<?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>assume </b><a NAME="E1:22"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">T</font> <a href="urysohn1.html#NR1" target="_self">is_T1</a> 
 ;<br/>

<b>let </b><font color="Maroon">s</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">FS2</font> be    <a href="seqfunc.html#M1">Functional_Sequence</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font></span><a href="zfmisc_1.html#K2">:]</a></span>, <a href="numbers.html#K1">REAL</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E2:22"/><i><font color="Green">E60</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">pmet</font> being  <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font></span><a href="zfmisc_1.html#K2">:]</a></span>, <a href="numbers.html#K1">REAL</a>  st <br/>( <font color="Maroon">FS2</font> <a href="seqfunc.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Olive">pmet</font> &amp; <font color="Olive">pmet</font> <a href="nagata_1.html#R1">is_a_pseudometric_of</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> &amp; ( for <font color="Olive">pq</font> being   <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font></span><a href="zfmisc_1.html#K2">:]</a></span> holds  <font color="Olive">pmet</font> <a href="funct_2.html#K3">.</a> <font color="Olive">pq</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font> ) &amp; ( for <font color="Olive">pmet'</font> being  <a href="pscomp_1.html#NM1">RealMap</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">T</font>,<font color="Maroon">T</font></span><a href="borsuk_1.html#K5">:]</a></span>  st <font color="Olive">pmet</font> <a href="funct_2.html#R1">=</a> <font color="Olive">pmet'</font> holds <br/><font color="Olive">pmet'</font> is <a href="pscomp_1.html#V9">continuous</a> ) )
 <b>and </b><br/><a NAME="E3:22"/><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><br/> for <font color="Olive">A'</font> being non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>  st not <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">A'</font> &amp; <font color="Olive">A'</font> is <a href="pre_topc.html#V4">closed</a> holds <br/> ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>for <font color="Olive">pmet</font> being  <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font></span><a href="zfmisc_1.html#K2">:]</a></span>, <a href="numbers.html#K1">REAL</a>   st <font color="Maroon">FS2</font> <a href="seqfunc.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Olive">pmet</font> holds <br/><span class="p1">(<span class="default"><a href="nagata_2.html#K4" target="_self">inf</a> <font color="Olive">pmet</font>,<font color="Olive">A'</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<b>set </b><font color="Maroon">cT</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>;<br/>
<b>set </b><font color="Maroon">Geo</font> = <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="prepower.html#K2">GeoSeq</a> ;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>,   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  =  <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="prepower.html#K2">GeoSeq</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">FS2</font> <a href="seqfunc.html#K10">#</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">a<sub>2</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span></span>)</span>;<br/>
<b>consider </b><font color="Maroon">pmet</font> being   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font></span><a href="zfmisc_1.html#K2">:]</a></span>, <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E5:22"/><i><font color="Green">E62</font></i>: 
for <font color="Olive">p</font>, <font color="Olive">q</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font> holds  <font color="Maroon">pmet</font> <a href="metric_1.html#K1">.</a> <font color="Olive">p</font>,<font color="Olive">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">p</font>,<font color="Olive">q</font>)
 <b>from </b><i><a class="ref" href="binop_1.html#S4">BINOP_1:sch 4</a>();<br/></i>
<a NAME="E6:22"/><i><font color="Green">E63</font></i>: 
for <font color="Olive">pq</font> being   <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font></span><a href="zfmisc_1.html#K2">:]</a></span> holds  <font color="Maroon">pmet</font> <a href="funct_2.html#K3">.</a> <font color="Olive">pq</font> <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"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="prepower.html#K2">GeoSeq</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">FS2</font> <a href="seqfunc.html#K10">#</a> <font color="Olive">pq</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">pq</font> be    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font></span><a href="zfmisc_1.html#K2">:]</a></span>;<br/>

