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

<b>let </b><font color="Maroon" title="c1">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="frechet.html#K2" title="FRECHET:func.2">REAL?</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="frechet.html#D5" target="_self" title="FRECHET:def.5">FRECHET:def 5</a><br/><b>let </b><font color="Maroon" title="c2">x</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="frechet.html#K2" title="FRECHET:func.2">REAL?</a> ;<br/>



<b>assume </b><a NAME="E1:46"/><i><font color="Green" title="E41">A1</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c1">A</font>
 ;<br/>

<a NAME="E2:46"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="frechet.html#K2" title="FRECHET:func.2">REAL?</a> 
 
;<br/>
<a NAME="E3:46"/><b>then </b>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="frechet.html#D7" target="_self" title="FRECHET:def.7">Def7</a></i>;<br/>
<a NAME="E4:46"/><b>then </b><i><font color="Green" title="E42">A2</font></i>: 
( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  or <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">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" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  or ( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp; <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> ) or ( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp; not <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> ) )
 </span><b>by </b><i><a class="txt" href="frechet.html#E4:46"><i><font color="Green" title="E42">A2</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">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" title="E43">A3</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 ;<br/><div class="add"><a NAME="E2:46_1_1"/><b>then </b><i><font color="Green" title="E44">A4</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 ;<br/><b>reconsider </b><font color="Maroon" title="c3">x'</font> = <font color="Maroon" title="c2">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  <b>by </b><i><a class="txt" href="frechet.html#E1:46_1_1"><i><font color="Green" title="E43">A3</font></i></a>, <a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></i>;<br/><a NAME="E4:46_1_1"/>
<font color="Maroon" title="c1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="frechet.html#K2" title="FRECHET:func.2">REAL?</a> 
 ;<br/><a NAME="E5:46_1_1"/><b>then </b><i><font color="Green" title="E45">A5</font></i>: 
<font color="Maroon" title="c1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="frechet.html#D7" target="_self" title="FRECHET:def.7">Def7</a></i>;<br/><a NAME="E6:46_1_1"/>
<font color="Maroon" title="c1">A</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">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" title="c4">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:46_1_1_1"/>
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/>

<a NAME="E2:46_1_1_1"/><b>then </b>
( <font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; not <font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E3:46_1_1_1"/><b>then </b>
( ( <font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  or <font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) &amp; not <font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<b>by </b><i><a class="txt" href="frechet.html#E5:46_1_1"><i><font color="Green" title="E45">A5</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E4:46_1_1_1"/>
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 ;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c4">A'</font> = <font color="Maroon" title="c1">A</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  <b>by </b><i><a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c5">A'</font> = <font color="Maroon" title="c4">A'</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  ;<br/><a NAME="E9:46_1_1"/>
for <font color="Olive" title="b1">B'</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>   st <font color="Olive" title="b1">B'</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c3">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">B'</font> holds <br/><font color="Maroon" title="c5">A'</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Olive" title="b1">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" title="c6">B'</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> ;<br/>

<b>assume </b><a NAME="E1:46_1_1_2"/><i><font color="Green" title="E46">A6</font></i>: 
<font color="Maroon" title="c6">B'</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 ;<br/>

<b>assume </b><a NAME="E2:46_1_1_2"/><i><font color="Green" title="E47">A7</font></i>: 
<font color="Maroon" title="c3">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B'</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c7">B1</font> = <font color="Maroon" title="c6">B'</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  ;<br/>
<b>reconsider </b><font color="Maroon" title="c8">C</font> =  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  <b>by </b><i><a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">C</font> = <font color="Maroon" title="c8">C</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  ;<br/>
<a NAME="E6:46_1_1_2"/>
<font color="Maroon" title="c9">C</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 
<b>by </b><i><a class="ref" href="frechet.html#T10" target="_self" title="FRECHET:th.10">Th10</a></i>;<br/>
<a NAME="E7:46_1_1_2"/><b>then </b><i><font color="Green" title="E48">A8</font></i>: 
<font color="Maroon" title="c7">B1</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c9">C</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="txt" href="frechet.html#E1:46_1_1_2"><i><font color="Green" title="E46">A6</font></i></a>, <a class="ref" href="frechet.html#T4" target="_self" title="FRECHET:th.4">Th4</a></i>;<br/>
<a NAME="E8:46_1_1_2"/>
<font color="Maroon" title="c6">B'</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="xboole_1.html#T79" title="XBOOLE_1:th.79">XBOOLE_1:79</a></i>;<br/>
<a NAME="E9:46_1_1_2"/><b>then </b><i><font color="Green" title="E49">A9</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">B'</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c10">D</font> = <font color="Maroon" title="c7">B1</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c9">C</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="frechet.html#K2" title="FRECHET:func.2">REAL?</a>  <b>by </b><i><a class="ref" href="frechet.html#T32" target="_self" title="FRECHET:th.32">Th32</a></i>;<br/>
<a NAME="E11:46_1_1_2"/><i><font color="Green" title="E50">A10</font></i>: 
<font color="Maroon" title="c10">D</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="txt" href="frechet.html#E7:46_1_1_2"><i><font color="Green" title="E48">A8</font></i></a>, <a class="txt" href="frechet.html#E9:46_1_1_2"><i><font color="Green" title="E49">A9</font></i></a>, <a class="ref" href="frechet.html#T33" target="_self" title="FRECHET:th.33">Th33</a></i>;<br/>
<a NAME="E12:46_1_1_2"/>
not <font color="Maroon" title="c3">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="txt" href="frechet.html#E1:46_1_1"><i><font color="Green" title="E43">A3</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E13:46_1_1_2"/><b>then </b><i><font color="Green" title="E51">A11</font></i>: 
<font color="Maroon" title="c3">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B'</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="txt" href="frechet.html#E2:46_1_1_2"><i><font color="Green" title="E47">A7</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">D</font> = <font color="Maroon" title="c10">D</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="frechet.html#K2" title="FRECHET:func.2">REAL?</a>  ;<br/>
<a NAME="E15:46_1_1_2"/>
<font color="Maroon" title="c1">A</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c11">D</font>
 
