</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" title="CSSPACE4:func.6">modetrans</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">vseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">x</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">tseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">x</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">e</font>
 <b>by </b><i><a class="txt" href="csspace4.html#E9:55"><i><font color="Green" title="E49">A31</font></i></a>, <a class="txt" href="csspace4.html#E1:55_5"><i><font color="Green" title="E51">A51</font></i></a></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" title="c10">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:55_5_1"/><i><font color="Green" title="E53">A53</font></i>: 
<font color="Maroon" title="c10">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c9">k</font>
 ;<br/><b>set </b><font color="Maroon" title="c11">f1</font> =  <a href="csspace4.html#K6" title="CSSPACE4:func.6">modetrans</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">vseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c10">n</font></span>)</span>,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font>;<br/><b>set </b><font color="Maroon" title="c12">g1</font> = <font color="Maroon" title="c6">tseq</font>;<br/><b>reconsider </b><font color="Maroon" title="c13">h1</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c3">vseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c7">tv</font> as  <a href="csspace4.html#V1" title="CSSPACE4:attr.1">bounded</a> <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font> <b>by </b><i><a class="ref" href="csspace4.html#D5" target="_self" title="CSSPACE4:def.5">Def5</a></i>;<br/><div><i><font color="Green" title="E54">A54</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" title="c14">t</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>;<br/><a NAME="E1:55_5_1_1"/>
<font color="Maroon" title="c3">vseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c10">n</font> is  <a href="csspace4.html#V1" title="CSSPACE4:attr.1">bounded</a> <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font>
 <b>by </b><i><a class="ref" href="csspace4.html#D5" target="_self" title="CSSPACE4:def.5">Def5</a></i>;<br/><a NAME="E2:55_5_1_1"/><b>then </b>
 <a href="csspace4.html#K6" title="CSSPACE4:func.6">modetrans</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">vseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c10">n</font></span>)</span>,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">vseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c10">n</font>
 <b>by </b><i><a class="ref" href="csspace4.html#T17" target="_self" title="CSSPACE4:th.17">Th17</a></i>;<br/><a NAME="E3:55_5_1_1"/><b>then </b>
<span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">h1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">t</font></span>)</span></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</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" title="CSSPACE4:func.6">modetrans</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">vseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c10">n</font></span>)</span>,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">t</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">tseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">t</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span>
 <b>by </b><i><a class="ref" href="csspace4.html#T28" target="_self" title="CSSPACE4:th.28">Th28</a></i>;<br/><b>hence </b><a NAME="E4:55_5_1_1"/>
<span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">h1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">t</font></span>)</span></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">e</font>
 <b>by </b><i><a class="txt" href="csspace4.html#E3:55_5"><i><font color="Green" title="E52">A52</font></i></a>, <a class="txt" href="csspace4.html#E1:55_5_1"><i><font color="Green" title="E53">A53</font></i></a></i>;<br/></div><b>end;</b></div><div><i><font color="Green" title="E55">A55</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" title="c14">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>assume </b><a NAME="E1:55_5_1_2"/><i><font color="Green" title="E56">A56</font></i>: 
<font color="Maroon" title="c14">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="csspace4.html#K7" title="CSSPACE4:func.7">PreNorms</a> <font color="Maroon" title="c13">h1</font>
 ;<br/><a NAME="E2:55_5_1_2"/>
<font color="Maroon" title="c14">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">h1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">t</font></span>)</span></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span> where <font color="Olive" title="b1">t</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> : verum </span>}</span> 
 <b>by </b><i><a class="txt" href="csspace4.html#E1:55_5_1_2"><i><font color="Green" title="E56">A56</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c15">t</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E4:55_5_1_2"/><i><font color="Green" title="E57">A57</font></i>: 
<font color="Maroon" title="c14">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">h1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">t</font></span>)</span></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span>
 ;<br/><b>thus </b><a NAME="E5:55_5_1_2"/>
<font color="Maroon" title="c14">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">e</font>
 <b>by </b><i><a class="txt" href="csspace4.html#E3:55_5_1"><i><font color="Green" title="E54">A54</font></i></a>, <a class="txt" href="csspace4.html#E4:55_5_1_2"><i><font color="Green" title="E57">A57</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E5:55_5_1"/><i><font color="Green" title="E56">A58</font></i>: 
( ( for <font color="Olive" title="b1">s</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="csspace4.html#K7" title="CSSPACE4:func.7">PreNorms</a> <font color="Maroon" title="c13">h1</font> holds <br/><font color="Olive" title="b1">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">e</font> ) implies  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="csspace4.html#K7" title="CSSPACE4:func.7">PreNorms</a> <font color="Maroon" title="c13">h1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">e</font> )
 <b>by </b><i><a class="ref" href="seq_4.html#T62" title="SEQ_4:th.62">SEQ_4:62</a></i>;<br/><a NAME="E6:55_5_1"/>
<span class="p1">(<span class="default"><a href="csspace4.html#K8" title="CSSPACE4:func.8">ComplexBoundedFunctionsNorm</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">vseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c7">tv</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="csspace4.html#K7" title="CSSPACE4:func.7">PreNorms</a> <font color="Maroon" title="c13">h1</font></span>)</span>
 <b>by </b><i><a class="ref" href="csspace4.html#T18" target="_self" title="CSSPACE4:th.18">Th18</a></i>;<br/><b>hence </b><a NAME="E7:55_5_1"/>
<span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">vseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c7">tv</font></span>)</span></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">e</font>
 <b>by </b><i><a class="txt" href="csspace4.html#E4:55_5_1"><i><font color="Green" title="E55">A55</font></i></a>, <a class="txt" href="csspace4.html#E5:55_5_1"><i><font color="Green" title="E56">A58</font></i></a>, <a class="ref" href="clvect_1.html#D9" title="CLVECT_1:def.9">CLVECT_1:def 9</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:55_5"/>
 ex <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>for <font color="Olive" title="b2">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">k</font> holds <br/><span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">vseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b2">n</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c7">tv</font></span>)</span></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">e</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E13:55"/>
for <font color="Olive" title="b1">e</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">e</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 holds <br/> ex <font color="Olive" title="b2">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>for <font color="Olive" title="b3">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b3">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b2">m</font> holds <br/><span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">vseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b3">n</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c7">tv</font></span>)</span></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">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" title="c8">e</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>