<b>consider </b><font color="Maroon">p</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">q</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:22_1"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> &amp; <font color="Maroon">pq</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">q</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<a NAME="E3:22_1"/>
<font color="Maroon">pmet</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">pq</font> <a href="hidden.html#R1">=</a> <font color="Maroon">pmet</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:22_1"/>
<font color="Maroon">pmet</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">pq</font> <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"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="prepower.html#K2">GeoSeq</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">FS2</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">pq</font></span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:22"/><i><font color="Green">E65</font></i>: 
for <font color="Olive">pq</font> being   <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font></span><a href="zfmisc_1.html#K2">:]</a></span><br/> for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds <br/> ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="prepower.html#K2">GeoSeq</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">FS2</font> <a href="seqfunc.html#K10">#</a> <font color="Olive">pq</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="prepower.html#K2">GeoSeq</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">FS2</font> <a href="seqfunc.html#K10">#</a> <font color="Olive">pq</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seq_1.html#K14">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="prepower.html#K2">GeoSeq</a> </span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">pq</font> be    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font></span><a href="zfmisc_1.html#K2">:]</a></span>;<br/><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<b>consider </b><font color="Maroon">pmet1</font> being   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font></span><a href="zfmisc_1.html#K2">:]</a></span>, <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E2:22_2"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">FS2</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">pmet1</font> &amp; <font color="Maroon">pmet1</font> <a href="nagata_1.html#R1">is_a_pseudometric_of</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> &amp; ( for <font color="Olive">pq</font> being   <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font></span><a href="zfmisc_1.html#K2">:]</a></span> holds  <font color="Maroon">pmet1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">pq</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font> ) &amp; ( for <font color="Olive">pmet1'</font> being  <a href="pscomp_1.html#NM1">RealMap</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">T</font>,<font color="Maroon">T</font></span><a href="borsuk_1.html#K5">:]</a></span>  st <font color="Maroon">pmet1</font> <a href="funct_2.html#R1">=</a> <font color="Olive">pmet1'</font> holds <br/><font color="Olive">pmet1'</font> is <a href="pscomp_1.html#V9">continuous</a> ) )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:22_2"/><i><font color="Green">E68</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">pq</font> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font>, <font color="Maroon">y</font> = <font color="Maroon">y</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font> <b>by </b><i/>;<br/>
<a NAME="E6:22_2"/>
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="prepower.html#K3">|^</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="newton.html#T102">NEWTON:102</a></i>;<br/>
<a NAME="E7:22_2"/><b>then </b>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">pmet1</font> <a href="metric_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> &amp; <font color="Maroon">pmet1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">pq</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font> &amp; <font color="Maroon">pmet1</font> <a href="metric_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">pmet1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">pq</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="prepower.html#K2">GeoSeq</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">pmet1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">pq</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">FS2</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">pq</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> )
 
<b>by </b><i>, , <a class="ref" href="nagata_1.html#T29">NAGATA_1:29</a>, <a class="ref" href="prepower.html#D1">PREPOWER:def 1</a>, <a class="ref" href="seqfunc.html#D11">SEQFUNC:def 11</a></i>;<br/>
<a NAME="E8:22_2"/><b>then </b>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="prepower.html#K2">GeoSeq</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">FS2</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">pq</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="prepower.html#K2">GeoSeq</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">FS2</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">pq</font></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"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="prepower.html#K2">GeoSeq</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">s</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="prepower.html#K2">GeoSeq</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seq_1.html#K14">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="prepower.html#K2">GeoSeq</a> </span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a>, <a class="ref" href="real_2.html#T121">REAL_2:121</a>, <a class="ref" href="seq_1.html#T13">SEQ_1:13</a></i>;<br/>
<b>hence </b><a NAME="E9:22_2"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="prepower.html#K2">GeoSeq</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">FS2</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">pq</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="prepower.html#K2">GeoSeq</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">FS2</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">pq</font></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">s</font> <a href="seq_1.html#K14">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="prepower.html#K2">GeoSeq</a> </span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> )
 <b>by </b><i><a class="ref" href="seq_1.html#T12">SEQ_1:12</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:22"/><i><font color="Green">E69</font></i>: 
for <font color="Olive">p</font>, <font color="Olive">q</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>  st <font color="Maroon">pmet</font> <a href="metric_1.html#K1">.</a> <font color="Olive">p</font>,<font color="Olive">q</font> <a href="hidden.html#R1">=</a> 0 holds <br/><font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Olive">q</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_3"><b>proof </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/><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>assume </b><a NAME="E1:22_3"/><i><font color="Green">E70</font></i>: 
( <font color="Maroon">pmet</font> <a href="metric_1.html#K1">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q</font> )
 ;<br/>

