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

<b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <a href="frechet.html#K2" target="_self">REAL?</a> ; <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 <a href="frechet.html#K2" target="_self">REAL?</a> ;<br/>



<b>assume </b><a NAME="E1:46"/><i><font color="Green">E19</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/>

<a NAME="E2:46"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <a href="frechet.html#K2" target="_self">REAL?</a> 
 
;<br/>
<a NAME="E3:46"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="numbers.html#K1">REAL</a>  <a href="subset_1.html#K6">\</a> <a href="numbers.html#K5">NAT</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="numbers.html#K1">REAL</a> </span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="txt" href="frechet.html#E6"><i><font color="Green">Lemma33</font></i></a></i>;<br/>
<a NAME="E4:46"/><b>then </b><i><font color="Green">E22</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <a href="numbers.html#K1">REAL</a>  <a href="subset_1.html#K6">\</a> <a href="numbers.html#K5">NAT</a>  or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="numbers.html#K1">REAL</a> </span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:46_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <a href="numbers.html#K1">REAL</a>  <a href="subset_1.html#K6">\</a> <a href="numbers.html#K5">NAT</a>  or ( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K1">REAL</a>  &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> ) or ( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K1">REAL</a>  &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> ) )
 </span><b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1_1"><b>suppose </b></a><a NAME="E1:46_1_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <a href="numbers.html#K1">REAL</a>  <a href="subset_1.html#K6">\</a> <a href="numbers.html#K5">NAT</a> 
 ;<br/><div class="add"><a NAME="E2:46_1_1"/><b>then </b><i><font color="Green">E39</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a> 
 ;<br/><b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K3">R^1</a>  <b>by </b><i>, <a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/><a NAME="E4:46_1_1"/>
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <a href="frechet.html#K2" target="_self">REAL?</a> 
 ;<br/><a NAME="E5:46_1_1"/><b>then </b><i><font color="Green">E40</font></i>: 
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="numbers.html#K1">REAL</a>  <a href="subset_1.html#K6">\</a> <a href="numbers.html#K5">NAT</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="numbers.html#K1">REAL</a> </span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="frechet.html#E6"><i><font color="Green">Lemma33</font></i></a></i>;<br/><a NAME="E6:46_1_1"/>
<font color="Maroon">A</font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="numbers.html#K1">REAL</a> </span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K1">REAL</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_1_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="E1:46_1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="numbers.html#K1">REAL</a> </span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<a NAME="E2:46_1_1_1"/><b>then </b>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="numbers.html#K1">REAL</a> </span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E3:46_1_1_1"/><b>then </b>
( ( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <a href="numbers.html#K1">REAL</a>  <a href="subset_1.html#K6">\</a> <a href="numbers.html#K5">NAT</a>  or <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="numbers.html#K1">REAL</a> </span><a href="tarski.html#K1">}</a></span> ) &amp; not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="numbers.html#K1">REAL</a> </span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E4:46_1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a> 
 ;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">A'</font> = <font color="Maroon">A</font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="numbers.html#K1">REAL</a> </span><a href="tarski.html#K1">}</a></span> as   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a>  <b>by </b><i><a class="ref" href="topmetr.html#T24">TOPMETR:24</a></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 <a href="topmetr.html#K3">R^1</a>  ;<br/><a NAME="E9:46_1_1"/>
for <font color="Olive">B'</font> being  <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a>   st <font color="Olive">B'</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">B'</font> holds <br/><font color="Maroon">A'</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Olive">B'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">B'</font> be   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a> ;<br/>

<b>assume </b><a NAME="E1:46_1_1_2"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">B'</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>