<b>by </b><i><a class="txt" href="frechet.html#E1:46"><i><font color="Green" title="E41">A1</font></i></a>, <a class="txt" href="frechet.html#E11:46_1_1_2"><i><font color="Green" title="E50">A10</font></i></a>, <a class="txt" href="frechet.html#E13:46_1_1_2"><i><font color="Green" title="E51">A11</font></i></a>, <a class="ref" href="pre_topc.html#D13" title="PRE_TOPC:def.13">PRE_TOPC:def 13</a></i>;<br/>
<a NAME="E16:46_1_1_2"/><b>then </b><i><font color="Green" title="E52">A12</font></i>: 
<font color="Maroon" title="c1">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c11">D</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E17:46_1_1_2"/>
<font color="Maroon" title="c5">A'</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c6">B'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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" title="c12">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c11">D</font>;<br/>
<a NAME="E2:46_1_1_2_1"/><i><font color="Green" title="E53">A13</font></i>: 
<font color="Maroon" title="c12">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">D</font>
 
<b>by </b><i><a class="txt" href="frechet.html#E16:46_1_1_2"><i><font color="Green" title="E52">A12</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E3:46_1_1_2_1"/><b>then </b><i><font color="Green" title="E54">A14</font></i>: 
<font color="Maroon" title="c12">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B'</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E4:46_1_1_2_1"/><i><font color="Green" title="E55">A15</font></i>: 
<font color="Maroon" title="c12">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<b>by </b><i><a class="txt" href="frechet.html#E2:46_1_1_2_1"><i><font color="Green" title="E53">A13</font></i></a>, <a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></i>;<br/>
<a NAME="E5:46_1_1_2_1"/><i><font color="Green" title="E56">A16</font></i>: 
not <font color="Maroon" title="c12">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</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" title="c12">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/>

<a NAME="E2:46_1_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c12">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">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" title="E55">A15</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:46_1_1_2_1"/>
<font color="Maroon" title="c12">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 
<b>by </b><i><a class="txt" href="frechet.html#E16:46_1_1_2"><i><font color="Green" title="E52">A12</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E7:46_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c12">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="frechet.html#E5:46_1_1_2_1"><i><font color="Green" title="E56">A16</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="E8:46_1_1_2_1"/>
<font color="Maroon" title="c5">A'</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c6">B'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="frechet.html#E3:46_1_1_2_1"><i><font color="Green" title="E54">A14</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">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" title="c5">A'</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c6">B'</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/>


</div><b>end;</b></div><a NAME="E10:46_1_1"/><b>then </b>
<font color="Maroon" title="c3">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c5">A'</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#D13" title="PRE_TOPC:def.13">PRE_TOPC:def 13</a></i>;<br/><b>then consider </b><font color="Maroon" title="c6">S'</font> being   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> <b> such that </b><br/><a NAME="E12:46_1_1"/><i><font color="Green" title="E46">A17</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">S'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">A'</font>
 <b>and </b><br/><a NAME="E13:46_1_1"/><i><font color="Green" title="E47">A18</font></i>: 
<font color="Maroon" title="c3">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="frechet.html#K1" title="FRECHET:func.1">Lim</a> <font color="Maroon" title="c6">S'</font>
 <b>by </b><i><a class="ref" href="frechet.html#D5" target="_self" title="FRECHET:def.5">Def5</a></i>;<br/><a NAME="E14:46_1_1"/><i><font color="Green" title="E48">A19</font></i>: 
<font color="Maroon" title="c6">S'</font> <a href="frechet.html#R1" title="FRECHET:pred.1">is_convergent_to</a> <font color="Maroon" title="c3">x'</font>
 <b>by </b><i><a class="txt" href="frechet.html#E13:46_1_1"><i><font color="Green" title="E47">A18</font></i></a>, <a class="ref" href="frechet.html#D4" target="_self" title="FRECHET:def.4">Def4</a></i>;<br/><a NAME="E15:46_1_1"/><i><font color="Green" title="E49">A20</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">S'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">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" title="c7">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:46_1_1_3"/>
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">S'</font>
 ;<br/>

<b>hence </b><a NAME="E2:46_1_1_3"/>
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 <b>by </b><i><a class="txt" href="frechet.html#E12:46_1_1"><i><font color="Green" title="E46">A17</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">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" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">S'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="frechet.html#K2" title="FRECHET:func.2">REAL?</a> 
 <b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c7">S</font> = <font color="Maroon" title="c6">S'</font> as   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <a href="frechet.html#K2" title="FRECHET:func.2">REAL?</a>  <b>by </b><i><a class="ref" href="frechet.html#T2" target="_self" title="FRECHET:th.2">Th2</a></i>;<br/><b>take </b>
<font color="Maroon" title="c7">S</font>
;<br/><b>thus </b><a NAME="E18:46_1_1"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c7">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">A</font>
 <b>by </b><i><a class="txt" href="frechet.html#E15:46_1_1"><i><font color="Green" title="E49">A20</font></i></a></i>;<br/><a NAME="E19:46_1_1"/>
<font color="Maroon" title="c7">S</font> <a href="frechet.html#R1" title="FRECHET:pred.1">is_convergent_to</a> <font color="Maroon" title="c2">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" title="c8">V</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="frechet.html#K2" title="FRECHET:func.2">REAL?</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="frechet.html#D2" target="_self" title="FRECHET:def.2">FRECHET:def 2</a><br/>

<b>assume </b><a NAME="E1:46_1_1_4"/><i><font color="Green" title="E50">A21</font></i>: 
( <font color="Maroon" title="c8">V</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">V</font> )
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c9">C</font> = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="frechet.html#K2" title="FRECHET:func.2">REAL?</a>  <b>by </b><i><a class="txt" href="frechet.html#E39"><i><font color="Green" title="E33">Lm4</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">C</font> = <font color="Maroon" title="c9">C</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="frechet.html#K2" title="FRECHET:func.2">REAL?</a>  ;<br/>
<a NAME="E4:46_1_1_4"/>
<font color="Maroon" title="c10">C</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 
<b>by </b><i><a class="ref" href="frechet.html#T34" target="_self" title="FRECHET:th.34">Th34</a></i>;<br/>
<a NAME="E5:46_1_1_4"/><b>then </b><i><font color="Green" title="E51">A22</font></i>: 
<font color="Maroon" title="c8">V</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c10">C</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="txt" href="frechet.html#E1:46_1_1_4"><i><font color="Green" title="E50">A21</font></i></a>, <a class="ref" href="frechet.html#T4" target="_self" title="FRECHET:th.4">Th4</a></i>;<br/>
<a NAME="E6:46_1_1_4"/>
 <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E7:46_1_1_4"/><b>then </b><i><font color="Green" title="E52">A23</font></i>: 
not  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">V</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c11">V'</font> = <font color="Maroon" title="c8">V</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c10">C</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  <b>by </b><i><a class="ref" href="frechet.html#T32" target="_self" title="FRECHET:th.32">Th32</a></i>;<br/>
<a NAME="E9:46_1_1_4"/><i><font color="Green" title="E53">A24</font></i>: 
<font color="Maroon" title="c11">V'</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="txt" href="frechet.html#E5:46_1_1_4"><i><font color="Green" title="E51">A22</font></i></a>, <a class="txt" href="frechet.html#E7:46_1_1_4"><i><font color="Green" title="E52">A23</font></i></a>, <a class="ref" href="frechet.html#T33" target="_self" title="FRECHET:th.33">Th33</a></i>;<br/>
<a NAME="E10:46_1_1_4"/>
not <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">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" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">C</font>
 ;<br/>

