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

<b>let </b><font color="Maroon">T</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">FS1</font> be    <a href="seqfunc.html#M1">Functional_Sequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>, <a href="numbers.html#K1">REAL</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:19"/><i><font color="Green">E58</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   ex <font color="Olive">f</font> being  <a href="pscomp_1.html#NM1">RealMap</a> of <font color="Maroon">T</font> st <br/>( <font color="Maroon">FS1</font> <a href="seqfunc.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> &amp; <font color="Olive">f</font> is <a href="pscomp_1.html#V9">continuous</a> &amp; ( for <font color="Olive">p</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font> holds  <font color="Olive">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) )
 <b>and </b><br/><a NAME="E2:19"/><i><font color="Green">E60</font></i>: 
 ex <font color="Olive">seq</font> being  <a href="seq_1.html#NM1">Real_Sequence</a> st <br/>( <font color="Olive">seq</font> is <a href="series_1.html#V1">summable</a> &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">p</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font> holds  <span class="p1">(<span class="default"><font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Olive">p</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">seq</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font> ) )
 ;<br/>

<b>let </b><font color="Maroon">f</font> be   <a href="pscomp_1.html#NM1">RealMap</a> of <font color="Maroon">T</font>;<br/>

<b>assume </b><a NAME="E3:19"/><i><font color="Green">E61</font></i>: 
for <font color="Olive">p</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font> holds  <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a>  <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Olive">p</font></span>)</span>
 ;<br/>

<b>reconsider </b><font color="Maroon">fR</font> = <font color="Maroon">f</font> as   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">T</font>,<a href="topmetr.html#K3">R^1</a>  <b>by </b><i><a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">p</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>;<br/><a NAME="E1:19_1"/>
for <font color="Olive">R</font> being  <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a>   st <font color="Olive">R</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">fR</font> <a href="nagata_2.html#K1" target="_self">.</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">R</font> holds <br/> ex <font color="Olive">U</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">U</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">U</font> &amp; <font color="Maroon">fR</font> <a href="relset_1.html#K10">.:</a> <font color="Olive">U</font> <a href="tarski.html#R1">c=</a> <font color="Olive">R</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">R</font> be   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a> ;<br/>

<b>assume </b><a NAME="E1:19_1_1"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">R</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">fR</font> <a href="nagata_2.html#K1" target="_self">.</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> )
 ;<br/>

<b>reconsider </b><font color="Maroon">fRp</font> = <font color="Maroon">fR</font> <a href="nagata_2.html#K1" target="_self">.</a> <font color="Maroon">p</font> as   <a href="pre_topc.html#NM2">Point</a> of <a href="metric_1.html#K8">RealSpace</a>  <b>by </b><i><a class="ref" href="metric_1.html#D14">METRIC_1:def 14</a></i>;<br/>
<b>consider </b><font color="Maroon">rn</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E4:19_1_1"/><i><font color="Green">E63</font></i>: 
( <font color="Maroon">rn</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">fRp</font>,<font color="Maroon">rn</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">R</font> )
 <b>by </b><i>, <a class="ref" href="topmetr.html#T22">TOPMETR:22</a>, <a class="ref" href="topmetr.html#D7">TOPMETR:def 7</a></i>;<br/>
<b>set </b><font color="Maroon">r2</font> = <font color="Maroon">rn</font> <a href="xcmplx_0.html#K7">/</a> 2;<br/>
<b>set </b><font color="Maroon">r4</font> = <font color="Maroon">rn</font> <a href="xcmplx_0.html#K7">/</a> 4;<br/>
<b>reconsider </b><font color="Maroon">r2</font> = <font color="Maroon">rn</font> <a href="xcmplx_0.html#K7">/</a> 2, <font color="Maroon">r4</font> = <font color="Maroon">rn</font> <a href="xcmplx_0.html#K7">/</a> 4 as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">seq</font> being   <a href="seq_1.html#NM1">Real_Sequence</a><b> such that </b><br/><a NAME="E7:19_1_1"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">seq</font> is <a href="series_1.html#V1">summable</a> &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">q</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font> holds  <span class="p1">(<span class="default"><font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Olive">q</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">seq</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font> ) )
 <b>by </b><i/>;<br/>
<a NAME="E8:19_1_1"/>
(  <a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">seq</font> is <a href="seq_2.html#V4">convergent</a> &amp; <font color="Maroon">r4</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 
<b>by </b><i>, , <a class="ref" href="seq_2.html#T3">SEQ_2:3</a>, <a class="ref" href="series_1.html#D2">SERIES_1:def 2</a></i>;<br/>
<b>then consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E10:19_1_1"/><i><font color="Green">E65</font></i>: 
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r4</font>
 <b>by </b><i><a class="ref" href="seq_2.html#D7">SEQ_2:def 7</a></i>;<br/>
<a NAME="E11:19_1_1"/><i><font color="Green">E66</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">q</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font> holds  <span class="p1">(<span class="default"><font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Olive">q</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">q</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>;<br/>