<b>assume </b><a NAME="E2:46_1_1_2"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B'</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">B1</font> = <font color="Maroon">B'</font> as   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a>  ;<br/>
<b>reconsider </b><font color="Maroon">C</font> =  <a href="numbers.html#K5">NAT</a>  as   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a>  <b>by </b><i><a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<b>reconsider </b><font color="Maroon">C</font> = <font color="Maroon">C</font> as   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a>  ;<br/>
<a NAME="E6:46_1_1_2"/>
<font color="Maroon">C</font> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i/>;<br/>
<a NAME="E7:46_1_1_2"/><b>then </b><i><font color="Green">E45</font></i>: 
<font color="Maroon">B1</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">C</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E8:46_1_1_2"/>
<font color="Maroon">B'</font> <a href="subset_1.html#K7">\</a> <a href="numbers.html#K5">NAT</a>  <a href="xboole_0.html#R1">misses</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="xboole_1.html#T79">XBOOLE_1:79</a></i>;<br/>
<a NAME="E9:46_1_1_2"/><b>then </b><i><font color="Green">E46</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">B'</font> <a href="subset_1.html#K7">\</a> <a href="numbers.html#K5">NAT</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <a href="numbers.html#K5">NAT</a>  <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">D</font> = <font color="Maroon">B1</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">C</font> as   <a href="struct_0.html#NM2">Subset</a> of <a href="frechet.html#K2" target="_self">REAL?</a>  <b>by </b><i/>;<br/>
<a NAME="E11:46_1_1_2"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">D</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="txt" href="frechet.html#E13:46_1_1_2"><i><font color="Green">E49</font></i></a>, <a class="txt" href="frechet.html#E16:46_1_1_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="frechet.html#E1:46_1_1"><i><font color="Green">E38</font></i></a></i>;<br/>
<a NAME="E12:46_1_1_2"/>
not <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="txt" href="frechet.html#E2:46_1_1_2"><i><font color="Green">E44</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E13:46_1_1_2"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B'</font> <a href="subset_1.html#K7">\</a> <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="txt" href="frechet.html#E11:46_1_1_2"><i><font color="Green">E48</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">D</font> = <font color="Maroon">D</font> as   <a href="struct_0.html#NM2">Subset</a> of <a href="frechet.html#K2" target="_self">REAL?</a>  ;<br/>
<a NAME="E15:46_1_1_2"/>
<font color="Maroon">A</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">D</font>
 
<b>by </b><i>, <a class="ref" href="frechet.html#T6" target="_self">Th6</a>, , <a class="ref" href="pre_topc.html#D13">PRE_TOPC:def 13</a></i>;<br/>
<a NAME="E16:46_1_1_2"/><b>then </b><i><font color="Green">E50</font></i>: 
<font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">D</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E17:46_1_1_2"/>
<font color="Maroon">A'</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_1_2_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">a</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">D</font>;<br/>
<a NAME="E2:46_1_1_2_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E3:46_1_1_2_1"/><b>then </b><i><font color="Green">E55</font></i>: 
<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="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E4:46_1_1_2_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="txt" href="frechet.html#E2:46_1_1_2_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<a NAME="E5:46_1_1_2_1"/><i><font color="Green">E57</font></i>: 
not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="numbers.html#K1">REAL</a> </span><a href="tarski.html#K1">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_1_2_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:46_1_1_2_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="numbers.html#K1">REAL</a> </span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<a NAME="E2:46_1_1_2_1_1"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:46_1_1_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="frechet.html#E4:46_1_1_2_1"><i><font color="Green">E56</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:46_1_1_2_1"/>
<font color="Maroon">a</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/>
<a NAME="E7:46_1_1_2_1"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="numbers.html#K1">REAL</a> </span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="txt" href="frechet.html#E5:46_1_1_2_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="E8:46_1_1_2_1"/>
<font color="Maroon">A'</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="frechet.html#E3:46_1_1_2_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E18:46_1_1_2"/>
<font color="Maroon">A'</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">B'</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>


</div><b>end;</b></div><a NAME="E10:46_1_1"/><b>then </b>
<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>
 <b>by </b><i><a class="ref" href="pre_topc.html#D13">PRE_TOPC:def 13</a></i>;<br/><b>then consider </b><font color="Maroon">S'</font> being   <a href="normsp_1.html#NM2">sequence</a> of <a href="topmetr.html#K3">R^1</a> <b> such that </b><br/><a NAME="E12:46_1_1"/><i><font color="Green">E58</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">S'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A'</font>
 <b>and </b><br/><a NAME="E13:46_1_1"/><i><font color="Green">E73</font></i>: 
<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><a class="ref" href="frechet.html#T3" target="_self">Th3</a></i>;<br/><a NAME="E14:46_1_1"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">S'</font> <a href="frechet.html#R1" target="_self">is_convergent_to</a> <font color="Maroon">x'</font>
 <b>by </b><i>, </i>;<br/><a NAME="E15:46_1_1"/><i><font color="Green">E76</font></i>: 
 <a href="relset_1.html#K5">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="46_1_1_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="E1:46_1_1_3"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">S'</font>
 ;<br/>

<b>hence </b><a NAME="E2:46_1_1_3"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>


