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

<b>let </b><font color="Maroon">T</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/>

<b>assume </b><a NAME="E1:32"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">T</font> is <a href="frechet.html#V1" target="_self">first-countable</a>
 ;<br/>

<b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="frechet.html#D5" target="_self">FRECHET:def 5</a><br/><b>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>;<br/>



<b>assume </b><a NAME="E2:32"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font>
 ;<br/>

<b>consider </b><font color="Maroon">B</font> being    <a href="yellow_8.html#M1">Basis</a> of <font color="Maroon">x</font><b> such that </b><br/><a NAME="E4:32"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">B</font> is <a href="card_4.html#V1">countable</a>
 <b>by </b><i>, <a class="txt" href="frechet.html#E2:32"><i><font color="Green">E22</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> ,<font color="Maroon">B</font><b> such that </b><br/><a NAME="E6:32"/><i><font color="Green">E39</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font>
 <b>by </b><i>, <a class="ref" href="waybel12.html#T4">WAYBEL12:4</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K8">/\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1">meet</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p3">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span></span>)</span>;<br/>
<a NAME="E7:32"/><i><font color="Green">E40</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">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>,<font color="Olive">y</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1"><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="E1:32_1"/>
<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>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b> ex <font color="Olive">H</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">H</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span> &amp; <font color="Olive">H</font> is <a href="pre_topc.html#V3">open</a> );<br/>
<a NAME="E3:32_1"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">n</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:32_1_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:32_1_1_1"/>
0 <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 
;<br/>
<a NAME="E2:32_1_1_1"/><b>then </b><i><font color="Green">E44</font></i>: 
0 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E3:32_1_1_1"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 
;<br/>
<b>then consider </b><font color="Maroon">H</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E5:32_1_1_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> 0
 ;<br/>
<b>reconsider </b><font color="Maroon">H</font> = <font color="Maroon">H</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> ;<br/>
<b>take </b>
<font color="Maroon">H</font>
;<br/>

 <a href="setfam_1.html#K1">meet</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><a href="ordinal1.html#K1">succ</a> 0</span>)</span></span>)</span> = 
 <a href="setfam_1.html#K1">meet</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> 0</span>)</span></span><a href="domain_1.html#K6">}</a></span>
<b>by </b><i><a class="ref" href="frechet.html#T5" target="_self">Th5</a>, <a class="ref" href="card_5.html#T1">CARD_5:1</a>, <a class="ref" href="funct_1.html#T117">FUNCT_1:117</a></i>
<br/>.= 
<font color="Maroon">H</font>
<b>by </b><i>, <a class="ref" href="setfam_1.html#T11">SETFAM_1:11</a></i>
;<br/>
<b>hence </b><a NAME="E8:32_1_1_1"/>
( <font color="Maroon">H</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><a href="ordinal1.html#K1">succ</a> 0</span>)</span></span>)</span> &amp; <font color="Maroon">H</font> is <a href="pre_topc.html#V3">open</a> )
 <b>by </b><i>, <a class="ref" href="yellow_8.html#T21">YELLOW_8:21</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:32_1_1"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1_2"><b>proof </b></a><div class="add">

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

<b>given </b><font color="Maroon">G</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><a NAME="E1:32_1_1_2"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E2:32_1_1_2"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">G</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>

<a NAME="E3:32_1_1_2"/>
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 
;<br/>
<a NAME="E4:32_1_1_2"/><b>then </b><i><font color="Green">E50</font></i>: 
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E5:32_1_1_2"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 
;<br/>
<b>then consider </b><font color="Maroon">H1</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E7:32_1_1_2"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">H1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/>
<b>reconsider </b><font color="Maroon">H1</font> = <font color="Maroon">H1</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> ;<br/>
<a NAME="E9:32_1_1_2"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">H1</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="txt" href="frechet.html#E7:32"><i><font color="Green">E40</font></i></a>, <a class="ref" href="yellow_8.html#T21">YELLOW_8:21</a></i>;<br/>
<a NAME="E10:32_1_1_2"/>
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">n</font> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a>  )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="ordinal1.html#T10">ORDINAL1:10</a></i>;<br/>
<a NAME="E11:32_1_1_2"/><b>then </b><i><font color="Green">E56</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>consider </b><font color="Maroon">H</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E13:32_1_1_2"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">G</font>
 ;<br/>
<b>reconsider </b><font color="Maroon">H</font> = <font color="Maroon">H</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> ;<br/>
<b>take </b>
<font color="Maroon">H</font>
;<br/>