<b>set </b><font color="Maroon">pq</font> = <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">q</font></span><a href="domain_1.html#K1">]</a></span>;<br/>
<a NAME="E2:22_3"/>
( 0 <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K6">/</a> 2 &amp; 1 <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#R1">&lt;</a> 1 )
 
;<br/>
<a NAME="E3:22_3"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> 1
 
<b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/>
<a NAME="E4:22_3"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="prepower.html#K2">GeoSeq</a>  is <a href="series_1.html#V1">summable</a>
 
<b>by </b><i><a class="ref" href="series_1.html#T28">SERIES_1:28</a></i>;<br/>
<a NAME="E5:22_3"/><b>then </b>
( <font color="Maroon">s</font> <a href="seq_1.html#K14">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="prepower.html#K2">GeoSeq</a> </span>)</span> is <a href="series_1.html#V1">summable</a> &amp; <font color="Maroon">pmet</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">q</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">pmet</font> <a href="metric_1.html#K1">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font> &amp; ( for <font color="Olive">n</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"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="prepower.html#K2">GeoSeq</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">FS2</font> <a href="seqfunc.html#K10">#</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">q</font></span><a href="domain_1.html#K1">]</a></span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="prepower.html#K2">GeoSeq</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">FS2</font> <a href="seqfunc.html#K10">#</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">q</font></span><a href="domain_1.html#K1">]</a></span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seq_1.html#K14">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="prepower.html#K2">GeoSeq</a> </span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font> ) ) )
 
<b>by </b><i>, <a class="ref" href="series_1.html#T13">SERIES_1:13</a></i>;<br/>
<a NAME="E6:22_3"/><b>then </b><i><font color="Green">E71</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="prepower.html#K2">GeoSeq</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">FS2</font> <a href="seqfunc.html#K10">#</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">q</font></span><a href="domain_1.html#K1">]</a></span></span>)</span> is <a href="series_1.html#V1">summable</a> &amp;  <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="prepower.html#K2">GeoSeq</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">FS2</font> <a href="seqfunc.html#K10">#</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">q</font></span><a href="domain_1.html#K1">]</a></span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0 &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="prepower.html#K2">GeoSeq</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">FS2</font> <a href="seqfunc.html#K10">#</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">q</font></span><a href="domain_1.html#K1">]</a></span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font> ) )
 
<b>by </b><i>, , <a class="ref" href="series_1.html#T24">SERIES_1:24</a></i>;<br/>
<b>set </b><font color="Maroon">Q</font> = <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">q</font></span><a href="domain_1.html#K6">}</a></span>;<br/>
<a NAME="E7:22_3"/>
( <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">q</font></span><a href="domain_1.html#K6">}</a></span> is <a href="pre_topc.html#V4">closed</a> &amp; not <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">q</font></span><a href="domain_1.html#K6">}</a></span> )
 