<b>consider </b><font color="Maroon">f1</font> being   <a href="pscomp_1.html#NM1">RealMap</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E2:19_1_1_1"/><i><font color="Green">E68</font></i>: 
( <font color="Maroon">FS1</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> &amp; <font color="Maroon">f1</font> is <a href="pscomp_1.html#V9">continuous</a> &amp; ( for <font color="Olive">q</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font> holds  <font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">q</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) )
 <b>by </b><i/>;<br/>
<a NAME="E3:19_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">FS1</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">q</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font>
 
<b>by </b><i><a class="ref" href="seqfunc.html#D11">SEQFUNC:def 11</a></i>;<br/>
<b>hence </b><a NAME="E4:19_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">q</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E12:19_1_1"/><i><font color="Green">E69</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_1_2"><b>proof </b></a><div class="add">

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

<a NAME="E1:19_1_1_2"/>
for <font color="Olive">q</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font> holds <br/> ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">p</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Olive">q</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">seq</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> &amp; <font color="Maroon">seq</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font> )
 
<b>by </b><i>, , <a class="ref" href="seqm_3.html#D9">SEQM_3:def 9</a></i>;<br/>
<b>hence </b><a NAME="E2:19_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 ;<br/>


</div><b>end;</b></div>
<a NAME="E13:19_1_1"/>
(  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r4</font> &amp;  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="series_1.html#K2">Sum</a> <font color="Maroon">seq</font> &amp;  <a href="series_1.html#K2">Sum</a> <font color="Maroon">seq</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i>, , <a class="ref" href="series_1.html#T18">SERIES_1:18</a>, <a class="ref" href="series_1.html#D3">SERIES_1:def 3</a></i>;<br/>
<a NAME="E14:19_1_1"/><b>then </b>
(  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p3">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r4</font> &amp; <font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> is <a href="series_1.html#V1">summable</a> )
 
<b>by </b><i>, <a class="ref" href="series_1.html#T15">SERIES_1:15</a></i>;<br/>
<a NAME="E15:19_1_1"/><b>then </b>
(  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r4</font> &amp;  <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 
<b>by </b><i>, <a class="ref" href="complex1.html#T138">COMPLEX1:138</a>, <a class="ref" href="series_1.html#T21">SERIES_1:21</a></i>;<br/>
<a NAME="E16:19_1_1"/><b>then </b><i><font color="Green">E70</font></i>: 
 <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r4</font>
 
<b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/>
<a NAME="E17:19_1_1"/><i><font color="Green">E71</font></i>: 
for <font color="Olive">q</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font> holds <br/> ( <font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Olive">q</font> is <a href="series_1.html#V1">summable</a> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Olive">q</font></span>)</span> <a href="seqm_3.html#K1">^\</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> &amp;  <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Olive">q</font></span>)</span> <a href="seqm_3.html#K1">^\</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r4</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">q</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>;<br/>

<b>set </b><font color="Maroon">F</font> = <font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">q</font>;<br/>
<a NAME="E1:19_1_1_3"/><i><font color="Green">E108</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds <br/> ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">q</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">k</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">q</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">seq</font> <a href="funct_2.html#K3">.</a> <font color="Olive">k</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E2:19_1_1_3"/><b>then </b><i><font color="Green">E109</font></i>: 
<font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">q</font> is <a href="series_1.html#V1">summable</a>
 
