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

<b>let </b><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</font>, <font color="Maroon" title="c3">c</font>, <font color="Maroon" title="c4">d</font>, <font color="Maroon" title="c5">x1</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>let </b><font color="Maroon" title="c6">f</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span>)</span>,<span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span>;<br/><b>let </b><font color="Maroon" title="c7">x</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span>)</span>;<br/><b>let </b><font color="Maroon" title="c8">g</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> , <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/>







<b>assume </b><a NAME="E1:21"/><i><font color="Green" title="E18">A1</font></i>: 
( <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c2">b</font> &amp; <font color="Maroon" title="c3">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">d</font> &amp; <font color="Maroon" title="c6">f</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c7">x</font> &amp; <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</font> &amp; <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">d</font> &amp; <font color="Maroon" title="c6">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp; <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">g</font> &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x1</font> )
 ;<br/>

<a NAME="E2:21"/><b>then </b>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E3:21"/><b>then </b>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<b>by </b><i><a class="txt" href="jordan5a.html#E1:21"><i><font color="Green" title="E18">A1</font></i></a>, <a class="ref" href="topmetr.html#T25" title="TOPMETR:th.25">TOPMETR:25</a></i>;<br/>
<a NAME="E4:21"/><b>then </b><i><font color="Green" title="E19">A2</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">g</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
;<br/>
<a NAME="E5:21"/>
for <font color="Olive" title="b1">c</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   st <font color="Olive" title="b1">c</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="c8">g</font> holds <br/><font color="Maroon" title="c8">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">c</font>
 
;<br/>
<a NAME="E6:21"/><b>then </b><i><font color="Green" title="E20">A3</font></i>: 
<font color="Maroon" title="c8">g</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c8">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<b>by </b><i><a class="txt" href="jordan5a.html#E4:21"><i><font color="Green" title="E19">A2</font></i></a>, <a class="ref" href="partfun2.html#T32" title="PARTFUN2:th.32">PARTFUN2:32</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:21_1"/>
( <font color="Maroon" title="c5">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font> or <font color="Maroon" title="c5">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font> or ( <font color="Maroon" title="c5">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">a</font> &amp; <font color="Maroon" title="c5">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">b</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1"><b>suppose </b></a><a NAME="E1:21_1_1"/><i><font color="Green" title="E21">A4</font></i>: 
<font color="Maroon" title="c5">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font>
 ;<br/><div class="add"><a NAME="E2:21_1_1"/><i><font color="Green" title="E22">A5</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E3:21_1_1"/>
for <font color="Olive" title="b1">N1</font> being   <a href="rcomp_1.html#M1" title="RCOMP_1:mode.1">Neighbourhood</a> of <font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x1</font>  ex <font color="Olive" title="b2">N</font> being   <a href="rcomp_1.html#M1" title="RCOMP_1:mode.1">Neighbourhood</a> of <font color="Maroon" title="c5">x1</font> st <font color="Maroon" title="c8">g</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Olive" title="b2">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">N1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">N1</font> be    <a href="rcomp_1.html#M1" title="RCOMP_1:mode.1">Neighbourhood</a> of <font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x1</font>;<br/>

<b>reconsider </b><font color="Maroon" title="c10">N2</font> = <font color="Maroon" title="c9">N1</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="metric_1.html#K8" title="METRIC_1:func.8">RealSpace</a>  <b>by </b><i><a class="ref" href="metric_1.html#D14" title="METRIC_1:def.14">METRIC_1:def 14</a></i>;<br/>
<a NAME="E2:21_1_1_1"/>
<font color="Maroon" title="c10">N2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">Family_open_set</a> <a href="metric_1.html#K8" title="METRIC_1:func.8">RealSpace</a> 
 
