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<div class="add">

<b>let </b><font color="Maroon" title="c1">T</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/>

<b>set </b><font color="Maroon" title="c2">cT</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>;<br/>
<b>set </b><font color="Maroon" title="c3">cTT</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c1">T</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>;<br/>
<b>set </b><font color="Maroon" title="c4">Geo</font> = <span class="p1">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> ;<br/>
<b>let </b><font color="Maroon" title="c5">s</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>let </b><font color="Maroon" title="c6">FS2</font> be    <a href="seqfunc.html#M1" title="SEQFUNC:mode.1">Functional_Sequence</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/>



<b>assume </b><a NAME="E1:20"/><i><font color="Green" title="E20">A1</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   ex <font color="Olive" title="b2">pmet</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <br/>( <font color="Maroon" title="c6">FS2</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b2">pmet</font> &amp; <font color="Olive" title="b2">pmet</font> <a href="nagata_1.html#R1" title="NAGATA_1:pred.1">is_a_pseudometric_of</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> &amp; ( for <font color="Olive" title="b3">pq</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> holds  <font color="Olive" title="b2">pmet</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">pq</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">s</font> ) &amp; ( for <font color="Olive" title="b3">pmet'</font> being  <a href="pscomp_1.html#NM1" title="PSCOMP_1:NM.1">RealMap</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c1">T</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>  st <font color="Olive" title="b2">pmet</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Olive" title="b3">pmet'</font> holds <br/><font color="Olive" title="b3">pmet'</font> is <a href="pscomp_1.html#V9" title="PSCOMP_1:attr.9">continuous</a> ) )
 ;<br/>

<b>let </b><font color="Maroon" title="c7">pmet</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/>

<b>assume </b><a NAME="E2:20"/><i><font color="Green" title="E21">A2</font></i>: 
for <font color="Olive" title="b1">pq</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> holds  <font color="Maroon" title="c7">pmet</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">pq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="series_1.html#K2" title="SERIES_1:func.2">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" title="REAL_1:func.6">/</a> 2</span>)</span> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">FS2</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Olive" title="b1">pq</font></span>)</span></span>)</span>
 ;<br/>