<b>by </b><i>, <a class="ref" href="series_1.html#T24">SERIES_1:24</a></i>;<br/>
<a NAME="E3:19_1_1_3"/>
( <span class="p1">(<span class="default"><font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">q</font></span>)</span> <a href="seqm_3.html#K1">^\</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> is <a href="series_1.html#V1">summable</a> &amp; <font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> is <a href="series_1.html#V1">summable</a> &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds <br/> ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">q</font></span>)</span> <a href="seqm_3.html#K1">^\</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">k</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">q</font></span>)</span> <a href="seqm_3.html#K1">^\</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">k</font> ) ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_1_3_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_1_3_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><a NAME="E1:19_1_1_3_1_1"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">q</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">q</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">seq</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> &amp; <font color="Maroon">seq</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">q</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">q</font></span>)</span> <a href="seqm_3.html#K1">^\</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font> )
 <b>by </b><i>, , <a class="ref" href="seqm_3.html#D9">SEQM_3:def 9</a></i>;<br/><b>hence </b><a NAME="E2:19_1_1_3_1_1"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">q</font></span>)</span> <a href="seqm_3.html#K1">^\</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">q</font></span>)</span> <a href="seqm_3.html#K1">^\</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font> )
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:19_1_1_3_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">q</font></span>)</span> <a href="seqm_3.html#K1">^\</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> is <a href="series_1.html#V1">summable</a> &amp; <font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> is <a href="series_1.html#V1">summable</a> &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds <br/> ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">q</font></span>)</span> <a href="seqm_3.html#K1">^\</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">k</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">q</font></span>)</span> <a href="seqm_3.html#K1">^\</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">k</font> ) ) )
 <b>by </b><i>, , <a class="ref" href="series_1.html#T15">SERIES_1:15</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:19_1_1_3"/><b>then </b>
( 0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">q</font></span>)</span> <a href="seqm_3.html#K1">^\</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> &amp;  <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">q</font></span>)</span> <a href="seqm_3.html#K1">^\</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="series_1.html#T21">SERIES_1:21</a>, <a class="ref" href="series_1.html#T24">SERIES_1:24</a></i>;<br/>
<b>hence </b><a NAME="E5:19_1_1_3"/>
( <font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">q</font> is <a href="series_1.html#V1">summable</a> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">q</font></span>)</span> <a href="seqm_3.html#K1">^\</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> &amp;  <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">q</font></span>)</span> <a href="seqm_3.html#K1">^\</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r4</font> )
 <b>by </b><i>, , , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="ref" href="series_1.html#T24">SERIES_1:24</a></i>;<br/>


</div><b>end;</b></div>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">SS</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">SS</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> &amp; <font color="Olive">SS</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">SS</font> &amp; ( for <font color="Olive">f1</font> being  <a href="pscomp_1.html#NM1">RealMap</a> of <font color="Maroon">T</font>  st <font color="Maroon">FS1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">f1</font> holds <br/>for <font color="Olive">f1p</font> being  <a href="pre_topc.html#NM2">Point</a> of <a href="metric_1.html#K8">RealSpace</a>   st <font color="Olive">f1p</font> <a href="hidden.html#R1">=</a> <font color="Olive">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font> holds <br/><font color="Olive">f1</font> <a href="relset_1.html#K10">.:</a> <font color="Olive">SS</font> <a href="tarski.html#R1">c=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Olive">f1p</font>,<span class="p1">(<span class="default"><font color="Maroon">r2</font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> ) );<br/>
<a NAME="E18:19_1_1"/><i><font color="Green">E110</font></i>: 
for <font color="Olive">k</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="seq_4.html#K1">{</a><span class="default">0</span><a href="seq_4.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span> holds <br/> ex <font color="Olive">U</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">U</font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>,<font color="Olive">U</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:19_1_1_4"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="seq_4.html#K1">{</a><span class="default">0</span><a href="seq_4.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span>
 ;<br/>