<b>by </b><i><a class="txt" href="jordan5a.html#E8"><i><font color="Green" title="E6">Lm3</font></i></a></i>;<br/>
<a NAME="E3:21_1_1_1"/><b>then </b><i><font color="Green" title="E23">A6</font></i>: 
<font color="Maroon" title="c10">N2</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 <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#T16" title="TOPMETR:th.16">TOPMETR:16</a></i>;<br/>
<a NAME="E4:21_1_1_1"/>
<font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">a</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 <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan5a.html#E1:21"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="jordan5a.html#E19"><i><font color="Green" title="E15">Lm6</font></i></a></i>;<br/>
<a NAME="E5:21_1_1_1"/><b>then </b><i><font color="Green" title="E24">A7</font></i>: 
( <font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">N1</font> &amp; <font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> )
 
<b>by </b><i><a class="txt" href="jordan5a.html#E1:21"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="jordan5a.html#E1:21_1_1"><i><font color="Green" title="E21">A4</font></i></a>, <a class="ref" href="rcomp_1.html#T37" title="RCOMP_1:th.37">RCOMP_1:37</a>, <a class="ref" href="topmetr.html#T25" title="TOPMETR:th.25">TOPMETR:25</a></i>;<br/>
<b>set </b><font color="Maroon" title="c11">NN1</font> = <font color="Maroon" title="c9">N1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>;<br/>
<a NAME="E6:21_1_1_1"/><i><font color="Green" title="E25">A8</font></i>: 
<font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">N1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<b>by </b><i><a class="txt" href="jordan5a.html#E5:21_1_1_1"><i><font color="Green" title="E24">A7</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="E7:21_1_1_1"/><i><font color="Green" title="E26">A9</font></i>: 
<font color="Maroon" title="c9">N1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">N1</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan5a.html#E1:21"><i><font color="Green" title="E18">A1</font></i></a>, <a class="ref" href="topmetr.html#T25" title="TOPMETR:th.25">TOPMETR:25</a></i>;<br/>
<a NAME="E8:21_1_1_1"/><i><font color="Green" title="E27">A10</font></i>: 
<font color="Maroon" title="c9">N1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">N1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan5a.html#E1:21"><i><font color="Green" title="E18">A1</font></i></a>, <a class="ref" href="topmetr.html#T25" title="TOPMETR:th.25">TOPMETR:25</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">NN</font> = <font color="Maroon" title="c9">N1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span> <b>by </b><i><a class="txt" href="jordan5a.html#E7:21_1_1_1"><i><font color="Green" title="E26">A9</font></i></a>, <a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E10:21_1_1_1"/>
<font color="Maroon" title="c12">NN</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 <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan5a.html#E3:21_1_1_1"><i><font color="Green" title="E23">A6</font></i></a>, <a class="txt" href="jordan5a.html#E8:21_1_1_1"><i><font color="Green" title="E27">A10</font></i></a>, <a class="ref" href="pre_topc.html#D9" title="PRE_TOPC:def.9">PRE_TOPC:def 9</a>, <a class="ref" href="topmetr.html#D7" title="TOPMETR:def.7">TOPMETR:def 7</a></i>;<br/>
<a NAME="E11:21_1_1_1"/><b>then </b><i><font color="Green" title="E28">A11</font></i>: 
<font color="Maroon" title="c12">NN</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">f0</font> = <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">a</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span> <b>by </b><i><a class="txt" href="jordan5a.html#E1:21"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="jordan5a.html#E19"><i><font color="Green" title="E15">Lm6</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">N1'</font> = <font color="Maroon" title="c12">NN</font> as    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c13">f0</font> <b>by </b><i><a class="txt" href="jordan5a.html#E1:21"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="jordan5a.html#E1:21_1_1"><i><font color="Green" title="E21">A4</font></i></a>, <a class="txt" href="jordan5a.html#E6:21_1_1_1"><i><font color="Green" title="E25">A8</font></i></a>, <a class="txt" href="jordan5a.html#E11:21_1_1_1"><i><font color="Green" title="E28">A11</font></i></a>, <a class="ref" href="connsp_2.html#T5" title="CONNSP_2:th.5">CONNSP_2:5</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c15">H</font> being    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c7">x</font><b> such that </b><br/><a NAME="E15:21_1_1_1"/><i><font color="Green" title="E29">A12</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c15">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c14">N1'</font>
 <b>by </b><i><a class="txt" href="jordan5a.html#E1:21"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="jordan5a.html#E1:21_1_1"><i><font color="Green" title="E21">A4</font></i></a>, <a class="ref" href="tmap_1.html#D2" title="TMAP_1:def.2">TMAP_1:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c16">H1</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span>)</span><b> such that </b><br/><a NAME="E17:21_1_1_1"/><i><font color="Green" title="E30">A13</font></i>: 
( <font color="Maroon" title="c16">H1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c16">H1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c15">H</font> &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">H1</font> )
 <b>by </b><i><a class="ref" href="connsp_2.html#T8" title="CONNSP_2:th.8">CONNSP_2:8</a></i>;<br/>
<a NAME="E18:21_1_1_1"/>
<font color="Maroon" title="c16">H1</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 <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan5a.html#E17:21_1_1_1"><i><font color="Green" title="E30">A13</font></i></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c17">Q</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="E20:21_1_1_1"/><i><font color="Green" title="E31">A14</font></i>: 
( <font color="Maroon" title="c17">Q</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>  &amp; <font color="Maroon" title="c16">H1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">Q</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="pre_topc.html#D9" title="PRE_TOPC:def.9">PRE_TOPC:def 9</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c18">Q'</font> = <font color="Maroon" title="c17">Q</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</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>, <a class="ref" href="topmetr.html#D7" title="TOPMETR:def.7">TOPMETR:def 7</a></i>;<br/>
<a NAME="E22:21_1_1_1"/><i><font color="Green" title="E32">A15</font></i>: 
<font color="Maroon" title="c18">Q'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">Family_open_set</a> <a href="metric_1.html#K8" title="METRIC_1:func.8">RealSpace</a> 
 