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ) <b>-&gt; </b>   <a href="relset_1.html#M2" title="RELSET_1:mode.2">Relation</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</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" title="REAL_1:func.6">/</a> 2</span>)</span> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1</span>)</span> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">FS2</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> $1</span>)</span>;<br/>
<b>consider </b><font color="Maroon" title="c8">GeoF</font> being    <a href="seqfunc.html#M1" title="SEQFUNC:mode.1">Functional_Sequence</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <b> such that </b><br/><a NAME="E4:20"/><i><font color="Green" title="E22">A3</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c8">GeoF</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">n</font>)
 <b>from </b><i><a class="ref" href="seqfunc.html#S1" title="SEQFUNC:sch.1">SEQFUNC:sch 1</a>();<br/></i>
<a NAME="E5:20"/><i><font color="Green" title="E23">A4</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   ex <font color="Olive" title="b2">pmet1</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <br/>( <font color="Maroon" title="c8">GeoF</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b2">pmet1</font> &amp; <font color="Olive" title="b2">pmet1</font> <a href="nagata_1.html#R1" title="NAGATA_1:pred.1">is_a_pseudometric_of</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>consider </b><font color="Maroon" title="c10">pmet1</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <b> such that </b><br/><a NAME="E2:20_1"/><i><font color="Green" title="E24">A5</font></i>: 
( <font color="Maroon" title="c6">FS2</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c9">n</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c10">pmet1</font> &amp; <font color="Maroon" title="c10">pmet1</font> <a href="nagata_1.html#R1" title="NAGATA_1:pred.1">is_a_pseudometric_of</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> &amp; ( for <font color="Olive" title="b1">pq</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> holds  <font color="Maroon" title="c10">pmet1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">pq</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">s</font> ) &amp; ( for <font color="Olive" title="b1">pmet1'</font> being  <a href="pscomp_1.html#NM1" title="PSCOMP_1:NM.1">RealMap</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c1">T</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>  st <font color="Maroon" title="c10">pmet1</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Olive" title="b1">pmet1'</font> holds <br/><font color="Olive" title="b1">pmet1'</font> is <a href="pscomp_1.html#V9" title="PSCOMP_1:attr.9">continuous</a> ) )
 <b>by </b><i><a class="txt" href="nagata_2.html#E1:20"><i><font color="Green" title="E20">A1</font></i></a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">pmet1</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> $1,$2</span>)</span>;<br/>
<b>consider </b><font color="Maroon" title="c11">GF</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <b> such that </b><br/><a NAME="E4:20_1"/><i><font color="Green" title="E25">A6</font></i>: 
for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font> holds  <font color="Maroon" title="c11">GF</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font>)
 <b>from </b><i><a class="ref" href="binop_1.html#S4" title="BINOP_1:sch.4">BINOP_1:sch 4</a>();<br/></i>
<a NAME="E5:20_1"/><i><font color="Green" title="E26">A7</font></i>: 
<font color="Maroon" title="c8">GeoF</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c9">n</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c11">GF</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:20_1_1"/><i><font color="Green" title="E27">A8</font></i>: 
<font color="Maroon" title="c8">GeoF</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c9">n</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</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" title="REAL_1:func.6">/</a> 2</span>)</span> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">FS2</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="nagata_2.html#E4:20"><i><font color="Green" title="E22">A3</font></i></a></i>;<br/>
<a NAME="E2:20_1_1"/><b>then </b><i><font color="Green" title="E28">A9</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">FS2</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">GeoF</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c10">pmet1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c11">GF</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="seq_1.html#D6" title="SEQ_1:def.6">SEQ_1:def 6</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c12">x</font>, <font color="Maroon" title="c13">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:20_1_1_1"/><i><font color="Green" title="E29">A10</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c13">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">GeoF</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c14">x'</font> = <font color="Maroon" title="c12">x</font>, <font color="Maroon" title="c15">y'</font> = <font color="Maroon" title="c13">y</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font> <b>by </b><i><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><a NAME="E3:20_1_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c8">GeoF</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c13">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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" title="REAL_1:func.6">/</a> 2</span>)</span> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">pmet1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c13">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span> &amp; <font color="Maroon" title="c11">GF</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c14">x'</font>,<font color="Maroon" title="c15">y'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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" title="REAL_1:func.6">/</a> 2</span>)</span> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">pmet1</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c14">x'</font>,<font color="Maroon" title="c15">y'</font></span>)</span> )
 <b>by </b><i><a class="txt" href="nagata_2.html#E2:20_1"><i><font color="Green" title="E24">A5</font></i></a>, <a class="txt" href="nagata_2.html#E4:20_1"><i><font color="Green" title="E25">A6</font></i></a>, <a class="txt" href="nagata_2.html#E1:20_1_1"><i><font color="Green" title="E27">A8</font></i></a>, <a class="txt" href="nagata_2.html#E1:20_1_1_1"><i><font color="Green" title="E29">A10</font></i></a>, <a class="ref" href="seq_1.html#D6" title="SEQ_1:def.6">SEQ_1:def 6</a></i>;<br/><b>hence </b><a NAME="E4:20_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">GeoF</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c13">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">GF</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c13">y</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:20_1_1"/>
<font color="Maroon" title="c8">GeoF</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c9">n</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c11">GF</font>
 <b>by </b><i><a class="txt" href="nagata_2.html#E2:20_1"><i><font color="Green" title="E24">A5</font></i></a>, <a class="txt" href="nagata_2.html#E2:20_1_1"><i><font color="Green" title="E28">A9</font></i></a>, <a class="ref" href="binop_1.html#T32" title="BINOP_1:th.32">BINOP_1:32</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:20_1"/>
<font color="Maroon" title="c11">GF</font> <a href="nagata_1.html#R1" title="NAGATA_1:pred.1">is_a_pseudometric_of</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c12">a</font>, <font color="Maroon" title="c13">b</font>, <font color="Maroon" title="c14">c</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font>;<br/><b>thus </b><a NAME="E1:20_1_2_1"/>
<font color="Maroon" title="c11">GF</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c12">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_2_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:20_1_2_1_1"/>
( <font color="Maroon" title="c11">GF</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c12">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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" title="REAL_1:func.6">/</a> 2</span>)</span> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">pmet1</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c12">a</font></span>)</span> &amp; <font color="Maroon" title="c10">pmet1</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c12">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 
<b>by </b><i><a class="txt" href="nagata_2.html#E2:20_1"><i><font color="Green" title="E24">A5</font></i></a>, <a class="txt" href="nagata_2.html#E4:20_1"><i><font color="Green" title="E25">A6</font></i></a>, <a class="ref" href="nagata_1.html#T28" title="NAGATA_1:th.28">NAGATA_1:28</a></i>;<br/>
<b>hence </b><a NAME="E2:20_1_2_1_1"/>
<font color="Maroon" title="c11">GF</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c12">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E2:20_1_2_1"/>
<font color="Maroon" title="c11">GF</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c14">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">GF</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c13">b</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">GF</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c14">c</font>,<font color="Maroon" title="c13">b</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_2_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:20_1_2_1_2"/>
<span class="p1">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c9">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="newton.html#T102" title="NEWTON:th.102">NEWTON:102</a></i>;<br/>
<a NAME="E2:20_1_2_1_2"/><b>then </b>
( <font color="Maroon" title="c10">pmet1</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c14">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">pmet1</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c13">b</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">pmet1</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c14">c</font>,<font color="Maroon" title="c13">b</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 )
 