<a NAME="E2:19_1_1_4"/><b>then </b>
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> or <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="seq_4.html#K1">{</a><span class="default">0</span><a href="seq_4.html#K1">}</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">k</font> = <font color="Maroon">k</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">f1</font> being   <a href="pscomp_1.html#NM1">RealMap</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E5:19_1_1_4"/><i><font color="Green">E111</font></i>: 
( <font color="Maroon">FS1</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> &amp; <font color="Maroon">f1</font> is <a href="pscomp_1.html#V9">continuous</a> &amp; ( for <font color="Olive">p</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font> holds  <font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) )
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">f1'</font> = <font color="Maroon">f1</font> as   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">T</font>,<a href="topmetr.html#K3">R^1</a>  <b>by </b><i><a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<b>reconsider </b><font color="Maroon">f1p</font> = <font color="Maroon">f1'</font> <a href="nagata_2.html#K1" target="_self">.</a> <font color="Maroon">p</font> as   <a href="pre_topc.html#NM2">Point</a> of <a href="metric_1.html#K8">RealSpace</a>  <b>by </b><i><a class="ref" href="metric_1.html#D14">METRIC_1:def 14</a></i>;<br/>
<b>set </b><font color="Maroon">B</font> =  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">f1p</font>,<span class="p1">(<span class="default"><font color="Maroon">r2</font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>;<br/>
<b>reconsider </b><font color="Maroon">B</font> =  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">f1p</font>,<span class="p1">(<span class="default"><font color="Maroon">r2</font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> as   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a>  <b>by </b><i><a class="ref" href="metric_1.html#D14">METRIC_1:def 14</a>, <a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<a NAME="E9:19_1_1_4"/>
(  <a href="metric_1.html#K4">dist</a> <font color="Maroon">f1p</font>,<font color="Maroon">f1p</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">r2</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 
<b>by </b><i>, <a class="ref" href="metric_1.html#T1">METRIC_1:1</a>, <a class="ref" href="seq_2.html#T3">SEQ_2:3</a></i>;<br/>
<a NAME="E10:19_1_1_4"/><b>then </b>
(  <a href="metric_1.html#K4">dist</a> <font color="Maroon">f1p</font>,<font color="Maroon">f1p</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r2</font> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">f1'</font> is <a href="pre_topc.html#V5">continuous</a> )
 
<b>by </b><i>, <a class="ref" href="real_2.html#T127">REAL_2:127</a>, <a class="ref" href="topreal6.html#T83">TOPREAL6:83</a></i>;<br/>
<a NAME="E11:19_1_1_4"/><b>then </b>
( <font color="Maroon">f1p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">B</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">f1'</font> <a href="tmap_1.html#R1">is_continuous_at</a> <font color="Maroon">p</font> )
 
<b>by </b><i><a class="ref" href="metric_1.html#T12">METRIC_1:12</a>, <a class="ref" href="tmap_1.html#T55">TMAP_1:55</a>, <a class="ref" href="topmetr.html#T21">TOPMETR:21</a>, <a class="ref" href="topmetr.html#D7">TOPMETR:def 7</a></i>;<br/>
<b>then consider </b><font color="Maroon">U</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E13:19_1_1_4"/><i><font color="Green">E112</font></i>: 
( <font color="Maroon">U</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">U</font> &amp; <font color="Maroon">f1'</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">U</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> )
 <b>by </b><i><a class="ref" href="tmap_1.html#T48">TMAP_1:48</a></i>;<br/>
<a NAME="E14:19_1_1_4"/>
for <font color="Olive">f1</font> being  <a href="pscomp_1.html#NM1">RealMap</a> of <font color="Maroon">T</font>  st <font color="Maroon">FS1</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Olive">f1</font> holds <br/>for <font color="Olive">f1p</font> being  <a href="pre_topc.html#NM2">Point</a> of <a href="metric_1.html#K8">RealSpace</a>   st <font color="Olive">f1p</font> <a href="hidden.html#R1">=</a> <font color="Olive">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font> holds <br/><font color="Olive">f1</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">U</font> <a href="tarski.html#R1">c=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Olive">f1p</font>,<span class="p1">(<span class="default"><font color="Maroon">r2</font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E15:19_1_1_4"/>
 ex <font color="Olive">U</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">U</font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>,<font color="Olive">U</font>] )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">FSn</font> being   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="seq_4.html#K1">{</a><span class="default">0</span><a href="seq_4.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span>, <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E20:19_1_1"/><i><font color="Green">E113</font></i>: 
for <font color="Olive">k</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="seq_4.html#K1">{</a><span class="default">0</span><a href="seq_4.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>,<font color="Maroon">FSn</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S1">FUNCT_2:sch 1</a>();<br/></i>
<b>reconsider </b><font color="Maroon">RNG</font> =  <a href="relset_1.html#K5">rng</a> <font color="Maroon">FSn</font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font> ;<br/>
<a NAME="E22:19_1_1"/>
( <span class="p1"><a href="seq_4.html#K1">{</a><span class="default">0</span><a href="seq_4.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span> is <a href="finset_1.html#V1">finite</a> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">FSn</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="seq_4.html#K1">{</a><span class="default">0</span><a href="seq_4.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E23:19_1_1"/><b>then </b><i><font color="Green">E114</font></i>: 
<font color="Maroon">RNG</font> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i><a class="ref" href="finset_1.html#T26">FINSET_1:26</a></i>;<br/>
<a NAME="E24:19_1_1"/>
<font color="Maroon">RNG</font> is <a href="tops_2.html#V1">open</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">Q</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="tops_2.html#D1">TOPS_2:def 1</a><br/>

