<?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>set </b><font color="Maroon">cT</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>;<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="E1:16"/><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">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> )
 <b>and </b><br/><a NAME="E2:16"/><i><font color="Green">E60</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">FS2</font> <a href="seqfunc.html#K10">#</a> <font color="Olive">pq</font> is <a href="series_1.html#V1">summable</a>
 ;<br/>

<b>let </b><font color="Maroon">pmet</font> be   <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> ;<br/>

<b>assume </b><a NAME="E3:16"/><i><font color="Green">E61</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"><font color="Maroon">FS2</font> <a href="seqfunc.html#K10">#</a> <font color="Olive">pq</font></span>)</span>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>, <font color="Maroon">b</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>, <font color="Maroon">c</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>;<br/><b>thus </b><a NAME="E1:16_1"/>
<font color="Maroon">pmet</font> <a href="metric_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">aa</font> = <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">a</font></span><a href="domain_1.html#K1">]</a></span>;<br/>
<b>set </b><font color="Maroon">F</font> = <font color="Maroon">FS2</font> <a href="seqfunc.html#K10">#</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">a</font></span><a href="domain_1.html#K1">]</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>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="E2:16_1_1_1"/><i><font color="Green">E62</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> )
 <b>by </b><i/>;<br/><a NAME="E3:16_1_1_1"/>
<font color="Maroon">pmet1</font> <a href="metric_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, <a class="ref" href="nagata_1.html#T28">NAGATA_1:28</a></i>;<br/><b>hence </b><a NAME="E4:16_1_1_1"/>
<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">a</font>,<font color="Maroon">a</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 <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">a</font>,<font color="Maroon">a</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>
 <b>by </b><i>, <a class="ref" href="seqfunc.html#D11">SEQFUNC:def 11</a></i>;<br/></div><b>end;</b></div>
<a NAME="E2:16_1_1"/><b>then </b>
( <font color="Maroon">FS2</font> <a href="seqfunc.html#K10">#</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">a</font></span><a href="domain_1.html#K1">]</a></span> <a href="funct_2.html#R2">=</a> 0 <a href="seq_1.html#K14">(#)</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">a</font>,<font color="Maroon">a</font></span><a href="domain_1.html#K1">]</a></span></span>)</span> &amp; <font color="Maroon">FS2</font> <a href="seqfunc.html#K10">#</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">a</font></span><a href="domain_1.html#K1">]</a></span> is <a href="series_1.html#V1">summable</a> )
 
<b>by </b><i>, <a class="ref" href="seq_1.html#T13">SEQ_1:13</a></i>;<br/>
<a NAME="E3:16_1_1"/><b>then </b>
 <a href="series_1.html#K2">Sum</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">a</font>,<font color="Maroon">a</font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="hidden.html#R1">=</a> 0 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="series_1.html#K2">Sum</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</font>,<font color="Maroon">a</font></span><a href="domain_1.html#K1">]</a></span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="series_1.html#T13">SERIES_1:13</a></i>;<br/>
<b>hence </b><a NAME="E4:16_1_1"/>
<font color="Maroon">pmet</font> <a href="metric_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E2:16_1"/>
<font color="Maroon">pmet</font> <a href="metric_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">c</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">pmet</font> <a href="metric_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">pmet</font> <a href="metric_1.html#K1">.</a> <font color="Maroon">c</font>,<font color="Maroon">b</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_2"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">ab</font> = <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K1">]</a></span>;<br/>
<b>set </b><font color="Maroon">cb</font> = <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K1">]</a></span>;<br/>
<b>set </b><font color="Maroon">ac</font> = <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="domain_1.html#K1">]</a></span>;<br/>
<b>set </b><font color="Maroon">Fac</font> = <font color="Maroon">FS2</font> <a href="seqfunc.html#K10">#</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="domain_1.html#K1">]</a></span>;<br/>
<b>set </b><font color="Maroon">Fab</font> = <font color="Maroon">FS2</font> <a href="seqfunc.html#K10">#</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K1">]</a></span>;<br/>
<b>set </b><font color="Maroon">Fcb</font> = <font color="Maroon">FS2</font> <a href="seqfunc.html#K10">#</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K1">]</a></span>;<br/>
<a NAME="E1:16_1_2"/>
( <font color="Maroon">FS2</font> <a href="seqfunc.html#K10">#</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K1">]</a></span> is <a href="series_1.html#V1">summable</a> &amp; <font color="Maroon">FS2</font> <a href="seqfunc.html#K10">#</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K1">]</a></span> is <a href="series_1.html#V1">summable</a> )
 