<b>by </b><i><a class="txt" href="nagata_2.html#E2:20_1"><i><font color="Green" title="E24">A5</font></i></a>, <a class="ref" href="nagata_1.html#T28" title="NAGATA_1:th.28">NAGATA_1:28</a>, <a class="ref" href="prepower.html#D1" title="PREPOWER:def.1">PREPOWER:def 1</a></i>;<br/>
<a NAME="E3:20_1_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c10">pmet1</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c14">c</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</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" title="REAL_1:func.6">/</a> 2</span>)</span> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">pmet1</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c13">b</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">pmet1</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c14">c</font>,<font color="Maroon" title="c13">b</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</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" title="REAL_1:func.6">/</a> 2</span>)</span> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></i>;<br/>
<a NAME="E4:20_1_2_1_2"/><b>then </b>
( <font color="Maroon" title="c11">GF</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c14">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&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" title="REAL_1:func.6">/</a> 2</span>)</span> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">pmet1</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c13">b</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</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" title="REAL_1:func.6">/</a> 2</span>)</span> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">pmet1</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c14">c</font>,<font color="Maroon" title="c13">b</font></span>)</span></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" title="REAL_1:func.6">/</a> 2</span>)</span> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">pmet1</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c13">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">GF</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c13">b</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" title="REAL_1:func.6">/</a> 2</span>)</span> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">pmet1</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c14">c</font>,<font color="Maroon" title="c13">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">GF</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c14">c</font>,<font color="Maroon" title="c13">b</font> )
 
<b>by </b><i><a class="txt" href="nagata_2.html#E4:20_1"><i><font color="Green" title="E25">A6</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:20_1_2_1_2"/>
<font color="Maroon" title="c11">GF</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c14">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">GF</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c13">b</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">GF</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c14">c</font>,<font color="Maroon" title="c13">b</font></span>)</span>
 ;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:20_1_2"/>
<font color="Maroon" title="c11">GF</font> <a href="nagata_1.html#R1" title="NAGATA_1:pred.1">is_a_pseudometric_of</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>
 <b>by </b><i><a class="ref" href="nagata_1.html#T28" title="NAGATA_1:th.28">NAGATA_1:28</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:20_1"/>
 ex <font color="Olive" title="b1">pmet1</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <br/>( <font color="Maroon" title="c8">GeoF</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c9">n</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b1">pmet1</font> &amp; <font color="Olive" title="b1">pmet1</font> <a href="nagata_1.html#R1" title="NAGATA_1:pred.1">is_a_pseudometric_of</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> )
 <b>by </b><i><a class="txt" href="nagata_2.html#E5:20_1"><i><font color="Green" title="E26">A7</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>set </b><font color="Maroon" title="c9">SGeo</font> = <font color="Maroon" title="c5">s</font> <a href="seq_1.html#K14" title="SEQ_1:func.14">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span>;<br/>
<a NAME="E6:20"/><i><font color="Green" title="E24">A11</font></i>: 
for <font color="Olive" title="b1">pq</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span><br/> for <font color="Olive" title="b2">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/> ( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">GeoF</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Olive" title="b1">pq</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">k</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c8">GeoF</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Olive" title="b1">pq</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="seq_1.html#K14" title="SEQ_1:func.14">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">k</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">pq</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>;<br/><b>let </b><font color="Maroon" title="c11">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>



<b>consider </b><font color="Maroon" title="c12">pmet1</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <b> such that </b><br/><a NAME="E2:20_2"/><i><font color="Green" title="E25">A12</font></i>: 
( <font color="Maroon" title="c6">FS2</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c11">k</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c12">pmet1</font> &amp; <font color="Maroon" title="c12">pmet1</font> <a href="nagata_1.html#R1" title="NAGATA_1:pred.1">is_a_pseudometric_of</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> &amp; ( for <font color="Olive" title="b1">pq</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> holds  <font color="Maroon" title="c12">pmet1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">pq</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">s</font> ) &amp; ( for <font color="Olive" title="b1">pmet1'</font> being  <a href="pscomp_1.html#NM1" title="PSCOMP_1:NM.1">RealMap</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c1">T</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>  st <font color="Maroon" title="c12">pmet1</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Olive" title="b1">pmet1'</font> holds <br/><font color="Olive" title="b1">pmet1'</font> is <a href="pscomp_1.html#V9" title="PSCOMP_1:attr.9">continuous</a> ) )
 <b>by </b><i><a class="txt" href="nagata_2.html#E1:20"><i><font color="Green" title="E20">A1</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c13">x</font>, <font color="Maroon" title="c14">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:20_2"/><i><font color="Green" title="E26">A13</font></i>: 
( <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> &amp; <font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c14">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">pq</font> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c15">x</font> = <font color="Maroon" title="c13">x</font>, <font color="Maroon" title="c16">y</font> = <font color="Maroon" title="c14">y</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font> <b>by </b><i><a class="txt" href="nagata_2.html#E4:20_2"><i><font color="Green" title="E26">A13</font></i></a></i>;<br/>
<a NAME="E6:20_2"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">FS2</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="txt" href="nagata_2.html#E2:20_2"><i><font color="Green" title="E25">A12</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E7:20_2"/><b>then </b>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</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" title="REAL_1:func.6">/</a> 2</span>)</span> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">FS2</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c11">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="seq_1.html#D6" title="SEQ_1:def.6">SEQ_1:def 6</a></i>;<br/>
<b>then </b><i><font color="Green" title="E27">A14</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" title="REAL_1:func.6">/</a> 2</span>)</span> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">FS2</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c10">pq</font></span>)</span> = 
<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" title="REAL_1:func.6">/</a> 2</span>)</span> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">FS2</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c11">k</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c10">pq</font>
<b>by </b><i><a class="ref" href="seq_1.html#D6" title="SEQ_1:def.6">SEQ_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">GeoF</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c10">pq</font>
<b>by </b><i><a class="txt" href="nagata_2.html#E4:20"><i><font color="Green" title="E22">A3</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">GeoF</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c10">pq</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">k</font>
<b>by </b><i><a class="ref" href="seqfunc.html#D11" title="SEQFUNC:def.11">SEQFUNC:def 11</a></i>
;<br/>
<a NAME="E9:20_2"/>
<span class="p1">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c11">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="newton.html#T102" title="NEWTON:th.102">NEWTON:102</a></i>;<br/>
<a NAME="E10:20_2"/><b>then </b>
( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">pmet1</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c15">x</font>,<font color="Maroon" title="c16">y</font> &amp; <font color="Maroon" title="c12">pmet1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">pq</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">s</font> &amp; <font color="Maroon" title="c12">pmet1</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c15">x</font>,<font color="Maroon" title="c16">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">pmet1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">pq</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 )
 