<b>by </b><i>, , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="urysohn1.html#T27">URYSOHN1:27</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="E9:22_3"/><i><font color="Green">E108</font></i>: 
for <font color="Olive">pmet1</font> being  <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font></span><a href="zfmisc_1.html#K2">:]</a></span>, <a href="numbers.html#K1">REAL</a>   st <font color="Maroon">FS2</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Olive">pmet1</font> holds <br/><span class="p1">(<span class="default"><a href="nagata_2.html#K4" target="_self">inf</a> <font color="Olive">pmet1</font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">q</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">pmet1</font> being   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font></span><a href="zfmisc_1.html#K2">:]</a></span>, <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E11:22_3"/><i><font color="Green">E109</font></i>: 
( <font color="Maroon">FS2</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">pmet1</font> &amp; <font color="Maroon">pmet1</font> <a href="nagata_1.html#R1">is_a_pseudometric_of</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> &amp; ( for <font color="Olive">pq</font> being   <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font></span><a href="zfmisc_1.html#K2">:]</a></span> holds  <font color="Maroon">pmet1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">pq</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font> ) &amp; ( for <font color="Olive">pmet1'</font> being  <a href="pscomp_1.html#NM1">RealMap</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">T</font>,<font color="Maroon">T</font></span><a href="borsuk_1.html#K5">:]</a></span>  st <font color="Maroon">pmet1</font> <a href="funct_2.html#R1">=</a> <font color="Olive">pmet1'</font> holds <br/><font color="Olive">pmet1'</font> is <a href="pscomp_1.html#V9">continuous</a> ) )
 <b>by </b><i/>;<br/>
<a NAME="E12:22_3"/>
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="nagata_2.html#K3" target="_self">dist</a> <font color="Maroon">pmet1</font>,<font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">q</font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">pmet1</font> is <a href="seq_2.html#V2">bounded_below</a> &amp; <span class="p1">(<span class="default"><a href="nagata_2.html#K4" target="_self">inf</a> <font color="Maroon">pmet1</font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">q</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 
<b>by </b><i>, , , <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E13:22_3"/><b>then </b>
( <span class="p1">(<span class="default"><a href="nagata_2.html#K3" target="_self">dist</a> <font color="Maroon">pmet1</font>,<font color="Maroon">p</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="nagata_2.html#K3" target="_self">dist</a> <font color="Maroon">pmet1</font>,<font color="Maroon">p</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">q</font></span><a href="domain_1.html#K6">}</a></span> &amp;  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nagata_2.html#K3" target="_self">dist</a> <font color="Maroon">pmet1</font>,<font color="Maroon">p</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">q</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; not <span class="p1">(<span class="default"><a href="nagata_2.html#K3" target="_self">dist</a> <font color="Maroon">pmet1</font>,<font color="Maroon">p</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">q</font></span><a href="domain_1.html#K6">}</a></span> is <a href="xboole_0.html#V1">empty</a> &amp; <span class="p1">(<span class="default"><a href="nagata_2.html#K3" target="_self">dist</a> <font color="Maroon">pmet1</font>,<font color="Maroon">p</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">q</font></span><a href="domain_1.html#K6">}</a></span> is <a href="seq_4.html#V2">bounded_below</a> )
 
<b>by </b><i>, , , <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E14:22_3"/><b>then </b><i><font color="Green">E110</font></i>: 
<span class="p1">(<span class="default"><a href="nagata_2.html#K3" target="_self">dist</a> <font color="Maroon">pmet1</font>,<font color="Maroon">p</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">q</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="seq_4.html#D5">SEQ_4:def 5</a></i>;<br/>
<a NAME="E15:22_3"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="prepower.html#K2">GeoSeq</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">FS2</font> <a href="seqfunc.html#K10">#</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">q</font></span><a href="domain_1.html#K1">]</a></span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0 &amp; <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="prepower.html#K3">|^</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 
<b>by </b><i>, <a class="ref" href="newton.html#T102">NEWTON:102</a>, <a class="ref" href="rsspace.html#T21">RSSPACE:21</a></i>;<br/>
<a NAME="E16:22_3"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="prepower.html#K2">GeoSeq</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">FS2</font> <a href="seqfunc.html#K10">#</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">q</font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> 0 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="prepower.html#K2">GeoSeq</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 
<b>by </b><i><a class="ref" href="prepower.html#D1">PREPOWER:def 1</a>, <a class="ref" href="seq_1.html#T12">SEQ_1:12</a></i>;<br/>
<a NAME="E17:22_3"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">FS2</font> <a href="seqfunc.html#K10">#</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">q</font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">FS2</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">q</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">pmet1</font> <a href="metric_1.html#K1">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font> &amp; <font color="Maroon">pmet1</font> <a href="metric_1.html#K1">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="nagata_2.html#K3" target="_self">dist</a> <font color="Maroon">pmet1</font>,<font color="Maroon">p</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">q</font> )
 
<b>by </b><i>, , <a class="ref" href="xcmplx_1.html#T6">XCMPLX_1:6</a></i>;<br/>
<b>hence </b><a NAME="E18:22_3"/>
contradiction
 <b>by </b><i>, <a class="ref" href="seqfunc.html#D11">SEQFUNC:def 11</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:22"/>
<font color="Maroon">pmet</font> <a href="nagata_1.html#R1">is_a_pseudometric_of</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E10:22"/><b>then </b>
( <font color="Maroon">pmet</font> is <a href="metric_1.html#V2">Reflexive</a> &amp; <font color="Maroon">pmet</font> is <a href="metric_1.html#V3">discerning</a> &amp; <font color="Maroon">pmet</font> is <a href="metric_1.html#V4">symmetric</a> &amp; <font color="Maroon">pmet</font> is <a href="metric_1.html#V5">triangle</a> )
 