<b>assume </b><a NAME="E1:19_1_1_5"/>
<font color="Maroon">Q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">RNG</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:19_1_1_5"/><i><font color="Green">E159</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">FSn</font> &amp; <font color="Maroon">FSn</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Q</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">SS</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E5:19_1_1_5"/><i><font color="Green">E160</font></i>: 
( <font color="Maroon">SS</font> <a href="hidden.html#R1">=</a> <font color="Maroon">FSn</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> &amp; <font color="Maroon">SS</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">SS</font> &amp; ( for <font color="Olive">f1</font> being  <a href="pscomp_1.html#NM1">RealMap</a> of <font color="Maroon">T</font>  st <font color="Maroon">FS1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">f1</font> holds <br/>for <font color="Olive">f1p</font> being  <a href="pre_topc.html#NM2">Point</a> of <a href="metric_1.html#K8">RealSpace</a>   st <font color="Olive">f1p</font> <a href="hidden.html#R1">=</a> <font color="Olive">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font> holds <br/><font color="Olive">f1</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">SS</font> <a href="tarski.html#R1">c=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Olive">f1p</font>,<span class="p1">(<span class="default"><font color="Maroon">r2</font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> ) )
 <b>by </b><i>, </i>;<br/>
<b>thus </b><a NAME="E6:19_1_1_5"/>
<font color="Maroon">Q</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E25:19_1_1"/><b>then </b><i><font color="Green">E161</font></i>: 
 <a href="setfam_1.html#K6">meet</a> <font color="Maroon">RNG</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i>, <a class="ref" href="tops_2.html#T27">TOPS_2:27</a></i>;<br/>
<a NAME="E26:19_1_1"/>
0 <a href="hidden.html#R2">in</a> <span class="p1"><a href="seq_4.html#K1">{</a><span class="default">0</span><a href="seq_4.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E27:19_1_1"/><b>then </b>
0 <a href="hidden.html#R2">in</a> <span class="p1"><a href="seq_4.html#K1">{</a><span class="default">0</span><a href="seq_4.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">RNG</font> = <font color="Maroon">RNG</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font> <b>by </b><i><a class="ref" href="funct_2.html#T6">FUNCT_2:6</a></i>;<br/>
<a NAME="E29:19_1_1"/><i><font color="Green">E162</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">RNG</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_1_6"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_1_6_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">Fx</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:19_1_1_6_1"/>
<font color="Maroon">Fx</font> <a href="hidden.html#R2">in</a> <font color="Maroon">RNG</font>
 ;<br/><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:19_1_1_6_1"/><i><font color="Green">E163</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">FSn</font> &amp; <font color="Maroon">FSn</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Fx</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>consider </b><font color="Maroon">SS</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E5:19_1_1_6_1"/><i><font color="Green">E164</font></i>: 
( <font color="Maroon">SS</font> <a href="hidden.html#R1">=</a> <font color="Maroon">FSn</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> &amp; <font color="Maroon">SS</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">SS</font> &amp; ( for <font color="Olive">f1</font> being  <a href="pscomp_1.html#NM1">RealMap</a> of <font color="Maroon">T</font>  st <font color="Maroon">FS1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">f1</font> holds <br/>for <font color="Olive">f1p</font> being  <a href="pre_topc.html#NM2">Point</a> of <a href="metric_1.html#K8">RealSpace</a>   st <font color="Olive">f1p</font> <a href="hidden.html#R1">=</a> <font color="Olive">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font> holds <br/><font color="Olive">f1</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">SS</font> <a href="tarski.html#R1">c=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Olive">f1p</font>,<span class="p1">(<span class="default"><font color="Maroon">r2</font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> ) )
 <b>by </b><i>, </i>;<br/><b>thus </b><a NAME="E6:19_1_1_6_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Fx</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:19_1_1_6"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">RNG</font>
 <b>by </b><i><a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E30:19_1_1"/>
<font color="Maroon">fR</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K6">meet</a> <font color="Maroon">RNG</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">R</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_1_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">fRq</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:19_1_1_7"/>
<font color="Maroon">fRq</font> <a href="hidden.html#R2">in</a> <font color="Maroon">fR</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K6">meet</a> <font color="Maroon">RNG</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">q</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:19_1_1_7"/><i><font color="Green">E195</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">fR</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">RNG</font> &amp; <font color="Maroon">fRq</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fR</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">q</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon">q</font> = <font color="Maroon">q</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">fRq</font> = <font color="Maroon">fR</font> <a href="nagata_2.html#K1" target="_self">.</a> <font color="Maroon">q</font> as   <a href="pre_topc.html#NM2">Point</a> of <a href="metric_1.html#K8">RealSpace</a>  <b>by </b><i><a class="ref" href="metric_1.html#D14">METRIC_1:def 14</a></i>;<br/>
<b>set </b><font color="Maroon">sp</font> = <font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">p</font>;<br/>
<b>set </b><font color="Maroon">sq</font> = <font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">q</font>;<br/>
<b>set </b><font color="Maroon">spn</font> = <span class="p1">(<span class="default"><font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">p</font></span>)</span> <a href="seqm_3.html#K1">^\</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/>
<b>set </b><font color="Maroon">sqn</font> = <span class="p1">(<span class="default"><font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">q</font></span>)</span> <a href="seqm_3.html#K1">^\</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/>
<b>set </b><font color="Maroon">PSp</font> =  <a href="series_1.html#K1">Partial_Sums</a> <span class="p1">(<span class="default"><font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">p</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">PSq</font> =  <a href="series_1.html#K1">Partial_Sums</a> <span class="p1">(<span class="default"><font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">q</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">PSpq</font> =  <a href="series_1.html#K1">Partial_Sums</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">p</font></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">q</font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">absPSpq</font> =  <a href="series_1.html#K1">Partial_Sums</a> <span class="p1">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">p</font></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p3">(<span class="default"><font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">q</font></span>)</span></span>)</span></span>)</span>;<br/>
<a NAME="E6:19_1_1_7"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">p</font></span>)</span> <a href="seqm_3.html#K1">^\</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">q</font></span>)</span> <a href="seqm_3.html#K1">^\</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> )
 