</div><b>end;</b></div><a NAME="E16:46_1_1"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">S'</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <a href="frechet.html#K2" target="_self">REAL?</a> 
 <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">S</font> = <font color="Maroon">S'</font> as   <a href="normsp_1.html#NM2">sequence</a> of <a href="frechet.html#K2" target="_self">REAL?</a>  <b>by </b><i/>;<br/><b>take </b>
<font color="Maroon">S</font>
;<br/><b>thus </b><a NAME="E18:46_1_1"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">S</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 <b>by </b><i/>;<br/><a NAME="E19:46_1_1"/>
<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="46_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">V</font> be   <a href="struct_0.html#NM2">Subset</a> of <a href="frechet.html#K2" target="_self">REAL?</a> ; <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><a NAME="E1:46_1_1_4"/><i><font color="Green">E77</font></i>: 
( <font color="Maroon">V</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">V</font> )
 ;<br/>

<b>reconsider </b><font color="Maroon">C</font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="numbers.html#K1">REAL</a> </span><a href="tarski.html#K1">}</a></span> as   <a href="struct_0.html#NM2">Subset</a> of <a href="frechet.html#K2" target="_self">REAL?</a>  <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">C</font> = <font color="Maroon">C</font> as   <a href="struct_0.html#NM2">Subset</a> of <a href="frechet.html#K2" target="_self">REAL?</a>  ;<br/>
<a NAME="E4:46_1_1_4"/>
<font color="Maroon">C</font> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i><a class="txt" href="frechet.html#E2:46_1_1"><i><font color="Green">E39</font></i></a></i>;<br/>
<a NAME="E5:46_1_1_4"/><b>then </b><i><font color="Green">E78</font></i>: 
<font color="Maroon">V</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">C</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="frechet.html#T6" target="_self">Th6</a>, </i>;<br/>
<a NAME="E6:46_1_1_4"/>
 <a href="numbers.html#K1">REAL</a>  <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="numbers.html#K1">REAL</a> </span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E7:46_1_1_4"/><b>then </b><i><font color="Green">E79</font></i>: 
not  <a href="numbers.html#K1">REAL</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="numbers.html#K1">REAL</a> </span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">V'</font> = <font color="Maroon">V</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">C</font> as   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a>  <b>by </b><i/>;<br/>
<a NAME="E9:46_1_1_4"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">V'</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i>, , <a class="txt" href="frechet.html#E1:46_1_1"><i><font color="Green">E38</font></i></a></i>;<br/>
<a NAME="E10:46_1_1_4"/>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_1_4_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:46_1_1_4_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 ;<br/>

<a NAME="E2:46_1_1_4_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:46_1_1_4_1"/>
contradiction
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E11:46_1_1_4"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="frechet.html#T6" target="_self">Th6</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<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="E13:46_1_1_4"/><i><font color="Green">E81</font></i>: 
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">V'</font>
 <b>by </b><i>, , </i>;<br/>
<b>take </b>
<font color="Maroon">n</font>
;<br/>

<b>thus </b><a NAME="E14:46_1_1_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">V</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_1_4_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> ;<br/>

<b>assume </b><a NAME="E1:46_1_1_4_2"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/>