<b>by </b><i>, <a class="ref" href="metric_1.html#D4">METRIC_1:def 4</a>, <a class="ref" href="nagata_1.html#D10">NAGATA_1:def 10</a></i>;<br/>
<a NAME="E11:22"/><b>then </b><i><font color="Green">E111</font></i>: 
<font color="Maroon">pmet</font> <a href="pcomps_1.html#R1">is_metric_of</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="metric_6.html#T3">METRIC_6:3</a></i>;<br/>
<b>hence </b><a NAME="E12:22"/>
 ex <font color="Olive">pmet</font> being  <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font></span><a href="zfmisc_1.html#K2">:]</a></span>, <a href="numbers.html#K1">REAL</a>  st <br/>( <font color="Olive">pmet</font> <a href="pcomps_1.html#R1">is_metric_of</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> &amp; ( for <font color="Olive">pq</font> being   <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font></span><a href="zfmisc_1.html#K2">:]</a></span> holds  <font color="Olive">pmet</font> <a href="funct_2.html#K3">.</a> <font color="Olive">pq</font> <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"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="prepower.html#K2">GeoSeq</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">FS2</font> <a href="seqfunc.html#K10">#</a> <font color="Olive">pq</font></span>)</span></span>)</span> ) )
 <b>by </b><i/>;<br/>

<a NAME="E13:22"/>
for <font color="Olive">A</font> being non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> holds   <a href="pre_topc.html#K4">Cl</a> <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font> : <span class="p1">(<span class="default"><a href="nagata_2.html#K4" target="_self">inf</a> <font color="Maroon">pmet</font>,<font color="Olive">A</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> 0 </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">A</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>;<br/>

<b>set </b><font color="Maroon">INF</font> = <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font> : <span class="p1">(<span class="default"><a href="nagata_2.html#K4" target="_self">inf</a> <font color="Maroon">pmet</font>,<font color="Maroon">A</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> 0 </span>}</span> ;<br/>
<a NAME="E1:22_4"/><i><font color="Green">E112</font></i>: 
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font> : <span class="p1">(<span class="default"><a href="nagata_2.html#K4" target="_self">inf</a> <font color="Maroon">pmet</font>,<font color="Maroon">A</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> 0 </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:22_4_1"/><i><font color="Green">E113</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">p</font> = <font color="Maroon">x</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font> ;<br/>
<a NAME="E3:22_4_1"/>
( <font color="Maroon">pmet</font> <a href="nagata_1.html#R1">is_a_pseudometric_of</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> &amp; ( for <font color="Olive">pmet'</font> being  <a href="pscomp_1.html#NM1">RealMap</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">T</font>,<font color="Maroon">T</font></span><a href="borsuk_1.html#K5">:]</a></span>  st <font color="Maroon">pmet</font> <a href="funct_2.html#R1">=</a> <font color="Olive">pmet'</font> holds <br/><font color="Olive">pmet'</font> is <a href="pscomp_1.html#V9">continuous</a> ) )
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E4:22_4_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="nagata_2.html#K4" target="_self">inf</a> <font color="Maroon">pmet</font>,<font color="Maroon">A</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E5:22_4_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font> : <span class="p1">(<span class="default"><a href="nagata_2.html#K4" target="_self">inf</a> <font color="Maroon">pmet</font>,<font color="Maroon">A</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> 0 </span>}</span> 
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:22_4"/>
<span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font> : <span class="p1">(<span class="default"><a href="nagata_2.html#K4" target="_self">inf</a> <font color="Maroon">pmet</font>,<font color="Maroon">A</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> 0 </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_4_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:22_4_2"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font> : <span class="p1">(<span class="default"><a href="nagata_2.html#K4" target="_self">inf</a> <font color="Maroon">pmet</font>,<font color="Maroon">A</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> 0 </span>}</span> 
 ;<br/>

<b>assume </b><a NAME="E2:22_4_2"/><i><font color="Green">E159</font></i>: 
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font>
 ;<br/>