<b>by </b><i><a class="txt" href="jordan5a.html#E20:21_1_1_1"><i><font color="Green" title="E31">A14</font></i></a>, <a class="ref" href="topmetr.html#T16" title="TOPMETR:th.16">TOPMETR:16</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="c19">Q1</font> = <font color="Maroon" title="c18">Q'</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <b>by </b><i><a class="ref" href="metric_1.html#D14" title="METRIC_1:def.14">METRIC_1:def 14</a></i>;<br/>
<a NAME="E24:21_1_1_1"/><i><font color="Green" title="E33">A16</font></i>: 
<font color="Maroon" title="c19">Q1</font> is <a href="rcomp_1.html#V3" title="RCOMP_1:attr.3">open</a>
 
<b>by </b><i><a class="txt" href="jordan5a.html#E22:21_1_1_1"><i><font color="Green" title="E32">A15</font></i></a>, <a class="txt" href="jordan5a.html#E9"><i><font color="Green" title="E7">Lm4</font></i></a></i>;<br/>
<a NAME="E25:21_1_1_1"/>
<font color="Maroon" title="c5">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">Q1</font>
 
<b>by </b><i><a class="txt" href="jordan5a.html#E1:21"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="jordan5a.html#E17:21_1_1_1"><i><font color="Green" title="E30">A13</font></i></a>, <a class="txt" href="jordan5a.html#E20:21_1_1_1"><i><font color="Green" title="E31">A14</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c20">N</font> being    <a href="rcomp_1.html#M1" title="RCOMP_1:mode.1">Neighbourhood</a> of <font color="Maroon" title="c5">x1</font><b> such that </b><br/><a NAME="E27:21_1_1_1"/><i><font color="Green" title="E34">A17</font></i>: 
<font color="Maroon" title="c20">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c19">Q1</font>
 <b>by </b><i><a class="txt" href="jordan5a.html#E24:21_1_1_1"><i><font color="Green" title="E33">A16</font></i></a>, <a class="ref" href="rcomp_1.html#T39" title="RCOMP_1:th.39">RCOMP_1:39</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c20">N</font>
;<br/>