<a NAME="E2:46_1_1_4_2"/><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">V'</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:46_1_1_4_2"/>
<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">V</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div><b>hence </b><a NAME="E20:46_1_1"/>
<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/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1_2"><b>suppose </b></a><a NAME="E1:46_1_2"/><i><font color="Green">E82</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K1">REAL</a>  &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 ;<br/><div class="add"><a NAME="E2:46_1_2"/><i><font color="Green">E83</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="numbers.html#K5">NAT</a>  <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="funcop_1.html#T14">FUNCOP_1:14</a></i>;<br/><a NAME="E3:46_1_2"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="numbers.html#K5">NAT</a>  <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/><b>then reconsider </b><font color="Maroon">S</font> = <a href="numbers.html#K5">NAT</a>  <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">x</font> as   <a href="normsp_1.html#NM2">sequence</a> of <a href="frechet.html#K2" target="_self">REAL?</a>  <b>by </b><i>, <a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/><b>take </b>
<font color="Maroon">S</font>
;<br/><a NAME="E5:46_1_2"/>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/><b>hence </b><a NAME="E6:46_1_2"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">S</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="funcop_1.html#T14">FUNCOP_1:14</a></i>;<br/><a NAME="E7:46_1_2"/>
<font color="Maroon">S</font> <a href="frechet.html#R1" target="_self">is_convergent_to</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="frechet.html#T2" target="_self">Th2</a></i>;<br/><b>hence </b><a NAME="E8:46_1_2"/>
<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/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1_3"><b>suppose </b></a><a NAME="E1:46_1_3"/><i><font color="Green">E84</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K1">REAL</a>  &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">A'</font> = <font color="Maroon">A</font> as   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a>  <b>by </b><i/>;<br/><a NAME="E3:46_1_3"/>
 ex <font color="Olive">k</font> being  <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K3">R^1</a>  st <br/>( <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  &amp;  ex <font color="Olive">S'</font> being  <a href="normsp_1.html#NM2">sequence</a> of <a href="topmetr.html#K3">R^1</a>  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="Olive">S'</font> <a href="frechet.html#R1" target="_self">is_convergent_to</a> <font color="Olive">k</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:46_1_3_1"/><i><font color="Green">E85</font></i>: 
for <font color="Olive">k</font> being  <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K3">R^1</a>   holds <br/>(  not <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  or for <font color="Olive">S'</font> being  <a href="normsp_1.html#NM2">sequence</a> of <a href="topmetr.html#K3">R^1</a>   holds <br/>(  not  <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> or  not <font color="Olive">S'</font> <a href="frechet.html#R1" target="_self">is_convergent_to</a> <font color="Olive">k</font> ) )
 ;<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>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>2</sub></font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <a href="topmetr.html#K3">R^1</a>  &amp; <font color="Maroon">a<sub>2</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A'</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  );<br/>
<a NAME="E2:46_1_3_1"/><i><font color="Green">E86</font></i>: 
for <font color="Olive">k</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  holds <br/> ex <font color="Olive">U1</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>,<font color="Olive">U1</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_3_1_1"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon">k</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E1:46_1_3_1_1"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 <b>and </b><br/><a NAME="E2:46_1_3_1_1"/><i><font color="Green">E88</font></i>: 
for <font color="Olive">U1</font> being   <a href="hidden.html#M1">set</a>   st <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Olive">U1</font> &amp; <font color="Olive">U1</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <a href="topmetr.html#K3">R^1</a>  holds <br/> not <font color="Olive">U1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A'</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>reconsider </b><font color="Maroon">k</font> = <font color="Maroon">k</font> as   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K3">R^1</a>  <b>by </b><i>, <a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<b>reconsider </b><font color="Maroon">k''</font> = <font color="Maroon">k</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <a href="metric_1.html#K8">RealSpace</a> </span>)</span> <b>by </b><i><a class="ref" href="topmetr.html#D7">TOPMETR:def 7</a></i>;<br/>
<b>reconsider </b><font color="Maroon">k'</font> = <font color="Maroon">k''</font> as   <a href="pre_topc.html#NM2">Point</a> of <a href="metric_1.html#K8">RealSpace</a>  <b>by </b><i><a class="ref" href="topmetr.html#T16">TOPMETR:16</a></i>;<br/>
<b>consider </b><font color="Maroon">Bs</font> being    <a href="yellow_8.html#M1">Basis</a> of <font color="Maroon">k''</font><b> such that </b><br/><a NAME="E7:46_1_3_1_1"/><i><font color="Green">E121</font></i>: 
<font color="Maroon">Bs</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">k'</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> 
 <b>and </b><br/><a NAME="E8:46_1_3_1_1"/>
<font color="Maroon">Bs</font> is <a href="card_4.html#V1">countable</a>
 <b>and </b><br/><a NAME="E9:46_1_3_1_1"/><i><font color="Green">E146</font></i>: 
 ex <font color="Olive">f</font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> ,<font color="Maroon">Bs</font> st <br/>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="Olive">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">k'</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/>;<br/>
<b>consider </b><font color="Maroon">h</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> ,<font color="Maroon">Bs</font><b> such that </b><br/><a NAME="E11:46_1_3_1_1"/><i><font color="Green">E147</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">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">h</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">k'</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/>;<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><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A'</font>;<br/>
<a NAME="E12:46_1_3_1_1"/><i><font color="Green">E149</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">z</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">z</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K3">R^1</a>  &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">n</font>,<font color="Olive">z</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_3_1_1_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="E1:46_1_3_1_1_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 ;<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="E3:46_1_3_1_1_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n'</font>
 <b>and </b><br/><a NAME="E4:46_1_3_1_1_1"/><i><font color="Green">E150</font></i>: 
<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">k'</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>
 <b>by </b><i><a class="ref" href="frechet.html#T6" target="_self">Th6</a></i>;<br/>
<a NAME="E5:46_1_3_1_1_1"/><i><font color="Green">E151</font></i>: 
<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Bs</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E6:46_1_3_1_1_1"/><i><font color="Green">E152</font></i>: 
<font color="Maroon">Bs</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> <a href="metric_1.html#K8">RealSpace</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_8.html#D2">YELLOW_8:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">H</font> = <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</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> <a href="metric_1.html#K8">RealSpace</a> </span>)</span> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">H</font> = <font color="Maroon">H</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> <a href="metric_1.html#K8">RealSpace</a> </span>)</span> ;<br/>
<a NAME="E9:46_1_3_1_1_1"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font>
 