<a NAME="E2:46_1_1_4_1"/><b>then </b>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:46_1_1_4_1"/>
contradiction
 <b>by </b><i><a class="txt" href="frechet.html#E2:46_1_1"><i><font color="Green" title="E44">A4</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:46_1_1_4"/><b>then </b>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">V</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c10">C</font>
 
<b>by </b><i><a class="txt" href="frechet.html#E1:46_1_1_4"><i><font color="Green" title="E50">A21</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c12">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E13:46_1_1_4"/><i><font color="Green" title="E54">A25</font></i>: 
for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> holds <br/><font color="Maroon" title="c6">S'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">V'</font>
 <b>by </b><i><a class="txt" href="frechet.html#E14:46_1_1"><i><font color="Green" title="E48">A19</font></i></a>, <a class="txt" href="frechet.html#E9:46_1_1_4"><i><font color="Green" title="E53">A24</font></i></a>, <a class="ref" href="frechet.html#D2" target="_self" title="FRECHET:def.2">Def2</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c12">n</font>
;<br/>

<b>thus </b><a NAME="E14:46_1_1_4"/>
for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> holds <br/><font color="Maroon" title="c7">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">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" title="c13">m</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:46_1_1_4_2"/>
<font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">m</font>
 ;<br/>

<a NAME="E2:46_1_1_4_2"/><b>then </b>
<font color="Maroon" title="c6">S'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">V'</font>
 
<b>by </b><i><a class="txt" href="frechet.html#E13:46_1_1_4"><i><font color="Green" title="E54">A25</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:46_1_1_4_2"/>
<font color="Maroon" title="c7">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">V</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">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" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="frechet.html#K1" title="FRECHET:func.1">Lim</a> <font color="Maroon" title="c7">S</font>
 <b>by </b><i><a class="ref" href="frechet.html#D4" target="_self" title="FRECHET:def.4">Def4</a></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" title="E43">A26</font></i>: 
( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp; <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> )
 ;<br/><div class="add"><a NAME="E2:46_1_2"/><i><font color="Green" title="E44">A27</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="funcop_1.html#K2" title="FUNCOP_1:func.2">--&gt;</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="funcop_1.html#T14" title="FUNCOP_1:th.14">FUNCOP_1:14</a></i>;<br/><a NAME="E3:46_1_2"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="funcop_1.html#K2" title="FUNCOP_1:func.2">--&gt;</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c3">S</font> = <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="funcop_1.html#K2" title="FUNCOP_1:func.2">--&gt;</a> <font color="Maroon" title="c2">x</font> as   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <a href="frechet.html#K2" title="FRECHET:func.2">REAL?</a>  <b>by </b><i><a class="txt" href="frechet.html#E2:46_1_2"><i><font color="Green" title="E44">A27</font></i></a>, <a class="ref" href="funct_2.html#T4" title="FUNCT_2:th.4">FUNCT_2:4</a></i>;<br/><b>take </b>
<font color="Maroon" title="c3">S</font>
;<br/><a NAME="E5:46_1_2"/>
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">A</font>
 <b>by </b><i><a class="txt" href="frechet.html#E1:46_1_2"><i><font color="Green" title="E43">A26</font></i></a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/><b>hence </b><a NAME="E6:46_1_2"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">A</font>
 <b>by </b><i><a class="ref" href="funcop_1.html#T14" title="FUNCOP_1:th.14">FUNCOP_1:14</a></i>;<br/><a NAME="E7:46_1_2"/>
<font color="Maroon" title="c3">S</font> <a href="frechet.html#R1" title="FRECHET:pred.1">is_convergent_to</a> <font color="Maroon" title="c2">x</font>
 <b>by </b><i><a class="ref" href="frechet.html#T23" target="_self" title="FRECHET:th.23">Th23</a></i>;<br/><b>hence </b><a NAME="E8:46_1_2"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="frechet.html#K1" title="FRECHET:func.1">Lim</a> <font color="Maroon" title="c3">S</font>
 <b>by </b><i><a class="ref" href="frechet.html#D4" target="_self" title="FRECHET:def.4">Def4</a></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" title="E43">A28</font></i>: 
( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp; not <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c3">A'</font> = <font color="Maroon" title="c1">A</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  <b>by </b><i><a class="ref" href="frechet.html#T32" target="_self" title="FRECHET:th.32">Th32</a></i>;<br/><a NAME="E3:46_1_3"/>
 ex <font color="Olive" title="b1">k</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  st <br/>( <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp;  ex <font color="Olive" title="b2">S'</font> being  <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  st <br/>(  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b2">S'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">A'</font> &amp; <font color="Olive" title="b2">S'</font> <a href="frechet.html#R1" title="FRECHET:pred.1">is_convergent_to</a> <font color="Olive" title="b1">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" title="E44">A29</font></i>: 
for <font color="Olive" title="b1">k</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>   holds <br/>(  not <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  or for <font color="Olive" title="b2">S'</font> being  <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>   holds <br/>(  not  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b2">S'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">A'</font> or  not <font color="Olive" title="b2">S'</font> <a href="frechet.html#R1" title="FRECHET:pred.1">is_convergent_to</a> <font color="Olive" title="b1">k</font> ) )
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>( $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> $2 &amp; $2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  &amp; $2 <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">A'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  );<br/>
<a NAME="E2:46_1_3_1"/><i><font color="Green" title="E45">A30</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/> ex <font color="Olive" title="b2">U1</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>,<font color="Olive" title="b2">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" title="c4">k</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><a NAME="E2:46_1_3_1_1"/><i><font color="Green" title="E46">A31</font></i>: 
<font color="Maroon" title="c4">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>and </b><br/><a NAME="E3:46_1_3_1_1"/><i><font color="Green" title="E47">A32</font></i>: 
for <font color="Olive" title="b1">U1</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Maroon" title="c4">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">U1</font> &amp; <font color="Olive" title="b1">U1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  holds <br/> not <font color="Olive" title="b1">U1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">A'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c5">k</font> = <font color="Maroon" title="c4">k</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  <b>by </b><i><a class="txt" href="frechet.html#E2:46_1_3_1_1"><i><font color="Green" title="E46">A31</font></i></a>, <a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">k''</font> = <font color="Maroon" title="c5">k</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <a href="metric_1.html#K8" title="METRIC_1:func.8">RealSpace</a> </span>)</span> <b>by </b><i><a class="ref" href="topmetr.html#D7" title="TOPMETR:def.7">TOPMETR:def 7</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">k'</font> = <font color="Maroon" title="c6">k''</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="metric_1.html#K8" title="METRIC_1:func.8">RealSpace</a>  <b>by </b><i><a class="ref" href="topmetr.html#T16" title="TOPMETR:th.16">TOPMETR:16</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">Bs</font> being    <a href="yellow_8.html#M1" title="YELLOW_8:mode.1">Basis</a> of <font color="Maroon" title="c6">k''</font><b> such that </b><br/><a NAME="E8:46_1_3_1_1"/><i><font color="Green" title="E48">A33</font></i>: 
<font color="Maroon" title="c8">Bs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c7">k'</font>,<span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span> where <font color="Olive" title="b1">n</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Olive" title="b1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  </span>}</span> 
 <b>and </b><br/><a NAME="E9:46_1_3_1_1"/>
<font color="Maroon" title="c8">Bs</font> is <a href="card_4.html#V1" title="CARD_4:attr.1">countable</a>
 <b>and </b><br/><a NAME="E10:46_1_3_1_1"/><i><font color="Green" title="E49">A34</font></i>: 
 ex <font color="Olive" title="b1">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<font color="Maroon" title="c8">Bs</font> st <br/>for <font color="Olive" title="b2">n</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/> ex <font color="Olive" title="b3">n'</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">n'</font> &amp; <font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c7">k'</font>,<span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">n'</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="frechet.html#T11" target="_self" title="FRECHET:th.11">Th11</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">h</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<font color="Maroon" title="c8">Bs</font><b> such that </b><br/><a NAME="E12:46_1_3_1_1"/><i><font color="Green" title="E50">A35</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/> ex <font color="Olive" title="b2">n'</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">n'</font> &amp; <font color="Maroon" title="c9">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c7">k'</font>,<span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">n'</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="frechet.html#E10:46_1_3_1_1"><i><font color="Green" title="E49">A34</font></i></a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>$2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">A'</font>;<br/>
<a NAME="E13:46_1_3_1_1"/><i><font color="Green" title="E51">A36</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/> ex <font color="Olive" title="b2">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Olive" title="b2">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" title="c10">n</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:46_1_3_1_1_1"/>
<font color="Maroon" title="c10">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c11">n'</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:46_1_3_1_1_1"/>
<font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">n'</font>
 <b>and </b><br/><a NAME="E4:46_1_3_1_1_1"/><i><font color="Green" title="E52">A37</font></i>: 
<font color="Maroon" title="c9">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c7">k'</font>,<span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">n'</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="frechet.html#E12:46_1_3_1_1"><i><font color="Green" title="E50">A35</font></i></a></i>;<br/>
<a NAME="E5:46_1_3_1_1_1"/><i><font color="Green" title="E53">A38</font></i>: 
<font color="Maroon" title="c9">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Bs</font>
 