<b>by </b><i/>;<br/>
<a NAME="E7:19_1_1_7"/><b>then </b>
(  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">p</font></span>)</span> <a href="seqm_3.html#K1">^\</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">p</font></span>)</span> <a href="seqm_3.html#K1">^\</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> &amp;  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">q</font></span>)</span> <a href="seqm_3.html#K1">^\</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">q</font></span>)</span> <a href="seqm_3.html#K1">^\</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> &amp;  <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">p</font></span>)</span> <a href="seqm_3.html#K1">^\</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r4</font> &amp;  <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">q</font></span>)</span> <a href="seqm_3.html#K1">^\</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r4</font> )
 
<b>by </b><i>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/>
<a NAME="E8:19_1_1_7"/><b>then </b><i><font color="Green">E204</font></i>: 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">p</font></span>)</span> <a href="seqm_3.html#K1">^\</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">q</font></span>)</span> <a href="seqm_3.html#K1">^\</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r4</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r4</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E9:19_1_1_7"/><i><font color="Green">E205</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> holds <br/><span class="p1">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">p</font></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p3">(<span class="default"><font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">q</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r2</font> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_1_7_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:19_1_1_7_1"/><i><font color="Green">E206</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 ;<br/>

<a NAME="E2:19_1_1_7_1"/>
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 )
 