<b>consider </b><font color="Maroon">p</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E4:22_4_2"/><i><font color="Green">E160</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> &amp; <span class="p1">(<span class="default"><a href="nagata_2.html#K4" target="_self">inf</a> <font color="Maroon">pmet</font>,<font color="Maroon">A</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i/>;<br/>
<a NAME="E5:22_4_2"/>
( <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font> &amp;  ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 
<b>by </b><i><a class="ref" href="pre_topc.html#T48">PRE_TOPC:48</a>, <a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</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="E7:22_4_2"/><i><font color="Green">E161</font></i>: 
for <font color="Olive">pmet1</font> being  <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font></span><a href="zfmisc_1.html#K2">:]</a></span>, <a href="numbers.html#K1">REAL</a>   st <font color="Maroon">FS2</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Olive">pmet1</font> holds <br/><span class="p1">(<span class="default"><a href="nagata_2.html#K4" target="_self">inf</a> <font color="Olive">pmet1</font>,<span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i>, , </i>;<br/>
<b>consider </b><font color="Maroon">pmet1</font> being   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font></span><a href="zfmisc_1.html#K2">:]</a></span>, <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E9:22_4_2"/><i><font color="Green">E162</font></i>: 
( <font color="Maroon">FS2</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">pmet1</font> &amp; <font color="Maroon">pmet1</font> <a href="nagata_1.html#R1">is_a_pseudometric_of</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> &amp; ( for <font color="Olive">pq</font> being   <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font></span><a href="zfmisc_1.html#K2">:]</a></span> holds  <font color="Maroon">pmet1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">pq</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font> ) &amp; ( for <font color="Olive">pmet1'</font> being  <a href="pscomp_1.html#NM1">RealMap</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">T</font>,<font color="Maroon">T</font></span><a href="borsuk_1.html#K5">:]</a></span>  st <font color="Maroon">pmet1</font> <a href="funct_2.html#R1">=</a> <font color="Olive">pmet1'</font> holds <br/><font color="Olive">pmet1'</font> is <a href="pscomp_1.html#V9">continuous</a> ) )
 <b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">r</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="prepower.html#K2">GeoSeq</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nagata_2.html#K4" target="_self">inf</a> <font color="Maroon">pmet1</font>,<span class="p3">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font></span>)</span>;<br/>
<a NAME="E10:22_4_2"/>
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="prepower.html#K3">|^</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="newton.html#T102">NEWTON:102</a></i>;<br/>
<a NAME="E11:22_4_2"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="prepower.html#K2">GeoSeq</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <span class="p1">(<span class="default"><a href="nagata_2.html#K4" target="_self">inf</a> <font color="Maroon">pmet1</font>,<span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 
<b>by </b><i>, , <a class="ref" href="prepower.html#D1">PREPOWER:def 1</a></i>;<br/>
<a NAME="E12:22_4_2"/><b>then </b><i><font color="Green">E163</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="prepower.html#K2">GeoSeq</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nagata_2.html#K4" target="_self">inf</a> <font color="Maroon">pmet1</font>,<span class="p3">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="real_2.html#T122">REAL_2:122</a></i>;<br/>
<a NAME="E13:22_4_2"/><b>then </b><i><font color="Green">E164</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="prepower.html#K2">GeoSeq</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="nagata_2.html#K4" target="_self">inf</a> <font color="Maroon">pmet1</font>,<span class="p4">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="seq_2.html#T3">SEQ_2:3</a></i>;<br/>
<a NAME="E14:22_4_2"/><i><font color="Green">E195</font></i>: 
<font color="Maroon">pmet</font> <a href="nagata_1.html#R1">is_a_pseudometric_of</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E15:22_4_2"/><b>then </b>
( not <span class="p1">(<span class="default"><a href="nagata_2.html#K3" target="_self">dist</a> <font color="Maroon">pmet</font>,<font color="Maroon">p</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">A</font> is <a href="xboole_0.html#V1">empty</a> &amp; <span class="p1">(<span class="default"><a href="nagata_2.html#K3" target="_self">dist</a> <font color="Maroon">pmet</font>,<font color="Maroon">p</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">A</font> is <a href="seq_4.html#V2">bounded_below</a> )
 