<b>by </b><i><a class="txt" href="nagata_2.html#E2:20_2"><i><font color="Green" title="E25">A12</font></i></a>, <a class="txt" href="nagata_2.html#E4:20_2"><i><font color="Green" title="E26">A13</font></i></a>, <a class="ref" href="nagata_1.html#T29" title="NAGATA_1:th.29">NAGATA_1:29</a>, <a class="ref" href="prepower.html#D1" title="PREPOWER:def.1">PREPOWER:def 1</a></i>;<br/>
<a NAME="E11:20_2"/><b>then </b>
( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&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" title="REAL_1:func.6">/</a> 2</span>)</span> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">pmet1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">pq</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" title="REAL_1:func.6">/</a> 2</span>)</span> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">pmet1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">pq</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&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" title="REAL_1:func.6">/</a> 2</span>)</span> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c5">s</font> )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a>, <a class="ref" href="real_2.html#T121" title="REAL_2:th.121">REAL_2:121</a></i>;<br/>
<b>hence </b><a NAME="E12:20_2"/>
( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">GeoF</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c10">pq</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">k</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c8">GeoF</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c10">pq</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="seq_1.html#K14" title="SEQ_1:func.14">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">k</font> )
 <b>by </b><i><a class="txt" href="nagata_2.html#E2:20_2"><i><font color="Green" title="E25">A12</font></i></a>, <a class="txt" href="nagata_2.html#E8:20_2"><i><font color="Green" title="E27">A14</font></i></a>, <a class="ref" href="seq_1.html#T13" title="SEQ_1:th.13">SEQ_1:13</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:20"/>
( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2 &amp; 1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 )
 
;<br/>
<a NAME="E8:20"/><b>then </b>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1
 
<b>by </b><i><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/>
<a NAME="E9:20"/><b>then </b><i><font color="Green" title="E25">A15</font></i>: 
<span class="p1">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a>  is <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a>
 
<b>by </b><i><a class="ref" href="series_1.html#T28" title="SERIES_1:th.28">SERIES_1:28</a></i>;<br/>
<a NAME="E10:20"/><i><font color="Green" title="E26">A16</font></i>: 
for <font color="Olive" title="b1">pq</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> holds  <font color="Maroon" title="c8">GeoF</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Olive" title="b1">pq</font> is <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">pq</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>;<br/>

<a NAME="E1:20_3"/>
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/> ( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">GeoF</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c10">pq</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">k</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c8">GeoF</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c10">pq</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="seq_1.html#K14" title="SEQ_1:func.14">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">k</font> &amp; <font color="Maroon" title="c5">s</font> <a href="seq_1.html#K14" title="SEQ_1:func.14">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span> is <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a> )
 