<b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/>
<a NAME="E3:19_1_1_7_1"/><b>then </b>
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="seq_4.html#K1">{</a><span class="default">0</span><a href="seq_4.html#K1">}</a></span> or <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> )
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E4:19_1_1_7_1"/><b>then </b><i><font color="Green">E207</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="seq_4.html#K1">{</a><span class="default">0</span><a href="seq_4.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon">f1</font> being   <a href="pscomp_1.html#NM1">RealMap</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E6:19_1_1_7_1"/><i><font color="Green">E208</font></i>: 
( <font color="Maroon">FS1</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> &amp; <font color="Maroon">f1</font> is <a href="pscomp_1.html#V9">continuous</a> &amp; ( for <font color="Olive">p</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font> holds  <font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) )
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">f1p</font> = <font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font>, <font color="Maroon">f1q</font> = <font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">q</font> as   <a href="pre_topc.html#NM2">Point</a> of <a href="metric_1.html#K8">RealSpace</a>  <b>by </b><i><a class="ref" href="metric_1.html#D14">METRIC_1:def 14</a></i>;<br/>
<a NAME="E8:19_1_1_7_1"/>
( <font color="Maroon">FSn</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">RNG</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">RNG</font> )
 
<b>by </b><i>, , <a class="ref" href="funct_2.html#T6">FUNCT_2:6</a></i>;<br/>
<a NAME="E9:19_1_1_7_1"/><b>then </b>
( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FSn</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>
<a NAME="E10:19_1_1_7_1"/><b>then </b><i><font color="Green">E210</font></i>: 
<font color="Maroon">f1q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f1</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">FSn</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>consider </b><font color="Maroon">SS</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E12:19_1_1_7_1"/><i><font color="Green">E211</font></i>: 
( <font color="Maroon">SS</font> <a href="hidden.html#R1">=</a> <font color="Maroon">FSn</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> &amp; <font color="Maroon">SS</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">SS</font> &amp; ( for <font color="Olive">f1</font> being  <a href="pscomp_1.html#NM1">RealMap</a> of <font color="Maroon">T</font>  st <font color="Maroon">FS1</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Olive">f1</font> holds <br/>for <font color="Olive">f1p</font> being  <a href="pre_topc.html#NM2">Point</a> of <a href="metric_1.html#K8">RealSpace</a>   st <font color="Olive">f1p</font> <a href="hidden.html#R1">=</a> <font color="Olive">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font> holds <br/><font color="Olive">f1</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">SS</font> <a href="tarski.html#R1">c=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Olive">f1p</font>,<span class="p1">(<span class="default"><font color="Maroon">r2</font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> ) )
 <b>by </b><i>, </i>;<br/>
<a NAME="E13:19_1_1_7_1"/>
<font color="Maroon">f1</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">FSn</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">f1p</font>,<span class="p1">(<span class="default"><font color="Maroon">r2</font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E14:19_1_1_7_1"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">f1p</font>,<font color="Maroon">f1q</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r2</font> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i>, <a class="ref" href="metric_1.html#T12">METRIC_1:12</a></i>;<br/>
<a NAME="E15:19_1_1_7_1"/><b>then </b>
(  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r2</font> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">p</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font> &amp; <font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">q</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">p</font></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a>  )
 
<b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="seqfunc.html#D11">SEQFUNC:def 11</a>, <a class="ref" href="topmetr.html#T15">TOPMETR:15</a></i>;<br/>
<a NAME="E16:19_1_1_7_1"/><b>then </b>
(  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">p</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">q</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r2</font> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">p</font></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">p</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">q</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="seq_1.html#D4">SEQ_1:def 4</a></i>;<br/>
<b>hence </b><a NAME="E17:19_1_1_7_1"/>
<span class="p1">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">p</font></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p3">(<span class="default"><font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">q</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r2</font> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="seq_1.html#T16">SEQ_1:16</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:19_1_1_7"/>
( <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="funct_2.html#R2">=</a>  <a href="series_1.html#K1">Partial_Sums</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">p</font></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">q</font></span>)</span></span>)</span> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">q</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a>  )
 
<b>by </b><i><a class="ref" href="seq_1.html#T3">SEQ_1:3</a>, <a class="ref" href="series_1.html#T9">SERIES_1:9</a></i>;<br/>
<a NAME="E11:19_1_1_7"/><b>then </b>
(  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">p</font></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p4">(<span class="default"><font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">q</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> &amp;  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">p</font></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p4">(<span class="default"><font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">q</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">p</font></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p4">(<span class="default"><font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">q</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> &amp; <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">p</font></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p4">(<span class="default"><font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">q</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">r2</font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 
<b>by </b><i>, , , <a class="ref" href="seq_1.html#D4">SEQ_1:def 4</a></i>;<br/>
<a NAME="E12:19_1_1_7"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">r2</font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E13:19_1_1_7"/><b>then </b><i><font color="Green">E212</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r2</font>
 