 <a href="setfam_1.html#K1">meet</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><a href="ordinal1.html#K1">succ</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> = 
 <a href="setfam_1.html#K1">meet</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="ordinal1.html#D1">ORDINAL1:def 1</a></i>
<br/>.= 
 <a href="setfam_1.html#K1">meet</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="card_1.html#T52">CARD_1:52</a></i>
<br/>.= 
 <a href="setfam_1.html#K1">meet</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p3">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T153">RELAT_1:153</a></i>
<br/>.= 
 <a href="setfam_1.html#K1">meet</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p3">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="frechet.html#E6:32"><i><font color="Green">E39</font></i></a>, <a class="ref" href="funct_1.html#T117">FUNCT_1:117</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="setfam_1.html#K1">meet</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1">meet</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p3">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="setfam_1.html#T10">SETFAM_1:10</a></i>
<br/>.= 
<font color="Maroon">H</font>
<b>by </b><i>, <a class="txt" href="frechet.html#E7:32"><i><font color="Green">E40</font></i></a>, <a class="txt" href="frechet.html#E1:32_1_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="setfam_1.html#T11">SETFAM_1:11</a></i>
;<br/>
<b>hence </b><a NAME="E16:32_1_1_2"/>
( <font color="Maroon">H</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><a href="ordinal1.html#K1">succ</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon">H</font> is <a href="pre_topc.html#V3">open</a> )
 <b>by </b><i><a class="txt" href="frechet.html#E4:32"><i><font color="Green">E38</font></i></a>, , <a class="txt" href="frechet.html#E1:32_1_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="tops_1.html#T38">TOPS_1:38</a></i>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E3:32_1_1"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">n</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="ref" href="frechet.html#T4" target="_self">Th4</a>, <a class="ref" href="frechet.html#T5" target="_self">Th5</a>);</i><br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">H</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E5:32_1"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">H</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">n</font></span>)</span></span>)</span> &amp; <font color="Maroon">H</font> is <a href="pre_topc.html#V3">open</a> )
 ;<br/>
<a NAME="E6:32_1"/>
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">n</font> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a>  )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="ordinal1.html#T10">ORDINAL1:10</a></i>;<br/>
<a NAME="E7:32_1"/><b>then </b><i><font color="Green">E73</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E8:32_1"/>
for <font color="Olive">G</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">G</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">n</font></span>)</span> holds <br/><font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">G</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:32_1_2"/>
<font color="Maroon">G</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">n</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">k</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:32_1_2"/><i><font color="Green">E74</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">n</font> &amp; <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E4:32_1_2"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E5:32_1_2"/><b>then </b><i><font color="Green">E76</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">G1</font> = <font color="Maroon">G</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">G1</font> = <font color="Maroon">G1</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> ;<br/>
<b>thus </b><a NAME="E8:32_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 <b>by </b><i>, <a class="txt" href="frechet.html#E4:32"><i><font color="Green">E38</font></i></a>, <a class="ref" href="yellow_8.html#T21">YELLOW_8:21</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:32_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="frechet.html#T5" target="_self">Th5</a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>
<a NAME="E10:32_1"/><b>then </b>
<font color="Maroon">A</font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="frechet.html#T4" target="_self">Th4</a>, <a class="ref" href="pre_topc.html#D13">PRE_TOPC:def 13</a></i>;<br/>
<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E12:32_1"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K8">/\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1">meet</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p3">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#T4">XBOOLE_0:4</a></i>;<br/>
<b>take </b>
<font color="Maroon">y</font>
;<br/>

<b>thus </b><a NAME="E13:32_1"/>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">y</font>] )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">S</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E9:32"/><i><font color="Green">E78</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a>  &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">S</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> &amp; ( 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>1</sub></font>[<font color="Olive">n</font>,<font color="Maroon">S</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>
<a NAME="E10:32"/><i><font color="Green">E79</font></i>: 
for <font color="Olive">m</font>, <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/><font color="Maroon">A</font> <a href="subset_1.html#K8">/\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1">meet</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p3">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Olive">m</font></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> <a href="subset_1.html#K8">/\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1">meet</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p3">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Olive">n</font></span>)</span></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:32_2"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/>