<a NAME="E28:21_1_1_1"/>
<font color="Maroon" title="c8">g</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c20">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">N1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c21">aa</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="E1:21_1_1_1_1"/><i><font color="Green" title="E35">A18</font></i>: 
<font color="Maroon" title="c21">aa</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">g</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c20">N</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c22">a'</font> = <font color="Maroon" title="c21">aa</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  ;<br/>
<b>consider </b><font color="Maroon" title="c23">cc</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <b> such that </b><br/><a NAME="E4:21_1_1_1_1"/><i><font color="Green" title="E36">A19</font></i>: 
( <font color="Maroon" title="c23">cc</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="c8">g</font> &amp; <font color="Maroon" title="c23">cc</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">N</font> &amp; <font color="Maroon" title="c22">a'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">cc</font> )
 <b>by </b><i><a class="txt" href="jordan5a.html#E1:21_1_1_1_1"><i><font color="Green" title="E35">A18</font></i></a>, <a class="ref" href="partfun2.html#T78" title="PARTFUN2:th.78">PARTFUN2:78</a></i>;<br/>
<a NAME="E5:21_1_1_1_1"/>
<font color="Maroon" title="c23">cc</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 <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan5a.html#E1:21"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="jordan5a.html#E4:21_1_1_1_1"><i><font color="Green" title="E36">A19</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E6:21_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c23">cc</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">H1</font>
 
<b>by </b><i><a class="txt" href="jordan5a.html#E20:21_1_1_1"><i><font color="Green" title="E31">A14</font></i></a>, <a class="txt" href="jordan5a.html#E27:21_1_1_1"><i><font color="Green" title="E34">A17</font></i></a>, <a class="txt" href="jordan5a.html#E4:21_1_1_1_1"><i><font color="Green" title="E36">A19</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="E7:21_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">cc</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c15">H</font>
 
<b>by </b><i><a class="txt" href="jordan5a.html#E1:21"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="jordan5a.html#E17:21_1_1_1"><i><font color="Green" title="E30">A13</font></i></a>, <a class="ref" href="funct_2.html#T43" title="FUNCT_2:th.43">FUNCT_2:43</a></i>;<br/>
<b>hence </b><a NAME="E8:21_1_1_1_1"/>
<font color="Maroon" title="c21">aa</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">N1</font>
 <b>by </b><i><a class="txt" href="jordan5a.html#E15:21_1_1_1"><i><font color="Green" title="E29">A12</font></i></a>, <a class="txt" href="jordan5a.html#E4:21_1_1_1_1"><i><font color="Green" title="E36">A19</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="E29:21_1_1_1"/>
<font color="Maroon" title="c8">g</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c20">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">N1</font>
 ;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E4:21_1_1"/>
<font color="Maroon" title="c8">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="fcont_1.html#R1" title="FCONT_1:pred.1">is_continuous_in</a> <font color="Maroon" title="c5">x1</font>
 <b>by </b><i><a class="txt" href="jordan5a.html#E1:21"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="jordan5a.html#E6:21"><i><font color="Green" title="E20">A3</font></i></a>, <a class="txt" href="jordan5a.html#E2:21_1_1"><i><font color="Green" title="E22">A5</font></i></a>, <a class="ref" href="fcont_1.html#T5" title="FCONT_1:th.5">FCONT_1:5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_2"><b>suppose </b></a><a NAME="E1:21_1_2"/><i><font color="Green" title="E21">A20</font></i>: 
<font color="Maroon" title="c5">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font>
 ;<br/><div class="add"><a NAME="E2:21_1_2"/><i><font color="Green" title="E22">A21</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E3:21_1_2"/>
for <font color="Olive" title="b1">N1</font> being   <a href="rcomp_1.html#M1" title="RCOMP_1:mode.1">Neighbourhood</a> of <font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x1</font>  ex <font color="Olive" title="b2">N</font> being   <a href="rcomp_1.html#M1" title="RCOMP_1:mode.1">Neighbourhood</a> of <font color="Maroon" title="c5">x1</font> st <font color="Maroon" title="c8">g</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Olive" title="b2">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">N1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">N1</font> be    <a href="rcomp_1.html#M1" title="RCOMP_1:mode.1">Neighbourhood</a> of <font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x1</font>;<br/>

