<?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="E1:20"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">T</font> is <a href="frechet.html#V1">first-countable</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>consider </b><font color="Maroon">B1</font> being    <a href="yellow_8.html#M1">Basis</a> of <font color="Maroon">x</font><b> such that </b><br/><a NAME="E3:20"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">B1</font> is <a href="card_4.html#V1">countable</a>
 <b>by </b><i>, <a class="ref" href="frechet.html#D1">FRECHET:def 1</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">B1</font><b> such that </b><br/><a NAME="E5:20"/><i><font color="Green">E32</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B1</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#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>;<br/>
<a NAME="E6:20"/><i><font color="Green">E53</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">A</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>,<font color="Olive">A</font>]
 
;<br/>
<b>consider </b><font color="Maroon">S</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E8:20"/><i><font color="Green">E64</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> 
 <b>and </b><br/><a NAME="E9:20"/><i><font color="Green">E67</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>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="zfrefle1.html#S1">ZFREFLE1:sch 1</a>();<br/></i>
<a NAME="E10:20"/><i><font color="Green">E70</font></i>: 
for <font color="Olive">n</font>, <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">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">n</font> holds <br/><font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Olive">m</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</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">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <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="E1:20_1"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n</font>
 ;<br/>

<a NAME="E2:20_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="E3:20_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="E4:20_1"/>
<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="E5:20_1"/><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="E6:20_1"/><b>then </b>
 <a href="ordinal1.html#K1">succ</a> <font color="Maroon">n</font> <a href="tarski.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="E7:20_1"/><b>then </b>
 <a href="ordinal1.html#K1">succ</a> <font color="Maroon">n</font> <a href="tarski.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="E8:20_1"/><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="E9:20_1"/><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="ref" href="setfam_1.html#T7">SETFAM_1:7</a></i>;<br/>
<a NAME="E10:20_1"/><b>then </b>
<font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <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/>;<br/>
<b>hence </b><a NAME="E11:20_1"/>
<font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E11:20"/><i><font color="Green">E74</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">S</font> <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 <font color="Maroon">T</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">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="E1:20_2"/>
<font color="Maroon">A</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">n</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:20_2"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">S</font>
 <b>and </b><br/><a NAME="E4:20_2"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E5:20_2"/>
<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>  <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">fsuccn</font> = <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> as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font> ;<br/>
<a NAME="E7:20_2"/>
 <a href="setfam_1.html#K6">meet</a> <font color="Maroon">fsuccn</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>
 
;<br/>
<a NAME="E8:20_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">n</font></span>)</span></span>)</span> <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 <font color="Maroon">T</font>
 
;<br/>
<b>hence </b><a NAME="E9:20_2"/>
<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 <font color="Maroon">T</font>
 <b>by </b><i>, , , <a class="txt" href="frechet2.html#E1:20"><i><font color="Green">E27</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">B</font> =  <a href="relat_1.html#K2">rng</a> <font color="Maroon">S</font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font> ;<br/>
<b>reconsider </b><font color="Maroon">B</font> = <font color="Maroon">B</font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font> ;<br/>
<a NAME="E14:20"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font>
 
<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">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="E1:20_3"/>
<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="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:20_3"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">S</font>
 <b>and </b><br/><a NAME="E4:20_3"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E5:20_3"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">A</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>by </b><i>, , <a class="txt" href="frechet2.html#E1:20"><i><font color="Green">E27</font></i></a>, <a class="txt" href="frechet2.html#E2"><i><font color="Green">Lemma28</font></i></a></i>;<br/>
<a NAME="E6:20_3"/>
<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>  <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">C</font> = <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> as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font> ;<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>, <a class="txt" href="frechet2.html#E1:20"><i><font color="Green">E27</font></i></a></i>;<br/>
<a NAME="E9:20_3"/>
<font color="Maroon">n'</font> <a href="nat_1.html#K1">+</a> 1 is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i><a class="ref" href="card_1.html#T69">CARD_1:69</a></i>;<br/>
<a NAME="E10:20_3"/><b>then </b>
 <a href="ordinal1.html#K1">succ</a> <font color="Maroon">n'</font> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i><a class="ref" href="card_1.html#T52">CARD_1:52</a></i>;<br/>
<a NAME="E11:20_3"/><b>then </b><i><font color="Green">E81</font></i>: 
<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> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i><a class="ref" href="finset_1.html#T17">FINSET_1:17</a></i>;<br/>
<a NAME="E12:20_3"/>
<font color="Maroon">C</font> is <a href="tops_2.html#V1">open</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">P</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="tops_2.html#D1">TOPS_2:def 1</a><br/>

<b>assume </b><a NAME="E1:20_3_1"/>
<font color="Maroon">P</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 ;<br/>

<b>hence </b><a NAME="E2:20_3_1"/>
<font color="Maroon">P</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="E13:20_3"/><b>then </b>
 <a href="setfam_1.html#K6">meet</a> <font color="Maroon">C</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i>, <a class="ref" href="tops_2.html#T27">TOPS_2:27</a></i>;<br/>
<b>hence </b><a NAME="E14:20_3"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font>
 <b>by </b><i>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E15:20"/><i><font color="Green">E83</font></i>: 
 ex <font color="Olive">A</font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_4"><b>proof </b></a><div class="add">