<b>by </b><i>, <a class="ref" href="yellow_8.html#T21">YELLOW_8:21</a></i>;<br/>
<a NAME="E10:46_1_3_1_1_1"/><b>then </b><i><font color="Green">E153</font></i>: 
<font color="Maroon">H</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, , , <a class="ref" href="topmetr.html#D7">TOPMETR:def 7</a></i>;<br/>
<b>consider </b><font color="Maroon">z</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">H</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A'</font>;<br/>
<a NAME="E12:46_1_3_1_1_1"/><i><font color="Green">E154</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>take </b>
<font color="Maroon">z</font>
;<br/>

<b>thus </b><a NAME="E13:46_1_3_1_1_1"/>
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K3">R^1</a>  &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A'</font> )
 <b>by </b><i>, , <a class="ref" href="topmetr.html#D7">TOPMETR:def 7</a></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="E14:46_1_3_1_1"/><i><font color="Green">E155</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 <a href="topmetr.html#K3">R^1</a>  )
 <b>and </b><br/><a NAME="E15:46_1_3_1_1"/><i><font color="Green">E166</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">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>
<b>reconsider </b><font color="Maroon">S'</font> = <font color="Maroon">S'</font> as   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> ,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K3">R^1</a>  <b>by </b><i>, <a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/>
<a NAME="E17:46_1_3_1_1"/><i><font color="Green">E167</font></i>: 
 <a href="relset_1.html#K5">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="46_1_3_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</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="E1:46_1_3_1_1_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">S'</font>
 ;<br/>

<b>then consider </b><font color="Maroon">z'</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:46_1_3_1_1_2"/><i><font color="Green">E169</font></i>: 
<font color="Maroon">z'</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">S'</font>
 <b>and </b><br/><a NAME="E4:46_1_3_1_1_2"/><i><font color="Green">E170</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z'</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E5:46_1_3_1_1_2"/>
<font color="Maroon">S'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z'</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z'</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A'</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E6:46_1_3_1_1_2"/>
<font color="Maroon">z</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>
<a NAME="E18:46_1_3_1_1"/>
<font color="Maroon">S'</font> <a href="frechet.html#R1" target="_self">is_convergent_to</a> <font color="Maroon">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_3_1_1_3"><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 <a href="topmetr.html#K3">R^1</a> ; <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><a NAME="E1:46_1_3_1_1_3"/><i><font color="Green">E171</font></i>: 
( <font color="Maroon">U1</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">U1</font> )
 ;<br/>

<b>reconsider </b><font color="Maroon">U2</font> = <font color="Maroon">U1</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> <a href="metric_1.html#K8">RealSpace</a> </span>)</span> <b>by </b><i><a class="ref" href="topmetr.html#D7">TOPMETR:def 7</a></i>;<br/>
<b>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="E4:46_1_3_1_1_3"/><i><font color="Green">E172</font></i>: 
<font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>and </b><br/><a NAME="E5:46_1_3_1_1_3"/><i><font color="Green">E173</font></i>: 
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">k'</font>,<font color="Maroon">r</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">U2</font>
 <b>by </b><i>, <a class="ref" href="topmetr.html#T22">TOPMETR:22</a>, <a class="ref" href="topmetr.html#D7">TOPMETR:def 7</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="E8:46_1_3_1_1_3"/><i><font color="Green">E174</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="E9:46_1_3_1_1_3"/><i><font color="Green">E175</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i>, </i>;<br/>
<b>take </b>
<font color="Maroon">n</font>
;<br/>

<b>thus </b><a NAME="E10:46_1_3_1_1_3"/>
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="46_1_3_1_1_3_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="E1:46_1_3_1_1_3_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/>

<a NAME="E2:46_1_3_1_1_3_1"/><b>then </b><i><font color="Green">E176</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E3:46_1_3_1_1_3_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E4:46_1_3_1_1_3_1"/><b>then </b>
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="real_1.html#K6">/</a> <font color="Maroon">n</font>
 