<b>by </b><i><a class="txt" href="nagata_2.html#E6:20"><i><font color="Green" title="E24">A11</font></i></a>, <a class="txt" href="nagata_2.html#E9:20"><i><font color="Green" title="E25">A15</font></i></a>, <a class="ref" href="series_1.html#T13" title="SERIES_1:th.13">SERIES_1:13</a></i>;<br/>
<b>hence </b><a NAME="E2:20_3"/>
<font color="Maroon" title="c8">GeoF</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c10">pq</font> is <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a>
 <b>by </b><i><a class="ref" href="series_1.html#T24" title="SERIES_1:th.24">SERIES_1:24</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:20"/><i><font color="Green" title="E27">A17</font></i>: 
for <font color="Olive" title="b1">pq</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> holds  <font color="Maroon" title="c7">pmet</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">pq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">GeoF</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Olive" title="b1">pq</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">pq</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_4_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c11">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:20_4_1"/>
<font color="Maroon" title="c11">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 ;<br/><b>then reconsider </b><font color="Maroon" title="c12">k</font> = <font color="Maroon" title="c11">z</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/><b>consider </b><font color="Maroon" title="c13">pmet1</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <b> such that </b><br/><a NAME="E4:20_4_1"/><i><font color="Green" title="E28">A18</font></i>: 
( <font color="Maroon" title="c6">FS2</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c12">k</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c13">pmet1</font> &amp; <font color="Maroon" title="c13">pmet1</font> <a href="nagata_1.html#R1" title="NAGATA_1:pred.1">is_a_pseudometric_of</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> &amp; ( for <font color="Olive" title="b1">pq</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> holds  <font color="Maroon" title="c13">pmet1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">pq</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">s</font> ) &amp; ( for <font color="Olive" title="b1">pmet1'</font> being  <a href="pscomp_1.html#NM1" title="PSCOMP_1:NM.1">RealMap</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c1">T</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>  st <font color="Maroon" title="c13">pmet1</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Olive" title="b1">pmet1'</font> holds <br/><font color="Olive" title="b1">pmet1'</font> is <a href="pscomp_1.html#V9" title="PSCOMP_1:attr.9">continuous</a> ) )
 <b>by </b><i><a class="txt" href="nagata_2.html#E1:20"><i><font color="Green" title="E20">A1</font></i></a></i>;<br/><a NAME="E5:20_4_1"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">FS2</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c12">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="txt" href="nagata_2.html#E4:20_4_1"><i><font color="Green" title="E28">A18</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E6:20_4_1"/><b>then </b>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</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" title="REAL_1:func.6">/</a> 2</span>)</span> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">k</font></span>)</span> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">FS2</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="ref" href="seq_1.html#D6" title="SEQ_1:def.6">SEQ_1:def 6</a></i>;<br/><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" title="REAL_1:func.6">/</a> 2</span>)</span> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">k</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">FS2</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c12">k</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c10">pq</font></span>)</span> = 
<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" title="REAL_1:func.6">/</a> 2</span>)</span> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">k</font></span>)</span> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">FS2</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c10">pq</font>
<b>by </b><i><a class="ref" href="seq_1.html#D6" title="SEQ_1:def.6">SEQ_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">GeoF</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c12">k</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c10">pq</font>
<b>by </b><i><a class="txt" href="nagata_2.html#E4:20"><i><font color="Green" title="E22">A3</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">GeoF</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c10">pq</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">k</font>
<b>by </b><i><a class="ref" href="seqfunc.html#D11" title="SEQFUNC:def.11">SEQFUNC:def 11</a></i>
;<br/><b>then </b><span class="p1">(<span class="default"><font color="Maroon" title="c8">GeoF</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c10">pq</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">k</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" title="REAL_1:func.6">/</a> 2</span>)</span> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">k</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">FS2</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c10">pq</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">k</font></span>)</span>
<b>by </b><i><a class="ref" href="seqfunc.html#D11" title="SEQFUNC:def.11">SEQFUNC:def 11</a></i>
<br/>.= 
<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" title="REAL_1:func.6">/</a> 2</span>)</span> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">FS2</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c10">pq</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">k</font>
<b>by </b><i><a class="ref" href="seq_1.html#T12" title="SEQ_1:th.12">SEQ_1:12</a></i>
;<br/><b>hence </b><a NAME="E9:20_4_1"/>
<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" title="REAL_1:func.6">/</a> 2</span>)</span> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">FS2</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c10">pq</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">GeoF</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c10">pq</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">z</font>
 ;<br/></div><b>end;</b></div>
<a NAME="E2:20_4"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">FS2</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c10">pq</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c8">GeoF</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c10">pq</font>
 
<b>by </b><i><a class="ref" href="seq_1.html#T8" title="SEQ_1:th.8">SEQ_1:8</a></i>;<br/>
<b>hence </b><a NAME="E3:20_4"/>
<font color="Maroon" title="c7">pmet</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">pq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">GeoF</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c10">pq</font></span>)</span>
 <b>by </b><i><a class="txt" href="nagata_2.html#E2:20"><i><font color="Green" title="E21">A2</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon" title="c10">GeoF'</font> = <font color="Maroon" title="c8">GeoF</font> as    <a href="seqfunc.html#M1" title="SEQFUNC:mode.1">Functional_Sequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c1">T</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <b>by </b><i><a class="ref" href="borsuk_1.html#D5" title="BORSUK_1:def.5">BORSUK_1:def 5</a></i>;<br/>
<a NAME="E13:20"/><i><font color="Green" title="E28">A19</font></i>: 
for <font color="Olive" title="b1">pq</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span><br/> for <font color="Olive" title="b2">pq'</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c1">T</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>  st <font color="Olive" title="b1">pq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">pq'</font> holds <br/><font color="Maroon" title="c8">GeoF</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Olive" title="b1">pq</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c10">GeoF'</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Olive" title="b2">pq'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">pq</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>;<br/><b>let </b><font color="Maroon" title="c12">pq'</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c1">T</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>;<br/>