<b>assume </b><a NAME="E1:55_6"/><i><font color="Green" title="E51">A59</font></i>: 
<font color="Maroon" title="c8">e</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 ;<br/>

<a NAME="E2:55_6"/><i><font color="Green" title="E52">A60</font></i>: 
<font color="Maroon" title="c8">e</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="csspace4.html#E1:55_6"><i><font color="Green" title="E51">A59</font></i></a>, <a class="ref" href="seq_2.html#T3" title="SEQ_2:th.3">SEQ_2:3</a></i>;<br/>
<a NAME="E3:55_6"/><i><font color="Green" title="E53">A61</font></i>: 
<font color="Maroon" title="c8">e</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c8">e</font>
 
<b>by </b><i><a class="txt" href="csspace4.html#E1:55_6"><i><font color="Green" title="E51">A59</font></i></a>, <a class="ref" href="xreal_1.html#T218" title="XREAL_1:th.218">XREAL_1:218</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">m</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E5:55_6"/><i><font color="Green" title="E54">A62</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c9">m</font> holds <br/><span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">vseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c7">tv</font></span>)</span></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">e</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2
 <b>by </b><i><a class="txt" href="csspace4.html#E12:55"><i><font color="Green" title="E50">A50</font></i></a>, <a class="txt" href="csspace4.html#E2:55_6"><i><font color="Green" title="E52">A60</font></i></a></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" title="c10">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:55_6_1"/><i><font color="Green" title="E55">A63</font></i>: 
<font color="Maroon" title="c10">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c9">m</font>
 ;<br/><a NAME="E2:55_6_1"/>
<span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">vseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c7">tv</font></span>)</span></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">e</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2
 <b>by </b><i><a class="txt" href="csspace4.html#E5:55_6"><i><font color="Green" title="E54">A62</font></i></a>, <a class="txt" href="csspace4.html#E1:55_6_1"><i><font color="Green" title="E55">A63</font></i></a></i>;<br/><b>hence </b><a NAME="E3:55_6_1"/>
<span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">vseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c7">tv</font></span>)</span></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c8">e</font>
 <b>by </b><i><a class="txt" href="csspace4.html#E3:55_6"><i><font color="Green" title="E53">A61</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:55_6"/>
 ex <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>for <font color="Olive" title="b2">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">m</font> holds <br/><span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">vseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b2">n</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c7">tv</font></span>)</span></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c8">e</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E14:55"/>
<font color="Maroon" title="c3">vseq</font> is <a href="clvect_1.html#V6" title="CLVECT_1:attr.6">convergent</a>
 <b>by </b><i><a class="ref" href="clvect_1.html#D15" title="CLVECT_1:def.15">CLVECT_1:def 15</a></i>;<br/>


</div>