<b>by </b><i>, <a class="ref" href="real_2.html#T151">REAL_2:151</a></i>;<br/>
<a NAME="E5:46_1_3_1_1_3_1"/><b>then </b>
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="frechet.html#T7" target="_self">Th7</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E6:46_1_3_1_1_3_1"/><b>then </b>
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">k'</font>,<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">k'</font>,<font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="pcomps_1.html#T1">PCOMPS_1:1</a></i>;<br/>
<a NAME="E7:46_1_3_1_1_3_1"/><b>then </b><i><font color="Green">E177</font></i>: 
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">k'</font>,<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">U2</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>consider </b><font color="Maroon">m'</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="E9:46_1_3_1_1_3_1"/><i><font color="Green">E178</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m'</font>
 <b>and </b><br/><a NAME="E10:46_1_3_1_1_3_1"/><i><font color="Green">E179</font></i>: 
<font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">k'</font>,<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">m'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="frechet.html#T6" target="_self">Th6</a></i>;<br/>
<a NAME="E11:46_1_3_1_1_3_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> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A'</font>
 
<b>by </b><i/>;<br/>
<a NAME="E12:46_1_3_1_1_3_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">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E13:46_1_3_1_1_3_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>, , </i>;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div>
<b>hence </b><a NAME="E19:46_1_3_1_1"/>
contradiction
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">g</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E4:46_1_3_1"/><i><font color="Green">E180</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 <b>and </b><br/><a NAME="E5:46_1_3_1"/><i><font color="Green">E181</font></i>: 
for <font color="Olive">k</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">k</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">k</font>,<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font>]
 <b>from </b><i><a class="ref" href="zfrefle1.html#S1">ZFREFLE1:sch 1</a>();<br/></i>
<a NAME="E6:46_1_3_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</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 <a href="topmetr.html#K3">R^1</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_3_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</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="E1:46_1_3_1_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font>
 ;<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="E3:46_1_3_1_2"/><i><font color="Green">E182</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font>
 <b>and </b><br/><a NAME="E4:46_1_3_1_2"/><i><font color="Green">E183</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E5:46_1_3_1_2"/>
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <a href="topmetr.html#K3">R^1</a> 
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E6:46_1_3_1_2"/>
<font color="Maroon">z</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 <a href="topmetr.html#K3">R^1</a> 
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">F</font> =  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <a href="topmetr.html#K3">R^1</a>  ;<br/>
<a NAME="E8:46_1_3_1"/>
<font color="Maroon">F</font> is <a href="tops_2.html#V1">open</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_3_1_3"><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 <a href="topmetr.html#K3">R^1</a> ; <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="E1:46_1_3_1_3"/>
<font color="Maroon">O</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<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="E3:46_1_3_1_3"/><i><font color="Green">E184</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font>
 <b>and </b><br/><a NAME="E4:46_1_3_1_3"/><i><font color="Green">E185</font></i>: 
<font color="Maroon">O</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E5:46_1_3_1_3"/>
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <a href="topmetr.html#K3">R^1</a> 
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E6:46_1_3_1_3"/>
<font color="Maroon">O</font> is <a href="pre_topc.html#V3">open</a>
 <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="E9:46_1_3_1"/><b>then </b><i><font color="Green">E186</font></i>: 
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="tops_2.html#T26">TOPS_2:26</a></i>;<br/>
<a NAME="E10:46_1_3_1"/><i><font color="Green">E187</font></i>: 
 <a href="numbers.html#K5">NAT</a>  <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_3_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</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="E1:46_1_3_1_4"/><i><font color="Green">E188</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 ;<br/>

<a NAME="E2:46_1_3_1_4"/><b>then </b><i><font color="Green">E189</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:46_1_3_1_4"/>
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font>
 