<b>reconsider </b><font color="Maroon" title="c10">N2</font> = <font color="Maroon" title="c9">N1</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="metric_1.html#K8" title="METRIC_1:func.8">RealSpace</a>  <b>by </b><i><a class="ref" href="metric_1.html#D14" title="METRIC_1:def.14">METRIC_1:def 14</a></i>;<br/>
<a NAME="E2:21_1_2_1"/>
<font color="Maroon" title="c10">N2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">Family_open_set</a> <a href="metric_1.html#K8" title="METRIC_1:func.8">RealSpace</a> 
 
<b>by </b><i><a class="txt" href="jordan5a.html#E8"><i><font color="Green" title="E6">Lm3</font></i></a></i>;<br/>
<a NAME="E3:21_1_2_1"/><b>then </b><i><font color="Green" title="E23">A22</font></i>: 
<font color="Maroon" title="c10">N2</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 <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#T16" title="TOPMETR:th.16">TOPMETR:16</a></i>;<br/>
<a NAME="E4:21_1_2_1"/><i><font color="Green" title="E24">A23</font></i>: 
( <font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">N1</font> &amp; <font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="jordan5a.html#E1:21"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="jordan5a.html#E1:21_1_2"><i><font color="Green" title="E21">A20</font></i></a>, <a class="txt" href="jordan5a.html#E19"><i><font color="Green" title="E15">Lm6</font></i></a>, <a class="ref" href="rcomp_1.html#T37" title="RCOMP_1:th.37">RCOMP_1:37</a></i>;<br/>
<b>set </b><font color="Maroon" title="c11">NN1</font> = <font color="Maroon" title="c9">N1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span></span>)</span>;<br/>
<a NAME="E5:21_1_2_1"/><i><font color="Green" title="E25">A24</font></i>: 
<font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">N1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan5a.html#E4:21_1_2_1"><i><font color="Green" title="E24">A23</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">NN</font> = <font color="Maroon" title="c9">N1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span></span>)</span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span> ;<br/>
<a NAME="E7:21_1_2_1"/>
<font color="Maroon" title="c12">NN</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 <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan5a.html#E3:21_1_2_1"><i><font color="Green" title="E23">A22</font></i></a>, <a class="ref" href="pre_topc.html#D9" title="PRE_TOPC:def.9">PRE_TOPC:def 9</a>, <a class="ref" href="topmetr.html#D7" title="TOPMETR:def.7">TOPMETR:def 7</a></i>;<br/>
<a NAME="E8:21_1_2_1"/><b>then </b><i><font color="Green" title="E26">A25</font></i>: 
<font color="Maroon" title="c12">NN</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">f0</font> = <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">b</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span> <b>by </b><i><a class="txt" href="jordan5a.html#E1:21"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="jordan5a.html#E19"><i><font color="Green" title="E15">Lm6</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">N1'</font> = <font color="Maroon" title="c12">NN</font> as    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c13">f0</font> <b>by </b><i><a class="txt" href="jordan5a.html#E1:21"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="jordan5a.html#E1:21_1_2"><i><font color="Green" title="E21">A20</font></i></a>, <a class="txt" href="jordan5a.html#E5:21_1_2_1"><i><font color="Green" title="E25">A24</font></i></a>, <a class="txt" href="jordan5a.html#E8:21_1_2_1"><i><font color="Green" title="E26">A25</font></i></a>, <a class="ref" href="connsp_2.html#T5" title="CONNSP_2:th.5">CONNSP_2:5</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c15">H</font> being    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c7">x</font><b> such that </b><br/><a NAME="E12:21_1_2_1"/><i><font color="Green" title="E27">A26</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c15">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c14">N1'</font>
 <b>by </b><i><a class="txt" href="jordan5a.html#E1:21"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="jordan5a.html#E1:21_1_2"><i><font color="Green" title="E21">A20</font></i></a>, <a class="ref" href="tmap_1.html#D2" title="TMAP_1:def.2">TMAP_1:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c16">H1</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span>)</span><b> such that </b><br/><a NAME="E14:21_1_2_1"/><i><font color="Green" title="E28">A27</font></i>: 
( <font color="Maroon" title="c16">H1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c16">H1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c15">H</font> &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">H1</font> )
 <b>by </b><i><a class="ref" href="connsp_2.html#T8" title="CONNSP_2:th.8">CONNSP_2:8</a></i>;<br/>
<a NAME="E15:21_1_2_1"/>
<font color="Maroon" title="c16">H1</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 <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan5a.html#E14:21_1_2_1"><i><font color="Green" title="E28">A27</font></i></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c17">Q</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="E17:21_1_2_1"/><i><font color="Green" title="E29">A28</font></i>: 
( <font color="Maroon" title="c17">Q</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>  &amp; <font color="Maroon" title="c16">H1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">Q</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="pre_topc.html#D9" title="PRE_TOPC:def.9">PRE_TOPC:def 9</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c18">Q'</font> = <font color="Maroon" title="c17">Q</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</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>, <a class="ref" href="topmetr.html#D7" title="TOPMETR:def.7">TOPMETR:def 7</a></i>;<br/>
<a NAME="E19:21_1_2_1"/><i><font color="Green" title="E30">A29</font></i>: 
<font color="Maroon" title="c18">Q'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">Family_open_set</a> <a href="metric_1.html#K8" title="METRIC_1:func.8">RealSpace</a> 
 