<b>by </b><i><a class="txt" href="frechet.html#E8:46_1_3_1_1"><i><font color="Green" title="E48">A33</font></i></a>, <a class="txt" href="frechet.html#E4:46_1_3_1_1_1"><i><font color="Green" title="E52">A37</font></i></a></i>;<br/>
<a NAME="E6:46_1_3_1_1_1"/><i><font color="Green" title="E54">A39</font></i>: 
<font color="Maroon" title="c8">Bs</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <a href="metric_1.html#K8" title="METRIC_1:func.8">RealSpace</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_8.html#D2" title="YELLOW_8:def.2">YELLOW_8:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">H</font> = <font color="Maroon" title="c9">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">n</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <a href="metric_1.html#K8" title="METRIC_1:func.8">RealSpace</a> </span>)</span> <b>by </b><i><a class="txt" href="frechet.html#E5:46_1_3_1_1_1"><i><font color="Green" title="E53">A38</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">H</font> = <font color="Maroon" title="c12">H</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <a href="metric_1.html#K8" title="METRIC_1:func.8">RealSpace</a> </span>)</span> ;<br/>
<a NAME="E9:46_1_3_1_1_1"/>
<font color="Maroon" title="c5">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">H</font>
 
<b>by </b><i><a class="txt" href="frechet.html#E5:46_1_3_1_1_1"><i><font color="Green" title="E53">A38</font></i></a>, <a class="ref" href="yellow_8.html#T21" title="YELLOW_8:th.21">YELLOW_8:21</a></i>;<br/>
<a NAME="E10:46_1_3_1_1_1"/><b>then </b><i><font color="Green" title="E55">A40</font></i>: 
<font color="Maroon" title="c13">H</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">A'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="frechet.html#E3:46_1_3_1_1"><i><font color="Green" title="E47">A32</font></i></a>, <a class="txt" href="frechet.html#E5:46_1_3_1_1_1"><i><font color="Green" title="E53">A38</font></i></a>, <a class="txt" href="frechet.html#E6:46_1_3_1_1_1"><i><font color="Green" title="E54">A39</font></i></a>, <a class="ref" href="topmetr.html#D7" title="TOPMETR:def.7">TOPMETR:def 7</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c14">z</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c13">H</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">A'</font>;<br/>
<a NAME="E12:46_1_3_1_1_1"/><i><font color="Green" title="E56">A41</font></i>: 
<font color="Maroon" title="c14">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">H</font>
 
<b>by </b><i><a class="txt" href="frechet.html#E10:46_1_3_1_1_1"><i><font color="Green" title="E55">A40</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c14">z</font>
;<br/>