<b>by </b><i/>;<br/>
<b>then 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="E17:22_4_2"/><i><font color="Green">E204</font></i>: 
( <font color="Maroon">rn</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="nagata_2.html#K3" target="_self">dist</a> <font color="Maroon">pmet</font>,<font color="Maroon">p</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">A</font> &amp; <font color="Maroon">rn</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="nagata_2.html#K3" target="_self">dist</a> <font color="Maroon">pmet</font>,<font color="Maroon">p</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="prepower.html#K2">GeoSeq</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="nagata_2.html#K4" target="_self">inf</a> <font color="Maroon">pmet1</font>,<span class="p5">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> )
 <b>by </b><i>, <a class="ref" href="seq_4.html#D5">SEQ_4:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E19:22_4_2"/><i><font color="Green">E205</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="nagata_2.html#K3" target="_self">dist</a> <font color="Maroon">pmet</font>,<font color="Maroon">p</font></span>)</span> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">rn</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="nagata_2.html#K3" target="_self">dist</a> <font color="Maroon">pmet</font>,<font color="Maroon">p</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</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">a</font> = <font color="Maroon">a</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">pa</font> = <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">a</font></span><a href="domain_1.html#K1">]</a></span> as    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font></span><a href="zfmisc_1.html#K2">:]</a></span> ;<br/>
<a NAME="E22:22_4_2"/>
( 0 <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K6">/</a> 2 &amp; 1 <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#R1">&lt;</a> 1 )
 
;<br/>
<a NAME="E23:22_4_2"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> 1
 
<b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/>
<a NAME="E24:22_4_2"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="prepower.html#K2">GeoSeq</a>  is <a href="series_1.html#V1">summable</a>
 
<b>by </b><i><a class="ref" href="series_1.html#T28">SERIES_1:28</a></i>;<br/>
<a NAME="E25:22_4_2"/><b>then </b>
( <font color="Maroon">s</font> <a href="seq_1.html#K14">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="prepower.html#K2">GeoSeq</a> </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"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="prepower.html#K2">GeoSeq</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">FS2</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">pa</font></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"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="prepower.html#K2">GeoSeq</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">FS2</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">pa</font></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">s</font> <a href="seq_1.html#K14">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="prepower.html#K2">GeoSeq</a> </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#T13">SERIES_1:13</a></i>;<br/>
<a NAME="E26:22_4_2"/><b>then </b><i><font color="Green">E206</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="prepower.html#K2">GeoSeq</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">FS2</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">pa</font></span>)</span> is <a href="series_1.html#V1">summable</a> &amp; <font color="Maroon">pmet</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">pa</font> <a href="hidden.html#R1">=</a> <font color="Maroon">pmet</font> <a href="metric_1.html#K1">.</a> <font color="Maroon">p</font>,<font color="Maroon">a</font> &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  0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="prepower.html#K2">GeoSeq</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">FS2</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">pa</font></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#T24">SERIES_1:24</a></i>;<br/>
<a NAME="E27:22_4_2"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="prepower.html#K2">GeoSeq</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">FS2</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">pa</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> <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"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="prepower.html#K2">GeoSeq</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">FS2</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">pa</font></span>)</span></span>)</span> &amp; <font color="Maroon">rn</font> <a href="hidden.html#R1">=</a> <font color="Maroon">pmet</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">pa</font> &amp; <font color="Maroon">pmet</font> is <a href="seq_2.html#V2">bounded_below</a> )
 
<b>by </b><i>, , , , <a class="ref" href="rsspace2.html#T4">RSSPACE2:4</a>, </i>;<br/>
<a NAME="E28:22_4_2"/><b>then </b>
( <font color="Maroon">rn</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="prepower.html#K2">GeoSeq</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">FS2</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">pa</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> &amp;  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nagata_2.html#K3" target="_self">dist</a> <font color="Maroon">pmet</font>,<font color="Maroon">p</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E29:22_4_2"/><b>then </b><i><font color="Green">E207</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="prepower.html#K2">GeoSeq</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">FS2</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">pa</font></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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="prepower.html#K2">GeoSeq</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="nagata_2.html#K4" target="_self">inf</a> <font color="Maroon">pmet1</font>,<span class="p4">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E30:22_4_2"/>
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T48">PRE_TOPC:48</a></i>;<br/>
<a NAME="E31:22_4_2"/><b>then </b>
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="nagata_2.html#K3" target="_self">dist</a> <font color="Maroon">pmet1</font>,<font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font> &amp; <font color="Maroon">pmet1</font> is <a href="seq_2.html#V2">bounded_below</a> )
 