<b>assume </b><a NAME="E1:20_5"/><i><font color="Green" title="E29">A20</font></i>: 
<font color="Maroon" title="c11">pq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">pq'</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_5_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c13">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:20_5_1"/><i><font color="Green" title="E30">A21</font></i>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 ;<br/><b>reconsider </b><font color="Maroon" title="c14">k</font> = <font color="Maroon" title="c13">x</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="nagata_2.html#E1:20_5_1"><i><font color="Green" title="E30">A21</font></i></a></i>;<br/><a NAME="E3:20_5_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c8">GeoF</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c11">pq</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">GeoF</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c14">k</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">pq</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c10">GeoF'</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c12">pq'</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">GeoF'</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c14">k</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c12">pq'</font> )
 <b>by </b><i><a class="ref" href="seqfunc.html#D11" title="SEQFUNC:def.11">SEQFUNC:def 11</a></i>;<br/><b>hence </b><a NAME="E4:20_5_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">GeoF</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c11">pq</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">GeoF'</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c12">pq'</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c13">x</font>
 <b>by </b><i><a class="txt" href="nagata_2.html#E1:20_5"><i><font color="Green" title="E29">A20</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:20_5"/>
<font color="Maroon" title="c8">GeoF</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c11">pq</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c10">GeoF'</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c12">pq'</font>
 <b>by </b><i><a class="ref" href="seq_1.html#T8" title="SEQ_1:th.8">SEQ_1:8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E14:20"/><i><font color="Green" title="E29">A22</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   ex <font color="Olive" title="b2">f</font> being  <a href="pscomp_1.html#NM1" title="PSCOMP_1:NM.1">RealMap</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c1">T</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> st <br/>( <font color="Maroon" title="c10">GeoF'</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b2">f</font> &amp; <font color="Olive" title="b2">f</font> is <a href="pscomp_1.html#V9" title="PSCOMP_1:attr.9">continuous</a> &amp; ( for <font color="Olive" title="b3">pq'</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c1">T</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> holds  <font color="Olive" title="b2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">pq'</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>consider </b><font color="Maroon" title="c12">pmet1</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <b> such that </b><br/><a NAME="E2:20_6"/><i><font color="Green" title="E30">A23</font></i>: 
( <font color="Maroon" title="c6">FS2</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c11">n</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c12">pmet1</font> &amp; <font color="Maroon" title="c12">pmet1</font> <a href="nagata_1.html#R1" title="NAGATA_1:pred.1">is_a_pseudometric_of</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> &amp; ( for <font color="Olive" title="b1">pq</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> holds  <font color="Maroon" title="c12">pmet1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">pq</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">s</font> ) &amp; ( for <font color="Olive" title="b1">pmet1'</font> being  <a href="pscomp_1.html#NM1" title="PSCOMP_1:NM.1">RealMap</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c1">T</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>  st <font color="Maroon" title="c12">pmet1</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Olive" title="b1">pmet1'</font> holds <br/><font color="Olive" title="b1">pmet1'</font> is <a href="pscomp_1.html#V9" title="PSCOMP_1:attr.9">continuous</a> ) )
 <b>by </b><i><a class="txt" href="nagata_2.html#E1:20"><i><font color="Green" title="E20">A1</font></i></a></i>;<br/>
<a NAME="E3:20_6"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c1">T</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D5" title="BORSUK_1:def.5">BORSUK_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c13">pmet1'</font> = <font color="Maroon" title="c12">pmet1</font> as   <a href="pscomp_1.html#NM1" title="PSCOMP_1:NM.1">RealMap</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c1">T</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> ;<br/>
<b>reconsider </b><font color="Maroon" title="c14">pR</font> = <font color="Maroon" title="c13">pmet1'</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c1">T</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  <b>by </b><i><a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></i>;<br/>
<a NAME="E6:20_6"/>
<font color="Maroon" title="c13">pmet1'</font> is <a href="pscomp_1.html#V9" title="PSCOMP_1:attr.9">continuous</a>
 