<b>thus </b><a NAME="E13:46_1_3_1_1_1"/>
( <font color="Maroon" title="c14">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  &amp; <font color="Maroon" title="c14">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">A'</font> )
 <b>by </b><i><a class="txt" href="frechet.html#E10:46_1_3_1_1_1"><i><font color="Green" title="E55">A40</font></i></a>, <a class="txt" href="frechet.html#E12:46_1_3_1_1_1"><i><font color="Green" title="E56">A41</font></i></a>, <a class="ref" href="topmetr.html#D7" title="TOPMETR:def.7">TOPMETR:def 7</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c10">S'</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E15:46_1_3_1_1"/><i><font color="Green" title="E52">A42</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c10">S'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">S'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  )
 <b>and </b><br/><a NAME="E16:46_1_3_1_1"/><i><font color="Green" title="E53">A43</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Maroon" title="c10">S'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font>]
 <b>from </b><i><a class="ref" href="wellord2.html#S1" title="WELLORD2:sch.1">WELLORD2:sch 1</a>(<a class="txt" href="frechet.html#E13:46_1_3_1_1"><i><font color="Green" title="E51">A36</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon" title="c11">S'</font> = <font color="Maroon" title="c10">S'</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  <b>by </b><i><a class="txt" href="frechet.html#E15:46_1_3_1_1"><i><font color="Green" title="E52">A42</font></i></a>, <a class="ref" href="funct_2.html#T4" title="FUNCT_2:th.4">FUNCT_2:4</a></i>;<br/>
<a NAME="E18:46_1_3_1_1"/><i><font color="Green" title="E54">A44</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c11">S'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">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" title="c12">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:46_1_3_1_1_2"/>
<font color="Maroon" title="c12">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c11">S'</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c13">z'</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:46_1_3_1_1_2"/><i><font color="Green" title="E55">A45</font></i>: 
<font color="Maroon" title="c13">z'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c11">S'</font>
 <b>and </b><br/><a NAME="E4:46_1_3_1_1_2"/><i><font color="Green" title="E56">A46</font></i>: 
<font color="Maroon" title="c12">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">S'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">z'</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E5:46_1_3_1_1_2"/>
<font color="Maroon" title="c11">S'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">z'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">z'</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">A'</font>
 
<b>by </b><i><a class="txt" href="frechet.html#E16:46_1_3_1_1"><i><font color="Green" title="E53">A43</font></i></a>, <a class="txt" href="frechet.html#E3:46_1_3_1_1_2"><i><font color="Green" title="E55">A45</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:46_1_3_1_1_2"/>
<font color="Maroon" title="c12">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A'</font>
 <b>by </b><i><a class="txt" href="frechet.html#E4:46_1_3_1_1_2"><i><font color="Green" title="E56">A46</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E19:46_1_3_1_1"/>
<font color="Maroon" title="c11">S'</font> <a href="frechet.html#R1" title="FRECHET:pred.1">is_convergent_to</a> <font color="Maroon" title="c5">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" title="c12">U1</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="frechet.html#D2" target="_self" title="FRECHET:def.2">FRECHET:def 2</a><br/>

<b>assume </b><a NAME="E1:46_1_3_1_1_3"/><i><font color="Green" title="E55">A47</font></i>: 
( <font color="Maroon" title="c12">U1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c5">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">U1</font> )
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c13">U2</font> = <font color="Maroon" title="c12">U1</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <a href="metric_1.html#K8" title="METRIC_1:func.8">RealSpace</a> </span>)</span> <b>by </b><i><a class="ref" href="topmetr.html#D7" title="TOPMETR:def.7">TOPMETR:def 7</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c14">r</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E4:46_1_3_1_1_3"/><i><font color="Green" title="E56">A48</font></i>: 
<font color="Maroon" title="c14">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>and </b><br/><a NAME="E5:46_1_3_1_1_3"/><i><font color="Green" title="E57">A49</font></i>: 
 <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c7">k'</font>,<font color="Maroon" title="c14">r</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">U2</font>
 <b>by </b><i><a class="txt" href="frechet.html#E1:46_1_3_1_1_3"><i><font color="Green" title="E55">A47</font></i></a>, <a class="ref" href="topmetr.html#T22" title="TOPMETR:th.22">TOPMETR:22</a>, <a class="ref" href="topmetr.html#D7" title="TOPMETR:def.7">TOPMETR:def 7</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c15">r</font> = <font color="Maroon" title="c14">r</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c16">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E8:46_1_3_1_1_3"/><i><font color="Green" title="E58">A50</font></i>: 
1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c16">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">r</font>
 <b>and </b><br/><a NAME="E9:46_1_3_1_1_3"/><i><font color="Green" title="E59">A51</font></i>: 
<font color="Maroon" title="c16">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="frechet.html#E4:46_1_3_1_1_3"><i><font color="Green" title="E56">A48</font></i></a>, <a class="txt" href="frechet.html#E2"><i><font color="Green" title="E2">Lm2</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c16">n</font>
;<br/>

<b>thus </b><a NAME="E10:46_1_3_1_1_3"/>
for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c16">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> holds <br/><font color="Maroon" title="c11">S'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">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" title="c17">m</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:46_1_3_1_1_3_1"/>
<font color="Maroon" title="c16">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">m</font>
 ;<br/>

<a NAME="E2:46_1_3_1_1_3_1"/><b>then </b>
<font color="Maroon" title="c16">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c17">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E3:46_1_3_1_1_3_1"/><b>then </b>
1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c16">n</font>
 
<b>by </b><i><a class="txt" href="frechet.html#E9:46_1_3_1_1_3"><i><font color="Green" title="E59">A51</font></i></a>, <a class="ref" href="real_2.html#T151" title="REAL_2:th.151">REAL_2:151</a></i>;<br/>
<a NAME="E4:46_1_3_1_1_3_1"/><b>then </b>
1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">r</font>
 
<b>by </b><i><a class="txt" href="frechet.html#E8:46_1_3_1_1_3"><i><font color="Green" title="E58">A50</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E5:46_1_3_1_1_3_1"/><b>then </b>
 <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c7">k'</font>,<span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c7">k'</font>,<font color="Maroon" title="c15">r</font>
 
<b>by </b><i><a class="ref" href="pcomps_1.html#T1" title="PCOMPS_1:th.1">PCOMPS_1:1</a></i>;<br/>
<a NAME="E6:46_1_3_1_1_3_1"/><b>then </b><i><font color="Green" title="E60">A52</font></i>: 
 <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c7">k'</font>,<span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">U2</font>
 