<b>by </b><i/>;<br/>
<a NAME="E2:16_1_2"/><b>then </b><i><font color="Green">E63</font></i>: 
( <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">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="seq_1.html#K9">+</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">c</font>,<font color="Maroon">b</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"><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</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="seq_1.html#K9">+</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">c</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K1">]</a></span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K2">Sum</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</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K1">]</a></span></span>)</span></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">FS2</font> <a href="seqfunc.html#K10">#</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K1">]</a></span></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="series_1.html#T10">SERIES_1:10</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><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="E2:16_1_2_1"/><i><font color="Green">E64</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> )
 <b>by </b><i/>;<br/><a NAME="E3:16_1_2_1"/>
( <font color="Maroon">pmet1</font> <a href="metric_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">c</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">pmet1</font> <a href="metric_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">pmet1</font> <a href="metric_1.html#K1">.</a> <font color="Maroon">c</font>,<font color="Maroon">b</font></span>)</span> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">pmet1</font> <a href="metric_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">c</font> &amp; <font color="Maroon">pmet1</font> <a href="metric_1.html#K1">.</a> <font color="Maroon">c</font>,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">pmet1</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Maroon">pmet1</font> <a href="metric_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">pmet1</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="domain_1.html#K1">]</a></span> )
 <b>by </b><i>, <a class="ref" href="nagata_1.html#T28">NAGATA_1:28</a>, <a class="ref" href="nagata_1.html#T29">NAGATA_1:29</a></i>;<br/><a NAME="E4:16_1_2_1"/><b>then </b>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <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">a</font>,<font color="Maroon">c</font></span><a href="domain_1.html#K1">]</a></span> &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">a</font>,<font color="Maroon">c</font></span><a href="domain_1.html#K1">]</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<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="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<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="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K1">]</a></span></span>)</span> &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">a</font>,<font color="Maroon">c</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</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">a</font>,<font color="Maroon">c</font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> &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">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</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">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> )
 <b>by </b><i>, <a class="ref" href="seqfunc.html#D11">SEQFUNC:def 11</a></i>;<br/><a NAME="E5:16_1_2_1"/><b>then </b>
( 0 <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> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> &amp; <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">a</font>,<font color="Maroon">c</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="xxreal_0.html#R1">&lt;=</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">a</font>,<font color="Maroon">b</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="real_1.html#K3">+</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">c</font>,<font color="Maroon">b</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> )
 <b>by </b><i><a class="ref" href="seqfunc.html#D11">SEQFUNC:def 11</a></i>;<br/><b>hence </b><a NAME="E6:16_1_2_1"/>
( 0 <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> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> &amp; <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">a</font>,<font color="Maroon">c</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="xxreal_0.html#R1">&lt;=</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">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="seq_1.html#K9">+</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">c</font>,<font color="Maroon">b</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> )
 <b>by </b><i><a class="ref" href="seq_1.html#T11">SEQ_1:11</a></i>;<br/></div><b>end;</b></div>
<a NAME="E4:16_1_2"/><b>then </b>
(  <a href="series_1.html#K2">Sum</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">a</font>,<font color="Maroon">c</font></span><a href="domain_1.html#K1">]</a></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"><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</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="seq_1.html#K9">+</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">c</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K1">]</a></span></span>)</span></span>)</span> &amp;  <a href="series_1.html#K2">Sum</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">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="hidden.html#R1">=</a> <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">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K1">]</a></span> &amp;  <a href="series_1.html#K2">Sum</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">c</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="hidden.html#R1">=</a> <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">c</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K1">]</a></span> &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">a</font>,<font color="Maroon">b</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">a</font>,<font color="Maroon">b</font> &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">c</font>,<font color="Maroon">b</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">c</font>,<font color="Maroon">b</font> )
 
<b>by </b><i>, , <a class="ref" href="series_1.html#T24">SERIES_1:24</a></i>;<br/>
<b>hence </b><a NAME="E5:16_1_2"/>
<font color="Maroon">pmet</font> <a href="metric_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">c</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">pmet</font> <a href="metric_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">pmet</font> <a href="metric_1.html#K1">.</a> <font color="Maroon">c</font>,<font color="Maroon">b</font></span>)</span>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E5:16"/>
<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><a class="ref" href="nagata_1.html#T28">NAGATA_1:28</a></i>;<br/>


</div>