<b>by </b><i><a class="txt" href="nagata_2.html#E2:20_6"><i><font color="Green" title="E30">A23</font></i></a></i>;<br/>
<a NAME="E7:20_6"/><b>then </b>
<font color="Maroon" title="c14">pR</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
<b>by </b><i><a class="ref" href="topreal6.html#T83" title="TOPREAL6:th.83">TOPREAL6:83</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c15">fR</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c1">T</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> <b> such that </b><br/><a NAME="E9:20_6"/><i><font color="Green" title="E31">A24</font></i>: 
( ( for <font color="Olive" title="b1">pq'</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c1">T</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span><br/> for <font color="Olive" title="b2">rn</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Maroon" title="c14">pR</font> <a href="nagata_2.html#K1" title="NAGATA_2:func.1">.</a> <font color="Olive" title="b1">pq'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">rn</font> holds <br/><font color="Maroon" title="c15">fR</font> <a href="nagata_2.html#K1" title="NAGATA_2:func.1">.</a> <font color="Olive" title="b1">pq'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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" title="REAL_1:func.6">/</a> 2</span>)</span> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b2">rn</font> ) &amp; <font color="Maroon" title="c15">fR</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> )
 <b>by </b><i><a class="ref" href="jgraph_2.html#T33" title="JGRAPH_2:th.33">JGRAPH_2:33</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c16">f</font> = <font color="Maroon" title="c15">fR</font> as   <a href="pscomp_1.html#NM1" title="PSCOMP_1:NM.1">RealMap</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c1">T</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <b>by </b><i><a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></i>;<br/>
<a NAME="E11:20_6"/><i><font color="Green" title="E32">A25</font></i>: 
<font color="Maroon" title="c10">GeoF'</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c11">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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" title="REAL_1:func.6">/</a> 2</span>)</span> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">FS2</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c11">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="nagata_2.html#E4:20"><i><font color="Green" title="E22">A3</font></i></a></i>;<br/>
<a NAME="E12:20_6"/><b>then </b><i><font color="Green" title="E33">A26</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">FS2</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">GeoF'</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c11">n</font></span>)</span> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c12">pmet1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c16">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c1">T</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> &amp; <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c1">T</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> )
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D5" title="BORSUK_1:def.5">BORSUK_1:def 5</a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="seq_1.html#D6" title="SEQ_1:def.6">SEQ_1:def 6</a></i>;<br/>
<div><i><font color="Green" title="E34">A27</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="20_6_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c17">pq'</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c1">T</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>;<br/><b>assume </b><a NAME="E1:20_6_1"/>
<font color="Maroon" title="c17">pq'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">GeoF'</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c11">n</font></span>)</span>
 ;<br/><a NAME="E2:20_6_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c10">GeoF'</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c17">pq'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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" title="REAL_1:func.6">/</a> 2</span>)</span> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">pmet1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c17">pq'</font></span>)</span> &amp; <font color="Maroon" title="c16">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">pq'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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" title="REAL_1:func.6">/</a> 2</span>)</span> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">pmet1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c17">pq'</font></span>)</span> )
 <b>by </b><i><a class="txt" href="nagata_2.html#E2:20_6"><i><font color="Green" title="E30">A23</font></i></a>, <a class="txt" href="nagata_2.html#E9:20_6"><i><font color="Green" title="E31">A24</font></i></a>, <a class="txt" href="nagata_2.html#E11:20_6"><i><font color="Green" title="E32">A25</font></i></a>, <a class="txt" href="nagata_2.html#E12:20_6"><i><font color="Green" title="E33">A26</font></i></a>, <a class="ref" href="seq_1.html#D6" title="SEQ_1:def.6">SEQ_1:def 6</a></i>;<br/><b>hence </b><a NAME="E3:20_6_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c10">GeoF'</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c17">pq'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">pq'</font>
 ;<br/></div><b>end;</b></div>
<b>take </b>
<font color="Maroon" title="c16">f</font>
;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_6_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c17">pq'</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c1">T</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>;<br/><b>reconsider </b><font color="Maroon" title="c18">pq</font> = <font color="Maroon" title="c17">pq'</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <b>by </b><i><a class="ref" href="borsuk_1.html#D5" title="BORSUK_1:def.5">BORSUK_1:def 5</a></i>;<br/><a NAME="E2:20_6_2"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c10">GeoF'</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c17">pq'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">GeoF</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c18">pq</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">n</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c8">GeoF</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c18">pq</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 )
 <b>by </b><i><a class="txt" href="nagata_2.html#E6:20"><i><font color="Green" title="E24">A11</font></i></a>, <a class="ref" href="seqfunc.html#D11" title="SEQFUNC:def.11">SEQFUNC:def 11</a></i>;<br/><b>hence </b><a NAME="E3:20_6_2"/>
<font color="Maroon" title="c16">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">pq'</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0
 <b>by </b><i><a class="txt" href="nagata_2.html#E2:20_6"><i><font color="Green" title="E30">A23</font></i></a>, <a class="txt" href="nagata_2.html#E12:20_6"><i><font color="Green" title="E33">A26</font></i></a>, <a class="txt" href="nagata_2.html#E13:20_6"><i><font color="Green" title="E34">A27</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E15:20_6"/>
( <font color="Maroon" title="c10">GeoF'</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c11">n</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c16">f</font> &amp; <font color="Maroon" title="c16">f</font> is <a href="pscomp_1.html#V9" title="PSCOMP_1:attr.9">continuous</a> &amp; ( for <font color="Olive" title="b1">pq'</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c1">T</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> holds  <font color="Maroon" title="c16">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">pq'</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 ) )
 <b>by </b><i><a class="txt" href="nagata_2.html#E2:20_6"><i><font color="Green" title="E30">A23</font></i></a>, <a class="txt" href="nagata_2.html#E9:20_6"><i><font color="Green" title="E31">A24</font></i></a>, <a class="txt" href="nagata_2.html#E12:20_6"><i><font color="Green" title="E33">A26</font></i></a>, <a class="txt" href="nagata_2.html#E13:20_6"><i><font color="Green" title="E34">A27</font></i></a>, <a class="ref" href="partfun1.html#T34" title="PARTFUN1:th.34">PARTFUN1:34</a>, <a class="ref" href="topreal6.html#T83" title="TOPREAL6:th.83">TOPREAL6:83</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E15:20"/><i><font color="Green" title="E30">A28</font></i>: 
 ex <font color="Olive" title="b1">seq</font> being  <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a> st <br/>( <font color="Olive" title="b1">seq</font> is <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a> &amp; ( for <font color="Olive" title="b2">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b3">pq'</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c1">T</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c10">GeoF'</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Olive" title="b3">pq'</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">seq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">n</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_7"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon" title="c5">s</font> <a href="seq_1.html#K14" title="SEQ_1:func.14">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span>
;<br/>