<b>by </b><i><a class="txt" href="frechet.html#E5:46_1_3_1_1_3"><i><font color="Green" title="E57">A49</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c18">m'</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E8:46_1_3_1_1_3_1"/><i><font color="Green" title="E61">A53</font></i>: 
<font color="Maroon" title="c17">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">m'</font>
 <b>and </b><br/><a NAME="E9:46_1_3_1_1_3_1"/><i><font color="Green" title="E62">A54</font></i>: 
<font color="Maroon" title="c9">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c7">k'</font>,<span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">m'</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="frechet.html#E12:46_1_3_1_1"><i><font color="Green" title="E50">A35</font></i></a></i>;<br/>
<a NAME="E10:46_1_3_1_1_3_1"/>
<font color="Maroon" title="c11">S'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">m</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">A'</font>
 
<b>by </b><i><a class="txt" href="frechet.html#E16:46_1_3_1_1"><i><font color="Green" title="E53">A43</font></i></a></i>;<br/>
<a NAME="E11:46_1_3_1_1_3_1"/><b>then </b>
<font color="Maroon" title="c11">S'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">m</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E12:46_1_3_1_1_3_1"/>
<font color="Maroon" title="c11">S'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">U1</font>
 <b>by </b><i><a class="txt" href="frechet.html#E6:46_1_3_1_1_3_1"><i><font color="Green" title="E60">A52</font></i></a>, <a class="txt" href="frechet.html#E8:46_1_3_1_1_3_1"><i><font color="Green" title="E61">A53</font></i></a>, <a class="txt" href="frechet.html#E9:46_1_3_1_1_3_1"><i><font color="Green" title="E62">A54</font></i></a></i>;<br/>


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


</div><b>end;</b></div>
<b>hence </b><a NAME="E20:46_1_3_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="frechet.html#E1:46_1_3_1"><i><font color="Green" title="E44">A29</font></i></a>, <a class="txt" href="frechet.html#E2:46_1_3_1_1"><i><font color="Green" title="E46">A31</font></i></a>, <a class="txt" href="frechet.html#E18:46_1_3_1_1"><i><font color="Green" title="E54">A44</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c4">g</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E4:46_1_3_1"/><i><font color="Green" title="E46">A55</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>and </b><br/><a NAME="E5:46_1_3_1"/><i><font color="Green" title="E47">A56</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>,<font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font>]
 <b>from </b><i><a class="ref" href="classes1.html#S1" title="CLASSES1:sch.1">CLASSES1:sch 1</a>(<a class="txt" href="frechet.html#E2:46_1_3_1"><i><font color="Green" title="E45">A30</font></i></a>);<br/></i>
<a NAME="E6:46_1_3_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">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" title="c5">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:46_1_3_1_2"/>
<font color="Maroon" title="c5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">g</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c6">k</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:46_1_3_1_2"/><i><font color="Green" title="E48">A57</font></i>: 
<font color="Maroon" title="c6">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">g</font>
 <b>and </b><br/><a NAME="E4:46_1_3_1_2"/><i><font color="Green" title="E49">A58</font></i>: 
<font color="Maroon" title="c5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E5:46_1_3_1_2"/>
<font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> 
 
<b>by </b><i><a class="txt" href="frechet.html#E4:46_1_3_1"><i><font color="Green" title="E46">A55</font></i></a>, <a class="txt" href="frechet.html#E5:46_1_3_1"><i><font color="Green" title="E47">A56</font></i></a>, <a class="txt" href="frechet.html#E3:46_1_3_1_2"><i><font color="Green" title="E48">A57</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:46_1_3_1_2"/>
<font color="Maroon" title="c5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> 
 <b>by </b><i><a class="txt" href="frechet.html#E4:46_1_3_1_2"><i><font color="Green" title="E49">A58</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c5">F</font> =  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">g</font> as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  ;<br/>
<a NAME="E8:46_1_3_1"/>
<font color="Maroon" title="c5">F</font> is <a href="tops_2.html#V1" title="TOPS_2:attr.1">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" title="c6">O</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tops_2.html#D1" title="TOPS_2:def.1">TOPS_2:def 1</a><br/>

<b>assume </b><a NAME="E1:46_1_3_1_3"/>
<font color="Maroon" title="c6">O</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">F</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">k</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:46_1_3_1_3"/><i><font color="Green" title="E48">A59</font></i>: 
<font color="Maroon" title="c7">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">g</font>
 <b>and </b><br/><a NAME="E4:46_1_3_1_3"/><i><font color="Green" title="E49">A60</font></i>: 
<font color="Maroon" title="c6">O</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E5:46_1_3_1_3"/>
<font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> 
 
<b>by </b><i><a class="txt" href="frechet.html#E4:46_1_3_1"><i><font color="Green" title="E46">A55</font></i></a>, <a class="txt" href="frechet.html#E5:46_1_3_1"><i><font color="Green" title="E47">A56</font></i></a>, <a class="txt" href="frechet.html#E3:46_1_3_1_3"><i><font color="Green" title="E48">A59</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:46_1_3_1_3"/>
<font color="Maroon" title="c6">O</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="txt" href="frechet.html#E4:46_1_3_1_3"><i><font color="Green" title="E49">A60</font></i></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">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" title="E48">A61</font></i>: 
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c5">F</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="ref" href="tops_2.html#T26" title="TOPS_2:th.26">TOPS_2:26</a></i>;<br/>
<a NAME="E10:46_1_3_1"/><i><font color="Green" title="E49">A62</font></i>: 
 <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c5">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" title="c6">k</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:46_1_3_1_4"/><i><font color="Green" title="E50">A63</font></i>: 
<font color="Maroon" title="c6">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 ;<br/>

<a NAME="E2:46_1_3_1_4"/><b>then </b><i><font color="Green" title="E51">A64</font></i>: 
<font color="Maroon" title="c6">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font>
 
<b>by </b><i><a class="txt" href="frechet.html#E5:46_1_3_1"><i><font color="Green" title="E47">A56</font></i></a></i>;<br/>
<a NAME="E3:46_1_3_1_4"/>
<font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">g</font>
 