<b>by </b><i><a class="txt" href="jordan5a.html#E17:21_1_2_1"><i><font color="Green" title="E29">A28</font></i></a>, <a class="ref" href="topmetr.html#T16" title="TOPMETR:th.16">TOPMETR:16</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="c19">Q1</font> = <font color="Maroon" title="c18">Q'</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <b>by </b><i><a class="ref" href="metric_1.html#D14" title="METRIC_1:def.14">METRIC_1:def 14</a></i>;<br/>
<a NAME="E21:21_1_2_1"/><i><font color="Green" title="E31">A30</font></i>: 
<font color="Maroon" title="c19">Q1</font> is <a href="rcomp_1.html#V3" title="RCOMP_1:attr.3">open</a>
 
<b>by </b><i><a class="txt" href="jordan5a.html#E19:21_1_2_1"><i><font color="Green" title="E30">A29</font></i></a>, <a class="txt" href="jordan5a.html#E9"><i><font color="Green" title="E7">Lm4</font></i></a></i>;<br/>
<a NAME="E22:21_1_2_1"/>
<font color="Maroon" title="c5">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">Q1</font>
 
<b>by </b><i><a class="txt" href="jordan5a.html#E1:21"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="jordan5a.html#E14:21_1_2_1"><i><font color="Green" title="E28">A27</font></i></a>, <a class="txt" href="jordan5a.html#E17:21_1_2_1"><i><font color="Green" title="E29">A28</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c20">N</font> being    <a href="rcomp_1.html#M1" title="RCOMP_1:mode.1">Neighbourhood</a> of <font color="Maroon" title="c5">x1</font><b> such that </b><br/><a NAME="E24:21_1_2_1"/><i><font color="Green" title="E32">A31</font></i>: 
<font color="Maroon" title="c20">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c19">Q1</font>
 <b>by </b><i><a class="txt" href="jordan5a.html#E21:21_1_2_1"><i><font color="Green" title="E31">A30</font></i></a>, <a class="ref" href="rcomp_1.html#T39" title="RCOMP_1:th.39">RCOMP_1:39</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c20">N</font>
;<br/>