<b>by </b><i>, <a class="ref" href="frechet.html#T6" target="_self">Th6</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E4:46_1_3_1_4"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:46_1_3_1"/>
<span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font></span>)</span> <a href="subset_1.html#K7">\</a> <a href="numbers.html#K5">NAT</a>  <a href="tarski.html#R1">c=</a> <a href="numbers.html#K1">REAL</a>  <a href="subset_1.html#K6">\</a> <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="topmetr.html#T24">TOPMETR:24</a>, <a class="ref" href="xboole_1.html#T33">XBOOLE_1:33</a></i>;<br/>
<a NAME="E12:46_1_3_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font></span>)</span> <a href="subset_1.html#K7">\</a> <a href="numbers.html#K5">NAT</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="numbers.html#K1">REAL</a> </span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="numbers.html#K1">REAL</a>  <a href="subset_1.html#K6">\</a> <a href="numbers.html#K5">NAT</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="numbers.html#K1">REAL</a> </span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T9">XBOOLE_1:9</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">B</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font></span>)</span> <a href="subset_1.html#K7">\</a> <a href="numbers.html#K5">NAT</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="numbers.html#K1">REAL</a> </span><a href="tarski.html#K1">}</a></span> as   <a href="struct_0.html#NM2">Subset</a> of <a href="frechet.html#K2" target="_self">REAL?</a>  <b>by </b><i><a class="txt" href="frechet.html#E6"><i><font color="Green">Lemma33</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">B</font> = <font color="Maroon">B</font> as   <a href="struct_0.html#NM2">Subset</a> of <a href="frechet.html#K2" target="_self">REAL?</a>  ;<br/>
<a NAME="E15:46_1_3_1"/>
( <font color="Maroon">B</font> is <a href="pre_topc.html#V3">open</a> &amp;  <a href="numbers.html#K1">REAL</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> )
 
<b>by </b><i>, , <a class="ref" href="frechet.html#T5" target="_self">Th5</a></i>;<br/>
<a NAME="E16:46_1_3_1"/><b>then </b><i><font color="Green">E190</font></i>: 
<font color="Maroon">A</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">B</font>
 
<b>by </b><i>, , <a class="ref" href="pre_topc.html#D13">PRE_TOPC:def 13</a></i>;<br/>
<a NAME="E17:46_1_3_1"/>
<font color="Maroon">B</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_3_1_5"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:46_1_3_1_5"/><i><font color="Green">E191</font></i>: 
<font color="Maroon">B</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>consider </b><font color="Maroon">z</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font>;<br/>
<a NAME="E3:46_1_3_1_5"/><i><font color="Green">E192</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">z</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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1_3_1_5_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:46_1_3_1_5_1"/>
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font></span>)</span> <a href="subset_1.html#K7">\</a> <a href="numbers.html#K5">NAT</a>  or <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="numbers.html#K1">REAL</a> </span><a href="tarski.html#K1">}</a></span> )
 </span><b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1_3_1_5_1_1"><b>suppose </b></a><a NAME="E1:46_1_3_1_5_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font></span>)</span> <a href="subset_1.html#K7">\</a> <a href="numbers.html#K5">NAT</a> 
 ;<br/><div class="add"><a NAME="E2:46_1_3_1_5_1_1"/><b>then </b>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>then consider </b><font color="Maroon">C</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:46_1_3_1_5_1_1"/><i><font color="Green">E193</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 <b>and </b><br/><a NAME="E5:46_1_3_1_5_1_1"/><i><font color="Green">E194</font></i>: 
<font color="Maroon">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon">l</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:46_1_3_1_5_1_1"/><i><font color="Green">E196</font></i>: 
<font color="Maroon">l</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font>
 <b>and </b><br/><a NAME="E8:46_1_3_1_5_1_1"/><i><font color="Green">E197</font></i>: 
<font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">l</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E9:46_1_3_1_5_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">l</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E10:46_1_3_1_5_1_1"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1_3_1_5_1_2"><b>suppose </b></a><a NAME="E1:46_1_3_1_5_1_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="numbers.html#K1">REAL</a> </span><a href="tarski.html#K1">}</a></span>
 ;<br/><div class="add"><a NAME="E2:46_1_3_1_5_1_2"/><b>then </b>
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K1">REAL</a> 
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:46_1_3_1_5_1_2"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E18:46_1_3_1"/>
contradiction
 <b>by </b><i><a class="ref" href="frechet.html#T6" target="_self">Th6</a>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon">k</font> being   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K3">R^1</a> <b> such that </b><br/><a NAME="E5:46_1_3"/><i><font color="Green">E198</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 <b>and </b><br/><a NAME="E6:46_1_3"/><i><font color="Green">E199</font></i>: 
 ex <font color="Olive">S'</font> being  <a href="normsp_1.html#NM2">sequence</a> of <a href="topmetr.html#K3">R^1</a>  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="Olive">S'</font> <a href="frechet.html#R1" target="_self">is_convergent_to</a> <font color="Maroon">k</font> )
 ;<br/><b>consider </b><font color="Maroon">S'</font> being   <a href="normsp_1.html#NM2">sequence</a> of <a href="topmetr.html#K3">R^1</a> <b> such that </b><br/><a NAME="E8:46_1_3"/><i><font color="Green">E200</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">S'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A'</font>
 <b>and </b><br/><a NAME="E9:46_1_3"/><i><font color="Green">E201</font></i>: 
<font color="Maroon">S'</font> <a href="frechet.html#R1" target="_self">is_convergent_to</a> <font color="Maroon">k</font>
 <b>by </b><i/>;<br/><a NAME="E10:46_1_3"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">S'</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <a href="frechet.html#K2" target="_self">REAL?</a> 
 <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">S</font> = <font color="Maroon">S'</font> as   <a href="normsp_1.html#NM2">sequence</a> of <a href="frechet.html#K2" target="_self">REAL?</a>  <b>by </b><i/>;<br/><b>take </b>
<font color="Maroon">S</font>
;<br/><b>thus </b><a NAME="E12:46_1_3"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">S</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 <b>by </b><i/>;<br/><a NAME="E13:46_1_3"/>
<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="46_1_3_2"><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 <a href="frechet.html#K2" target="_self">REAL?</a> ; <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><a NAME="E1:46_1_3_2"/>
( <font color="Maroon">U1</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">U1</font> )
 ;<br/>