<b>by </b><i><a class="txt" href="frechet.html#E4:46_1_3_1"><i><font color="Green" title="E46">A55</font></i></a>, <a class="txt" href="frechet.html#E1:46_1_3_1_4"><i><font color="Green" title="E50">A63</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E4:46_1_3_1_4"/>
<font color="Maroon" title="c6">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c5">F</font>
 <b>by </b><i><a class="txt" href="frechet.html#E2:46_1_3_1_4"><i><font color="Green" title="E51">A64</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">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" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a>, <a class="ref" href="xboole_1.html#T33" title="XBOOLE_1:th.33">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" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T9" title="XBOOLE_1:th.9">XBOOLE_1:9</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c6">B</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="frechet.html#K2" title="FRECHET:func.2">REAL?</a>  <b>by </b><i><a class="ref" href="frechet.html#D7" target="_self" title="FRECHET:def.7">Def7</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">B</font> = <font color="Maroon" title="c6">B</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="frechet.html#K2" title="FRECHET:func.2">REAL?</a>  ;<br/>
<a NAME="E15:46_1_3_1"/>
( <font color="Maroon" title="c7">B</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp;  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font> )
 
<b>by </b><i><a class="txt" href="frechet.html#E9:46_1_3_1"><i><font color="Green" title="E48">A61</font></i></a>, <a class="txt" href="frechet.html#E10:46_1_3_1"><i><font color="Green" title="E49">A62</font></i></a>, <a class="ref" href="frechet.html#T31" target="_self" title="FRECHET:th.31">Th31</a></i>;<br/>
<a NAME="E16:46_1_3_1"/><b>then </b><i><font color="Green" title="E50">A65</font></i>: 
<font color="Maroon" title="c1">A</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c7">B</font>
 
<b>by </b><i><a class="txt" href="frechet.html#E1:46"><i><font color="Green" title="E41">A1</font></i></a>, <a class="txt" href="frechet.html#E1:46_1_3"><i><font color="Green" title="E43">A28</font></i></a>, <a class="ref" href="pre_topc.html#D13" title="PRE_TOPC:def.13">PRE_TOPC:def 13</a></i>;<br/>
<a NAME="E17:46_1_3_1"/>
<font color="Maroon" title="c7">B</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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" title="E51">A66</font></i>: 
<font color="Maroon" title="c7">B</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/>

<b>consider </b><font color="Maroon" title="c8">z</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c7">B</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c1">A</font>;<br/>
<a NAME="E3:46_1_3_1_5"/><i><font color="Green" title="E52">A67</font></i>: 
( <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font> &amp; <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> )
 
<b>by </b><i><a class="txt" href="frechet.html#E1:46_1_3_1_5"><i><font color="Green" title="E51">A66</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">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" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  or <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 </span><b>by </b><i><a class="txt" href="frechet.html#E3:46_1_3_1_5"><i><font color="Green" title="E52">A67</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">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" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 ;<br/><div class="add"><a NAME="E2:46_1_3_1_5_1_1"/><b>then </b>
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c5">F</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><b>then consider </b><font color="Maroon" title="c9">C</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:46_1_3_1_5_1_1"/><i><font color="Green" title="E53">A68</font></i>: 
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</font>
 <b>and </b><br/><a NAME="E5:46_1_3_1_5_1_1"/><i><font color="Green" title="E54">A69</font></i>: 
<font color="Maroon" title="c9">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">F</font>
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon" title="c10">l</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E7:46_1_3_1_5_1_1"/><i><font color="Green" title="E55">A70</font></i>: 
<font color="Maroon" title="c10">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">g</font>
 <b>and </b><br/><a NAME="E8:46_1_3_1_5_1_1"/><i><font color="Green" title="E56">A71</font></i>: 
<font color="Maroon" title="c9">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">l</font>
 <b>by </b><i><a class="txt" href="frechet.html#E5:46_1_3_1_5_1_1"><i><font color="Green" title="E54">A69</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">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" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">l</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="frechet.html#E4:46_1_3_1"><i><font color="Green" title="E46">A55</font></i></a>, <a class="txt" href="frechet.html#E5:46_1_3_1"><i><font color="Green" title="E47">A56</font></i></a>, <a class="txt" href="frechet.html#E7:46_1_3_1_5_1_1"><i><font color="Green" title="E55">A70</font></i></a></i>;<br/><b>hence </b><a NAME="E10:46_1_3_1_5_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="frechet.html#E3:46_1_3_1_5"><i><font color="Green" title="E52">A67</font></i></a>, <a class="txt" href="frechet.html#E4:46_1_3_1_5_1_1"><i><font color="Green" title="E53">A68</font></i></a>, <a class="txt" href="frechet.html#E8:46_1_3_1_5_1_1"><i><font color="Green" title="E56">A71</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">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" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/><div class="add"><a NAME="E2:46_1_3_1_5_1_2"/><b>then </b>
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">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="txt" href="frechet.html#E1:46_1_3"><i><font color="Green" title="E43">A28</font></i></a>, <a class="txt" href="frechet.html#E1:46_1_3_1_5"><i><font color="Green" title="E51">A66</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">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="txt" href="frechet.html#E16:46_1_3_1"><i><font color="Green" title="E50">A65</font></i></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon" title="c4">k</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> <b> such that </b><br/><a NAME="E5:46_1_3"/><i><font color="Green" title="E44">A72</font></i>: 
<font color="Maroon" title="c4">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>and </b><br/><a NAME="E6:46_1_3"/><i><font color="Green" title="E45">A73</font></i>: 
 ex <font color="Olive" title="b1">S'</font> being  <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  st <br/>(  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b1">S'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">A'</font> &amp; <font color="Olive" title="b1">S'</font> <a href="frechet.html#R1" title="FRECHET:pred.1">is_convergent_to</a> <font color="Maroon" title="c4">k</font> )
 ;<br/><b>consider </b><font color="Maroon" title="c5">S'</font> being   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> <b> such that </b><br/><a NAME="E8:46_1_3"/><i><font color="Green" title="E46">A74</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">S'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">A'</font>
 <b>and </b><br/><a NAME="E9:46_1_3"/><i><font color="Green" title="E47">A75</font></i>: 
<font color="Maroon" title="c5">S'</font> <a href="frechet.html#R1" title="FRECHET:pred.1">is_convergent_to</a> <font color="Maroon" title="c4">k</font>
 <b>by </b><i><a class="txt" href="frechet.html#E6:46_1_3"><i><font color="Green" title="E45">A73</font></i></a></i>;<br/><a NAME="E10:46_1_3"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">S'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="frechet.html#K2" title="FRECHET:func.2">REAL?</a> 
 <b>by </b><i><a class="txt" href="frechet.html#E8:46_1_3"><i><font color="Green" title="E46">A74</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c6">S</font> = <font color="Maroon" title="c5">S'</font> as   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <a href="frechet.html#K2" title="FRECHET:func.2">REAL?</a>  <b>by </b><i><a class="ref" href="frechet.html#T2" target="_self" title="FRECHET:th.2">Th2</a></i>;<br/><b>take </b>
<font color="Maroon" title="c6">S</font>
;<br/><b>thus </b><a NAME="E12:46_1_3"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">A</font>
 <b>by </b><i><a class="txt" href="frechet.html#E8:46_1_3"><i><font color="Green" title="E46">A74</font></i></a></i>;<br/><a NAME="E13:46_1_3"/>
<font color="Maroon" title="c6">S</font> <a href="frechet.html#R1" title="FRECHET:pred.1">is_convergent_to</a> <font color="Maroon" title="c2">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" title="c7">U1</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="frechet.html#K2" title="FRECHET:func.2">REAL?</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="frechet.html#D2" target="_self" title="FRECHET:def.2">FRECHET:def 2</a><br/>