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

<a NAME="E1:20_4"/>
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> 0
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E2:20_4"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E16:20"/><b>then </b><i><font color="Green">E98</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="E17:20"/>
for <font color="Olive">A</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">A</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">A</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">A</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:20_5"/>
<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="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:20_5"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">S</font>
 <b>and </b><br/><a NAME="E4:20_5"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E5:20_5"/><i><font color="Green">E101</font></i>: 
<font color="Maroon">A</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>by </b><i>, , , </i>;<br/>
<a NAME="E6:20_5"/><i><font color="Green">E102</font></i>: 
for <font color="Olive">A1</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">A1</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">A1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_5_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">A1</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:20_5_1"/>
<font color="Maroon">A1</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">m</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:20_5_1"/><i><font color="Green">E103</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E4:20_5_1"/>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">n</font>
 <b>and </b><br/><a NAME="E5:20_5_1"/><i><font color="Green">E104</font></i>: 
<font color="Maroon">A1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E6:20_5_1"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="txt" href="frechet2.html#E3"><i><font color="Green">Lemma33</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">A2</font> = <font color="Maroon">A1</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> <b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">A2</font> = <font color="Maroon">A2</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> ;<br/>
<a NAME="E9:20_5_1"/>
<font color="Maroon">A2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B1</font>
 
<b>by </b><i>, <a class="txt" href="frechet2.html#E3"><i><font color="Green">Lemma33</font></i></a>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E10:20_5_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font>
 <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="E7:20_5"/><i><font color="Green">E105</font></i>: 
 <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></i>;<br/>
<a NAME="E8:20_5"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#T10">ORDINAL1:10</a></i>;<br/>
<a NAME="E9:20_5"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</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="txt" href="frechet2.html#E3"><i><font color="Green">Lemma33</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>hence </b><a NAME="E10:20_5"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="txt" href="frechet2.html#E3:20"><i><font color="Green">E31</font></i></a>, <a class="txt" href="frechet2.html#E5:20"><i><font color="Green">E32</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="E18:20"/><b>then </b><i><font color="Green">E106</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">B</font>
 
<b>by </b><i><a class="txt" href="frechet2.html#E1:20"><i><font color="Green">E27</font></i></a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>
<a NAME="E19:20"/>
for <font color="Olive">O</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>  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">A</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Olive">A</font> <a href="tarski.html#R1">c=</a> <font color="Olive">O</font> )
 
<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">O</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>;<br/>

<b>assume </b><a NAME="E1:20_6"/>
( <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">A1</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E3:20_6"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">A1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B1</font>
 <b>and </b><br/><a NAME="E4:20_6"/><i><font color="Green">E109</font></i>: 
<font color="Maroon">A1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">O</font>
 <b>by </b><i><a class="ref" href="yellow_8.html#D2">YELLOW_8:def 2</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="E6:20_6"/><i><font color="Green">E110</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>
 <b>and </b><br/><a NAME="E7:20_6"/><i><font color="Green">E111</font></i>: 
<font color="Maroon">A1</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="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E8:20_6"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="txt" href="frechet2.html#E5:20"><i><font color="Green">E32</font></i></a></i>;<br/>
<a NAME="E9:20_6"/><b>then </b>
<font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">S</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">A</font> = <font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</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">A</font> = <font color="Maroon">A</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> ;<br/>
<b>take </b>
<font color="Maroon">A</font>
;<br/>

<b>thus </b><a NAME="E12:20_6"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i>, <a class="txt" href="frechet2.html#E5:20"><i><font color="Green">E32</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>

<a NAME="E13:20_6"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#T10">ORDINAL1:10</a></i>;<br/>
<a NAME="E14:20_6"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</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="txt" href="frechet2.html#E5:20"><i><font color="Green">E32</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E15:20_6"/><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">n</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font>
 
<b>by </b><i><a class="txt" href="frechet2.html#E3"><i><font color="Green">Lemma33</font></i></a>, <a class="ref" href="setfam_1.html#T4">SETFAM_1:4</a></i>;<br/>
<a NAME="E16:20_6"/><b>then </b>
<font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font>
 
<b>by </b><i>, <a class="txt" href="frechet2.html#E5:20"><i><font color="Green">E32</font></i></a></i>;<br/>
<b>hence </b><a NAME="E17:20_6"/>
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">O</font>
 <b>by </b><i><a class="txt" href="frechet2.html#E3:20"><i><font color="Green">E31</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="txt" href="frechet2.html#E2"><i><font color="Green">Lemma28</font></i></a>, , <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>take </b>
<font color="Maroon">S</font>
;<br/>

<b>thus </b><a NAME="E21:20"/>
 <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> 
 <b>by </b><i/>;<br/>

<b>thus </b><a NAME="E22:20"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font>
 ;<br/>

<b>thus </b><a NAME="E23:20"/>
for <font color="Olive">n</font>, <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">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">n</font> holds <br/><font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Olive">m</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font>
 <b>by </b><i><a class="txt" href="frechet2.html#E1"><i><font color="Green">Lemma24</font></i></a></i>;<br/>


</div>