<a NAME="E2:32_2"/>
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">n</font> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a>  )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="ordinal1.html#T10">ORDINAL1:10</a></i>;<br/>
<a NAME="E3:32_2"/><b>then </b><i><font color="Green">E81</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E4:32_2"/>
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E5:32_2"/><b>then </b>
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="ordinal1.html#R1">c=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="card_1.html#T56">CARD_1:56</a></i>;<br/>
<a NAME="E6:32_2"/><b>then </b>
 <a href="ordinal1.html#K1">succ</a> <font color="Maroon">n</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="card_1.html#T52">CARD_1:52</a></i>;<br/>
<a NAME="E7:32_2"/><b>then </b>
 <a href="ordinal1.html#K1">succ</a> <font color="Maroon">n</font> <a href="ordinal1.html#R1">c=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="ref" href="card_1.html#T52">CARD_1:52</a></i>;<br/>
<a NAME="E8:32_2"/><b>then </b>
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">n</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">m</font></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T156">RELAT_1:156</a></i>;<br/>
<a NAME="E9:32_2"/><b>then </b>
 <a href="setfam_1.html#K1">meet</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="frechet.html#E4:32"><i><font color="Green">E38</font></i></a>, <a class="ref" href="setfam_1.html#T7">SETFAM_1:7</a></i>;<br/>
<b>hence </b><a NAME="E10:32_2"/>
<font color="Maroon">A</font> <a href="subset_1.html#K8">/\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1">meet</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p3">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> <a href="subset_1.html#K8">/\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1">meet</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p3">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T26">XBOOLE_1:26</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:32"/><i><font color="Green">E82</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">S</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</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="E1:32_3"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">S</font>
 ;<br/>

<b>then consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:32_3"/><i><font color="Green">E83</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">S</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:32_3"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K8">/\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1">meet</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p3">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="frechet.html#T4" target="_self">Th4</a>, <a class="txt" href="frechet.html#E4:32"><i><font color="Green">E38</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:32_3"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">S</font> = <font color="Maroon">S</font> as   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">T</font> <b>by </b><i><a class="ref" href="frechet.html#T4" target="_self">Th4</a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<a NAME="E13:32"/>
<font color="Maroon">S</font> <a href="frechet.html#R1" target="_self">is_convergent_to</a> <font color="Maroon">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">U1</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="frechet.html#D2" target="_self">FRECHET:def 2</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:32_4"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">U1</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E2:32_4"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">U1</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">U1</font> = <font color="Maroon">U1</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> ;<br/>
<b>consider </b><font color="Maroon">U2</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E5:32_4"/><i><font color="Green">E86</font></i>: 
( <font color="Maroon">U2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">U2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">U1</font> )
 <b>by </b><i><a class="txt" href="frechet.html#E4:32"><i><font color="Green">E38</font></i></a>, <a class="txt" href="frechet.html#E6:32"><i><font color="Green">E39</font></i></a>, <a class="ref" href="yellow_8.html#T22">YELLOW_8:22</a></i>;<br/>
<b>consider </b><font color="Maroon">n</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:32_4"/><i><font color="Green">E87</font></i>: 
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">U2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> )
 <b>by </b><i><a class="txt" href="frechet.html#E6"><i><font color="Green">Lemma33</font></i></a>, <a class="txt" href="frechet.html#E7:32"><i><font color="Green">E40</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>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>  <b>by </b><i/>;<br/>
<a NAME="E9:32_4"/>
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/><font color="Maroon">S</font> <a href="funct_2.html#K3">.</a> <font color="Olive">m</font> <a href="hidden.html#R2">in</a> <font color="Maroon">U1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_4_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:32_4_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/>

<a NAME="E2:32_4_1"/><b>then </b><i><font color="Green">E88</font></i>: 
<font color="Maroon">A</font> <a href="subset_1.html#K8">/\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1">meet</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p3">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> <a href="subset_1.html#K8">/\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1">meet</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p3">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="frechet.html#T5" target="_self">Th5</a></i>;<br/>
<a NAME="E3:32_4_1"/>
<font color="Maroon">S</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K8">/\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1">meet</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p3">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="frechet.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E4:32_4_1"/><b>then </b><i><font color="Green">E121</font></i>: 
<font color="Maroon">S</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E5:32_4_1"/>
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">n</font> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a>  )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="ordinal1.html#T10">ORDINAL1:10</a></i>;<br/>
<a NAME="E6:32_4_1"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E7:32_4_1"/><b>then </b>
<font color="Maroon">S</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font>
 
<b>by </b><i>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E8:32_4_1"/>
<font color="Maroon">S</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <font color="Maroon">U1</font>
 <b>by </b><i><a class="txt" href="frechet.html#E7:32"><i><font color="Green">E40</font></i></a>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E10:32_4"/>
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/><font color="Maroon">S</font> <a href="funct_2.html#K3">.</a> <font color="Olive">m</font> <a href="hidden.html#R2">in</a> <font color="Maroon">U1</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E14:32"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="frechet.html#K1" target="_self">Lim</a> <font color="Maroon">S</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E15:32"/>
 ex <font color="Olive">S</font> being  <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">T</font> st <br/>(  <a href="relset_1.html#K5">rng</a> <font color="Olive">S</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="frechet.html#K1" target="_self">Lim</a> <font color="Olive">S</font> )
 <b>by </b><i/>;<br/>


</div>