<b>assume </b><a NAME="E1:46_1_3_2"/>
( <font color="Maroon" title="c7">U1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">U1</font> )
 ;<br/>

<b>then consider </b><font color="Maroon" title="c8">O</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> <b> such that </b><br/><a NAME="E3:46_1_3_2"/><i><font color="Green" title="E48">A76</font></i>: 
<font color="Maroon" title="c8">O</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>and </b><br/><a NAME="E4:46_1_3_2"/><i><font color="Green" title="E49">A77</font></i>: 
 <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">O</font>
 <b>and </b><br/><a NAME="E5:46_1_3_2"/><i><font color="Green" title="E50">A78</font></i>: 
<font color="Maroon" title="c7">U1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">O</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="frechet.html#E1:46_1_3"><i><font color="Green" title="E43">A28</font></i></a>, <a class="ref" href="frechet.html#T31" target="_self" title="FRECHET:th.31">Th31</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E7:46_1_3_2"/><i><font color="Green" title="E51">A79</font></i>: 
for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c9">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> holds <br/><font color="Maroon" title="c5">S'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">O</font>
 <b>by </b><i><a class="txt" href="frechet.html#E5:46_1_3"><i><font color="Green" title="E44">A72</font></i></a>, <a class="txt" href="frechet.html#E9:46_1_3"><i><font color="Green" title="E47">A75</font></i></a>, <a class="txt" href="frechet.html#E3:46_1_3_2"><i><font color="Green" title="E48">A76</font></i></a>, <a class="txt" href="frechet.html#E4:46_1_3_2"><i><font color="Green" title="E49">A77</font></i></a>, <a class="ref" href="frechet.html#D2" target="_self" title="FRECHET:def.2">Def2</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c9">n</font>
;<br/>

<b>thus </b><a NAME="E8:46_1_3_2"/>
for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c9">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> holds <br/><font color="Maroon" title="c6">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">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" title="c10">m</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:46_1_3_2_1"/>
<font color="Maroon" title="c9">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">m</font>
 ;<br/>

<a NAME="E2:46_1_3_2_1"/><b>then </b><i><font color="Green" title="E52">A80</font></i>: 
<font color="Maroon" title="c5">S'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">O</font>
 
<b>by </b><i><a class="txt" href="frechet.html#E7:46_1_3_2"><i><font color="Green" title="E51">A79</font></i></a></i>;<br/>
<a NAME="E3:46_1_3_2_1"/>
<font color="Maroon" title="c10">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
;<br/>
<a NAME="E4:46_1_3_2_1"/><b>then </b>
<font color="Maroon" title="c10">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">S'</font>
 
<b>by </b><i><a class="ref" href="normsp_1.html#T17" title="NORMSP_1:th.17">NORMSP_1:17</a></i>;<br/>
<a NAME="E5:46_1_3_2_1"/><b>then </b>
<font color="Maroon" title="c5">S'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">S'</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E6:46_1_3_2_1"/><b>then </b>
<font color="Maroon" title="c5">S'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A'</font>
 
<b>by </b><i><a class="txt" href="frechet.html#E8:46_1_3"><i><font color="Green" title="E46">A74</font></i></a></i>;<br/>
<a NAME="E7:46_1_3_2_1"/><b>then </b>
<font color="Maroon" title="c5">S'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="frechet.html#K2" title="FRECHET:func.2">REAL?</a> 
 
;<br/>
<a NAME="E8:46_1_3_2_1"/><b>then </b>
<font color="Maroon" title="c5">S'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="frechet.html#D7" target="_self" title="FRECHET:def.7">Def7</a></i>;<br/>
<a NAME="E9:46_1_3_2_1"/><b>then </b><i><font color="Green" title="E53">A81</font></i>: 
( <font color="Maroon" title="c5">S'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  or <font color="Maroon" title="c5">S'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E10:46_1_3_2_1"/>
<font color="Maroon" title="c5">S'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">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" title="E54">A82</font></i>: 
<font color="Maroon" title="c5">S'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 ;<br/>

<a NAME="E2:46_1_3_2_1_1"/>
<font color="Maroon" title="c5">S'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<b>by </b><i><a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></i>;<br/>
<b>hence </b><a NAME="E3:46_1_3_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="frechet.html#E1:46_1_3_2_1_1"><i><font color="Green" title="E54">A82</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:46_1_3_2_1"/><b>then </b>
not <font color="Maroon" title="c5">S'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="txt" href="frechet.html#E9:46_1_3_2_1"><i><font color="Green" title="E53">A81</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E12:46_1_3_2_1"/><b>then </b>
<font color="Maroon" title="c5">S'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">O</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="txt" href="frechet.html#E2:46_1_3_2_1"><i><font color="Green" title="E52">A80</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="E13:46_1_3_2_1"/>
<font color="Maroon" title="c6">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">U1</font>
 <b>by </b><i><a class="txt" href="frechet.html#E5:46_1_3_2"><i><font color="Green" title="E50">A78</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">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" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="frechet.html#K1" title="FRECHET:func.1">Lim</a> <font color="Maroon" title="c6">S</font>
 <b>by </b><i><a class="ref" href="frechet.html#D4" target="_self" title="FRECHET:def.4">Def4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