<b>by </b><i>, , , <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E32:22_4_2"/><b>then </b>
( <span class="p1">(<span class="default"><a href="nagata_2.html#K3" target="_self">dist</a> <font color="Maroon">pmet1</font>,<font color="Maroon">p</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="nagata_2.html#K3" target="_self">dist</a> <font color="Maroon">pmet1</font>,<font color="Maroon">p</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font></span>)</span> &amp; not <span class="p1">(<span class="default"><a href="nagata_2.html#K3" target="_self">dist</a> <font color="Maroon">pmet1</font>,<font color="Maroon">p</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font></span>)</span> is <a href="xboole_0.html#V1">empty</a> &amp; <span class="p1">(<span class="default"><a href="nagata_2.html#K3" target="_self">dist</a> <font color="Maroon">pmet1</font>,<font color="Maroon">p</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font></span>)</span> is <a href="seq_4.html#V2">bounded_below</a> &amp;  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nagata_2.html#K3" target="_self">dist</a> <font color="Maroon">pmet1</font>,<font color="Maroon">p</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="nagata_2.html#K4" target="_self">inf</a> <font color="Maroon">pmet1</font>,<span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font> )
 
<b>by </b><i>, , , <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E33:22_4_2"/><b>then </b>
( <span class="p1">(<span class="default"><a href="nagata_2.html#K4" target="_self">inf</a> <font color="Maroon">pmet1</font>,<span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="nagata_2.html#K3" target="_self">dist</a> <font color="Maroon">pmet1</font>,<font color="Maroon">p</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> &amp; <span class="p1">(<span class="default"><a href="nagata_2.html#K3" target="_self">dist</a> <font color="Maroon">pmet1</font>,<font color="Maroon">p</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">pmet1</font> <a href="metric_1.html#K1">.</a> <font color="Maroon">p</font>,<font color="Maroon">a</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">FS2</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">pa</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">pmet1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">pa</font> &amp; <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="prepower.html#K3">|^</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 
<b>by </b><i>, , <a class="ref" href="newton.html#T102">NEWTON:102</a>, <a class="ref" href="seqfunc.html#D11">SEQFUNC:def 11</a>, <a class="ref" href="seq_4.html#D5">SEQ_4:def 5</a></i>;<br/>
<a NAME="E34:22_4_2"/><b>then </b>
( <span class="p1">(<span class="default"><a href="nagata_2.html#K4" target="_self">inf</a> <font color="Maroon">pmet1</font>,<span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">FS2</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">pa</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="prepower.html#K2">GeoSeq</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 
<b>by </b><i><a class="ref" href="prepower.html#D1">PREPOWER:def 1</a></i>;<br/>
<a NAME="E35:22_4_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="prepower.html#K2">GeoSeq</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nagata_2.html#K4" target="_self">inf</a> <font color="Maroon">pmet1</font>,<span class="p3">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="prepower.html#K2">GeoSeq</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">FS2</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">pa</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<a NAME="E36:22_4_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="prepower.html#K2">GeoSeq</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nagata_2.html#K4" target="_self">inf</a> <font color="Maroon">pmet1</font>,<span class="p3">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="prepower.html#K2">GeoSeq</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">FS2</font> <a href="seqfunc.html#K10">#</a> <font color="Maroon">pa</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="seq_1.html#T12">SEQ_1:12</a></i>;<br/>
<a NAME="E37:22_4_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="prepower.html#K2">GeoSeq</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nagata_2.html#K4" target="_self">inf</a> <font color="Maroon">pmet1</font>,<span class="p3">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</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">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="prepower.html#K2">GeoSeq</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="nagata_2.html#K4" target="_self">inf</a> <font color="Maroon">pmet1</font>,<span class="p4">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E38:22_4_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T218">XREAL_1:218</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:22_4"/>
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font> : <span class="p1">(<span class="default"><a href="nagata_2.html#K4" target="_self">inf</a> <font color="Maroon">pmet</font>,<font color="Maroon">A</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> 0 </span>}</span> 
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E14:22"/>
<font color="Maroon">T</font> is <a href="pcomps_1.html#V3">metrizable</a>
 <b>by </b><i>, </i>;<br/>


</div>
