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

<b>let </b><font color="Maroon">M</font> be  non <a href="struct_0.html#V3">empty</a> <a href="metric_1.html#NM1">MetrSpace</a>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">M</font></span>)</span>;<br/><b>let </b><font color="Maroon">x'</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">M</font>;<br/>





<b>assume </b><a NAME="E1:14"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x'</font>
 ;<br/>

<b>set </b><font color="Maroon">B</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Maroon">x'</font>,<span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Olive">n</font></span>)</span></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 </span>}</span> ;<br/>
<a NAME="E2:14"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Maroon">x'</font>,<span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Olive">n</font></span>)</span></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">M</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:14_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Maroon">x'</font>,<span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Olive">n</font></span>)</span></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:14_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">x'</font>,<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">n</font></span>)</span>
 <b>and </b><br/><a NAME="E4:14_1"/>
<font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/>
<a NAME="E5:14_1"/>
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">x'</font>,<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">n</font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M</font>
 
;<br/>
<a NAME="E6:14_1"/><b>then </b>
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">x'</font>,<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">n</font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">M</font></span>)</span>
 
<b>by </b><i><a class="ref" href="topmetr.html#T16">TOPMETR:16</a></i>;<br/>
<b>hence </b><a NAME="E7:14_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">M</font></span>)</span>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">B</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Maroon">x'</font>,<span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Olive">n</font></span>)</span></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 </span>}</span>  as   <a href="struct_0.html#NM3">Subset-Family</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">M</font></span>)</span> ;<br/>
<a NAME="E4:14"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">M</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:14_2"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/>

<b>then consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:14_2"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">x'</font>,<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">n</font></span>)</span>
 <b>and </b><br/><a NAME="E4:14_2"/>
<font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/>
<a NAME="E5:14_2"/>
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">x'</font>,<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K3">Family_open_set</a> <font color="Maroon">M</font>
 
<b>by </b><i><a class="ref" href="pcomps_1.html#T33">PCOMPS_1:33</a></i>;<br/>
<b>hence </b><a NAME="E6:14_2"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">M</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="topmetr.html#T16">TOPMETR:16</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:14"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">B</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_3"><b>proof </b></a><div class="add">

<a NAME="E1:14_3"/><i><font color="Green">E43</font></i>: 
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">x'</font>,<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 
;<br/>
<a NAME="E2:14_3"/><b>then </b><i><font color="Green">E44</font></i>: 
 <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">B</font>
 
<b>by </b><i><a class="ref" href="setfam_1.html#D10">SETFAM_1:def 10</a></i>;<br/>
<a NAME="E3:14_3"/>
for <font color="Olive">O</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">O</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> holds <br/><font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">O</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_3_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:14_3_1"/>
<font color="Maroon">O</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/>

<b>then consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:14_3_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">O</font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">x'</font>,<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">n</font></span>)</span>
 <b>and </b><br/><a NAME="E4:14_3_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/>
<a NAME="E5:14_3_1"/>
1 <a href="real_1.html#K6">/</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="frechet.html#E1:14"><i><font color="Green">E19</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E6:14_3_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O</font>
 <b>by </b><i>, , <a class="ref" href="goboard6.html#T4">GOBOARD6:4</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:14_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">B</font>
 <b>by </b><i>, <a class="txt" href="frechet.html#E1"><i><font color="Green">Lemma17</font></i></a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:14"/>
for <font color="Olive">O</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">M</font></span>)</span>  st <font color="Olive">O</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">O</font> holds <br/> ex <font color="Olive">V</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">M</font></span>)</span> st <br/>( <font color="Olive">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Olive">V</font> <a href="tarski.html#R1">c=</a> <font color="Olive">O</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">O</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">M</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:14_4"/>
( <font color="Maroon">O</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O</font> )
 ;<br/>

<b>then consider </b><font color="Maroon">r</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E3:14_4"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>and </b><br/><a NAME="E4:14_4"/><i><font color="Green">E49</font></i>: 
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">x'</font>,<font color="Maroon">r</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">O</font>
 <b>by </b><i>, <a class="ref" href="topmetr.html#T22">TOPMETR:22</a></i>;<br/>
<b>reconsider </b><font color="Maroon">r</font> = <font color="Maroon">r</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E7:14_4"/><i><font color="Green">E50</font></i>: 
1 <a href="real_1.html#K6">/</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 <b>and </b><br/><a NAME="E8:14_4"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i>, </i>;<br/>
<a NAME="E9:14_4"/><i><font color="Green">E55</font></i>: 
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">x'</font>,<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">n</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">x'</font>,<font color="Maroon">r</font>
 