<b>by </b><i><a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>;<br/>
<a NAME="E14:19_1_1_7"/>
( <font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">p</font> is <a href="series_1.html#V1">summable</a> &amp; <font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">q</font> is <a href="series_1.html#V1">summable</a> )
 
<b>by </b><i/>;<br/>
<a NAME="E15:19_1_1_7"/><b>then </b>
(  <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">p</font></span>)</span> <a href="seqm_3.html#K1">^\</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> &amp;  <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">q</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">q</font></span>)</span> <a href="seqm_3.html#K1">^\</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon">fRp</font> <a href="hidden.html#R1">=</a>  <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">p</font></span>)</span> &amp; <font color="Maroon">fRq</font> <a href="hidden.html#R1">=</a>  <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">q</font></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="series_1.html#T18">SERIES_1:18</a></i>;<br/>
<a NAME="E16:19_1_1_7"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">fR</font> <a href="nagata_2.html#K1" target="_self">.</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">fR</font> <a href="nagata_2.html#K1" target="_self">.</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">p</font></span>)</span> <a href="seqm_3.html#K1">^\</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">q</font></span>)</span> <a href="seqm_3.html#K1">^\</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 
;<br/>
<a NAME="E17:19_1_1_7"/><b>then </b><i><font color="Green">E213</font></i>: 
(  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">fR</font> <a href="nagata_2.html#K1" target="_self">.</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">fR</font> <a href="nagata_2.html#K1" target="_self">.</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">p</font></span>)</span> <a href="seqm_3.html#K1">^\</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">q</font></span>)</span> <a href="seqm_3.html#K1">^\</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> &amp;  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">p</font></span>)</span> <a href="seqm_3.html#K1">^\</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">q</font></span>)</span> <a href="seqm_3.html#K1">^\</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">p</font></span>)</span> <a href="seqm_3.html#K1">^\</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">q</font></span>)</span> <a href="seqm_3.html#K1">^\</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="complex1.html#T142">COMPLEX1:142</a>, <a class="ref" href="complex1.html#T143">COMPLEX1:143</a></i>;<br/>
<a NAME="E18:19_1_1_7"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">p</font></span>)</span> <a href="seqm_3.html#K1">^\</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">q</font></span>)</span> <a href="seqm_3.html#K1">^\</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r2</font>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E19:19_1_1_7"/><b>then </b>
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">p</font></span>)</span> <a href="seqm_3.html#K1">^\</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">FS1</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">q</font></span>)</span> <a href="seqm_3.html#K1">^\</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r2</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r2</font>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E20:19_1_1_7"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">fR</font> <a href="nagata_2.html#K1" target="_self">.</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">fR</font> <a href="nagata_2.html#K1" target="_self">.</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">rn</font>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E21:19_1_1_7"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">fRp</font>,<font color="Maroon">fRq</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">rn</font>
 
<b>by </b><i><a class="ref" href="topmetr.html#T15">TOPMETR:15</a></i>;<br/>
<a NAME="E22:19_1_1_7"/><b>then </b>
<font color="Maroon">fRq</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">fRp</font>,<font color="Maroon">rn</font>
 
<b>by </b><i><a class="ref" href="metric_1.html#T12">METRIC_1:12</a></i>;<br/>
<b>hence </b><a NAME="E23:19_1_1_7"/>
<font color="Maroon">fRq</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E31:19_1_1"/>
 ex <font color="Olive">U</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">U</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">U</font> &amp; <font color="Maroon">fR</font> <a href="relset_1.html#K10">.:</a> <font color="Olive">U</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">R</font> )
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E2:19_1"/>
<font color="Maroon">fR</font> <a href="tmap_1.html#R1">is_continuous_at</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="tmap_1.html#T48">TMAP_1:48</a></i>;<br/></div><b>end;</b></div>
<a NAME="E6:19"/><b>then </b>
<font color="Maroon">fR</font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="ref" href="tmap_1.html#T55">TMAP_1:55</a></i>;<br/>
<b>hence </b><a NAME="E7:19"/>
<font color="Maroon">f</font> is <a href="pscomp_1.html#V9">continuous</a>
 <b>by </b><i><a class="ref" href="topreal6.html#T83">TOPREAL6:83</a></i>;<br/>


</div>