<a NAME="E25:21_1_2_1"/>
<font color="Maroon" title="c8">g</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c20">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">N1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c21">aa</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="E1:21_1_2_1_1"/><i><font color="Green" title="E33">A32</font></i>: 
<font color="Maroon" title="c21">aa</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">g</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c20">N</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c22">a'</font> = <font color="Maroon" title="c21">aa</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  ;<br/>
<b>consider </b><font color="Maroon" title="c23">cc</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <b> such that </b><br/><a NAME="E4:21_1_2_1_1"/><i><font color="Green" title="E34">A33</font></i>: 
( <font color="Maroon" title="c23">cc</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="c8">g</font> &amp; <font color="Maroon" title="c23">cc</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">N</font> &amp; <font color="Maroon" title="c22">a'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">cc</font> )
 <b>by </b><i><a class="txt" href="jordan5a.html#E1:21_1_2_1_1"><i><font color="Green" title="E33">A32</font></i></a>, <a class="ref" href="partfun2.html#T78" title="PARTFUN2:th.78">PARTFUN2:78</a></i>;<br/>
<a NAME="E5:21_1_2_1_1"/>
<font color="Maroon" title="c23">cc</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 <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan5a.html#E1:21"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="jordan5a.html#E4:21_1_2_1_1"><i><font color="Green" title="E34">A33</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E6:21_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c23">cc</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">H1</font>
 
<b>by </b><i><a class="txt" href="jordan5a.html#E17:21_1_2_1"><i><font color="Green" title="E29">A28</font></i></a>, <a class="txt" href="jordan5a.html#E24:21_1_2_1"><i><font color="Green" title="E32">A31</font></i></a>, <a class="txt" href="jordan5a.html#E4:21_1_2_1_1"><i><font color="Green" title="E34">A33</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="E7:21_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">cc</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c15">H</font>
 
<b>by </b><i><a class="txt" href="jordan5a.html#E1:21"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="jordan5a.html#E14:21_1_2_1"><i><font color="Green" title="E28">A27</font></i></a>, <a class="ref" href="funct_2.html#T43" title="FUNCT_2:th.43">FUNCT_2:43</a></i>;<br/>
<b>hence </b><a NAME="E8:21_1_2_1_1"/>
<font color="Maroon" title="c21">aa</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">N1</font>
 <b>by </b><i><a class="txt" href="jordan5a.html#E12:21_1_2_1"><i><font color="Green" title="E27">A26</font></i></a>, <a class="txt" href="jordan5a.html#E4:21_1_2_1_1"><i><font color="Green" title="E34">A33</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="E26:21_1_2_1"/>
<font color="Maroon" title="c8">g</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c20">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">N1</font>
 ;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E4:21_1_2"/>
<font color="Maroon" title="c8">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="fcont_1.html#R1" title="FCONT_1:pred.1">is_continuous_in</a> <font color="Maroon" title="c5">x1</font>
 <b>by </b><i><a class="txt" href="jordan5a.html#E1:21"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="jordan5a.html#E6:21"><i><font color="Green" title="E20">A3</font></i></a>, <a class="txt" href="jordan5a.html#E2:21_1_2"><i><font color="Green" title="E22">A21</font></i></a>, <a class="ref" href="fcont_1.html#T5" title="FCONT_1:th.5">FCONT_1:5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_3"><b>suppose </b></a><a NAME="E1:21_1_3"/>
( <font color="Maroon" title="c5">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">a</font> &amp; <font color="Maroon" title="c5">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">b</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:21_1_3"/>
<font color="Maroon" title="c8">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="fcont_1.html#R1" title="FCONT_1:pred.1">is_continuous_in</a> <font color="Maroon" title="c5">x1</font>
 <b>by </b><i><a class="txt" href="jordan5a.html#E1:21"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="jordan5a.html#E6:21"><i><font color="Green" title="E20">A3</font></i></a>, <a class="txt" href="jordan5a.html#E20"><i><font color="Green" title="E16">Lm7</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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