<b>then consider </b><font color="Maroon">O</font> being   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a> <b> such that </b><br/><a NAME="E3:46_1_3_2"/><i><font color="Green">E202</font></i>: 
<font color="Maroon">O</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E4:46_1_3_2"/><i><font color="Green">E203</font></i>: 
 <a href="numbers.html#K5">NAT</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">O</font>
 <b>and </b><br/><a NAME="E5:46_1_3_2"/><i><font color="Green">E204</font></i>: 
<font color="Maroon">U1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">O</font> <a href="subset_1.html#K7">\</a> <a href="numbers.html#K5">NAT</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="numbers.html#K1">REAL</a> </span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, <a class="ref" href="frechet.html#T5" target="_self">Th5</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="E7:46_1_3_2"/><i><font color="Green">E205</font></i>: 
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">O</font>
 <b>by </b><i>, , , , </i>;<br/>
<b>take </b>
<font color="Maroon">n</font>
;<br/>

<b>thus </b><a NAME="E8:46_1_3_2"/>
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="46_1_3_2_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="E1:46_1_3_2_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/>

<a NAME="E2:46_1_3_2_1"/><b>then </b><i><font color="Green">E206</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> <font color="Maroon">O</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:46_1_3_2_1"/>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 
;<br/>
<a NAME="E4:46_1_3_2_1"/><b>then </b>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">S'</font>
 
<b>by </b><i><a class="ref" href="normsp_1.html#T17">NORMSP_1:17</a></i>;<br/>
<a NAME="E5:46_1_3_2_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>  <a href="relset_1.html#K5">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/>
<a NAME="E6:46_1_3_2_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">A'</font>
 
<b>by </b><i/>;<br/>
<a NAME="E7:46_1_3_2_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> the <a href="struct_0.html#U1">carrier</a> of <a href="frechet.html#K2" target="_self">REAL?</a> 
 
;<br/>
<a NAME="E8:46_1_3_2_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> <span class="p1">(<span class="default"><a href="numbers.html#K1">REAL</a>  <a href="subset_1.html#K6">\</a> <a href="numbers.html#K5">NAT</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="numbers.html#K1">REAL</a> </span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="txt" href="frechet.html#E6"><i><font color="Green">Lemma33</font></i></a></i>;<br/>
<a NAME="E9:46_1_3_2_1"/><b>then </b><i><font color="Green">E207</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="numbers.html#K1">REAL</a>  <a href="subset_1.html#K6">\</a> <a href="numbers.html#K5">NAT</a>  or <font color="Maroon">S'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="numbers.html#K1">REAL</a> </span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E10:46_1_3_2_1"/>
<font color="Maroon">S'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="numbers.html#K1">REAL</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_3_2_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:46_1_3_2_1_1"/><i><font color="Green">E208</font></i>: 
<font color="Maroon">S'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K1">REAL</a> 
 ;<br/>

<a NAME="E2:46_1_3_2_1_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>  <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<b>hence </b><a NAME="E3:46_1_3_2_1_1"/>
contradiction
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E11:46_1_3_2_1"/><b>then </b>
not <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="numbers.html#K5">NAT</a> 
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E12:46_1_3_2_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">O</font> <a href="subset_1.html#K7">\</a> <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="E13:46_1_3_2_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="ref" href="frechet.html#T6" target="_self">Th6</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div><b>hence </b><a NAME="E14:46_1_3"/>
<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/></div><b>end;</b></div></div><b>end;</b></div>

</div>