<b>by </b><i>, <a class="ref" href="pcomps_1.html#T1">PCOMPS_1:1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">Ba</font> =  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">x'</font>,<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">n</font></span>)</span> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">M</font></span>)</span> <b>by </b><i><a class="ref" href="topmetr.html#T16">TOPMETR:16</a></i>;<br/>
<b>reconsider </b><font color="Maroon">Ba</font> = <font color="Maroon">Ba</font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">M</font></span>)</span> ;<br/>
<b>take </b>
<font color="Maroon">Ba</font>
;<br/>

<b>thus </b><a NAME="E12:14_4"/>
<font color="Maroon">Ba</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="txt" href="frechet.html#E1:14"><i><font color="Green">E19</font></i></a></i>;<br/>

<b>thus </b><a NAME="E13:14_4"/>
<font color="Maroon">Ba</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">O</font>
 <b>by </b><i><a class="txt" href="frechet.html#E1"><i><font color="Green">Lemma17</font></i></a>, <a class="txt" href="frechet.html#E2"><i><font color="Green">Lemma20</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">B</font> = <font color="Maroon">B</font> as    <a href="yellow_8.html#M1">Basis</a> of <font color="Maroon">x</font> <b>by </b><i>, , <a class="ref" href="yellow_8.html#D2">YELLOW_8:def 2</a></i>;<br/>
<b>take </b>
<font color="Maroon">B</font>
;<br/>

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of <a href="zfmisc_1.html#K1">K10</a>(the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M</font>) =  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">x'</font>,<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b><font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0;<br/>
<b>thus </b><a NAME="E8:14"/>
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Maroon">x'</font>,<span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Olive">n</font></span>)</span></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 </span>}</span> 
 ;<br/>

<a NAME="E9:14"/>
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">n</font>) where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] </span>}</span>  is <a href="card_4.html#V1">countable</a>
 
<b>from </b><i><a class="ref" href="card_4.html#S1">CARD_4:sch 1</a>();<br/></i>
<b>hence </b><a NAME="E10:14"/>
<font color="Maroon">B</font> is <a href="card_4.html#V1">countable</a>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">n'</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">n'</font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">x'</font>,<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Olive">n'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> );<br/>
<a NAME="E11:14"/><i><font color="Green">E56</font></i>: 
for <font color="Olive">n</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  holds <br/> ex <font color="Olive">O</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">O</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">n</font>,<font color="Olive">O</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_5"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:14_5"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 ;<br/>

<b>then reconsider </b><font color="Maroon">n'</font> = <font color="Maroon">n</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<b>reconsider </b><font color="Maroon">O</font> =  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">x'</font>,<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">n'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> as    <a href="hidden.html#M1">set</a>  ;<br/>
<b>take </b>
<font color="Maroon">O</font>
;<br/>

<b>thus </b><a NAME="E4:14_5"/>
<font color="Maroon">O</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/>

<b>take </b>
<font color="Maroon">n'</font>
;<br/>

<b>thus </b><a NAME="E5:14_5"/>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n'</font>
 ;<br/>

<b>thus </b><a NAME="E6:14_5"/>
<font color="Maroon">O</font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">x'</font>,<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">n'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E13:14"/><i><font color="Green">E57</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 <b>and </b><br/><a NAME="E14:14"/><i><font color="Green">E58</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font>
 <b>and </b><br/><a NAME="E15:14"/><i><font color="Green">E73</font></i>: 
for <font color="Olive">n</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">n</font>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font>]
 <b>from </b><i><a class="ref" href="wellord2.html#S1">WELLORD2:sch 1</a>();<br/></i>
<b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> ,<font color="Maroon">B</font> <b>by </b><i><a class="txt" href="frechet.html#E1"><i><font color="Green">Lemma17</font></i></a>, , <a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/>
<b>take </b>
<font color="Maroon">f</font>
;<br/>

<b>thus </b><a NAME="E17:14"/>
for <font color="Olive">n</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  holds <br/> ex <font color="Olive">n'</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Olive">n'</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">x'</font>,<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Olive">n'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="frechet.html#E1:14"><i><font color="Green">E19</font></i></a></i>;<br/>


</div>