<b>thus </b><a NAME="E1:20_7"/>
<font color="Maroon" title="c5">s</font> <a href="seq_1.html#K14" title="SEQ_1:func.14">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span> is <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a>
 <b>by </b><i><a class="txt" href="nagata_2.html#E9:20"><i><font color="Green" title="E25">A15</font></i></a>, <a class="ref" href="series_1.html#T13" title="SERIES_1:th.13">SERIES_1:13</a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_7_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c11">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c12">pq'</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c1">T</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>;<br/><b>reconsider </b><font color="Maroon" title="c13">pq</font> = <font color="Maroon" title="c12">pq'</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <b>by </b><i><a class="ref" href="borsuk_1.html#D5" title="BORSUK_1:def.5">BORSUK_1:def 5</a></i>;<br/><a NAME="E2:20_7_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c10">GeoF'</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c12">pq'</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">GeoF</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c13">pq</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">n</font>
 <b>by </b><i><a class="txt" href="nagata_2.html#E13:20"><i><font color="Green" title="E28">A19</font></i></a></i>;<br/><b>hence </b><a NAME="E3:20_7_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c10">GeoF'</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c12">pq'</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="seq_1.html#K14" title="SEQ_1:func.14">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">n</font>
 <b>by </b><i><a class="txt" href="nagata_2.html#E6:20"><i><font color="Green" title="E24">A11</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:20_7"/>
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b2">pq'</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c1">T</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c10">GeoF'</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Olive" title="b2">pq'</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="seq_1.html#K14" title="SEQ_1:func.14">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font>
 ;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_8"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c11">pmet1'</font> be   <a href="pscomp_1.html#NM1" title="PSCOMP_1:NM.1">RealMap</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c1">T</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>;<br/><b>assume </b><a NAME="E1:20_8"/><i><font color="Green" title="E31">A29</font></i>: 
<font color="Maroon" title="c7">pmet</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c11">pmet1'</font>
 ;<br/><a NAME="E2:20_8"/>
for <font color="Olive" title="b1">pq'</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c1">T</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> holds  <font color="Maroon" title="c11">pmet1'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">pq'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">GeoF'</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Olive" title="b1">pq'</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_8_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">pq'</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c1">T</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>;<br/>

<b>reconsider </b><font color="Maroon" title="c13">pq</font> = <font color="Maroon" title="c12">pq'</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <b>by </b><i><a class="ref" href="borsuk_1.html#D5" title="BORSUK_1:def.5">BORSUK_1:def 5</a></i>;<br/>
<a NAME="E2:20_8_1"/>
( <font color="Maroon" title="c7">pmet</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">pq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">GeoF</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c13">pq</font></span>)</span> &amp; <font color="Maroon" title="c8">GeoF</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c13">pq</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c10">GeoF'</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c12">pq'</font> )
 
<b>by </b><i><a class="txt" href="nagata_2.html#E11:20"><i><font color="Green" title="E27">A17</font></i></a>, <a class="txt" href="nagata_2.html#E13:20"><i><font color="Green" title="E28">A19</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:20_8_1"/>
<font color="Maroon" title="c11">pmet1'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">pq'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">GeoF'</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c12">pq'</font></span>)</span>
 <b>by </b><i><a class="txt" href="nagata_2.html#E1:20_8"><i><font color="Green" title="E31">A29</font></i></a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:20_8"/>
<font color="Maroon" title="c11">pmet1'</font> is <a href="pscomp_1.html#V9" title="PSCOMP_1:attr.9">continuous</a>
 <b>by </b><i><a class="txt" href="nagata_2.html#E14:20"><i><font color="Green" title="E29">A22</font></i></a>, <a class="txt" href="nagata_2.html#E15:20"><i><font color="Green" title="E30">A28</font></i></a>, <a class="ref" href="nagata_2.html#T14" target="_self" title="NAGATA_2:th.14">Th14</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E17:20"/>
( <font color="Maroon" title="c7">pmet</font> <a href="nagata_1.html#R1" title="NAGATA_1:pred.1">is_a_pseudometric_of</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> &amp; ( for <font color="Olive" title="b1">pmet'</font> being  <a href="pscomp_1.html#NM1" title="PSCOMP_1:NM.1">RealMap</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c1">T</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>  st <font color="Maroon" title="c7">pmet</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Olive" title="b1">pmet'</font> holds <br/><font color="Olive" title="b1">pmet'</font> is <a href="pscomp_1.html#V9" title="PSCOMP_1:attr.9">continuous</a> ) )
 <b>by </b><i><a class="txt" href="nagata_2.html#E5:20"><i><font color="Green" title="E23">A4</font></i></a>, <a class="txt" href="nagata_2.html#E10:20"><i><font color="Green" title="E26">A16</font></i></a>, <a class="txt" href="nagata_2.html#E11:20"><i><font color="Green" title="E27">A17</font></i></a>, <a class="ref" href="nagata_2.html#T11" target="_self" title="NAGATA_2:th.11">Th11</a></i>;<br/>


</div>
