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

<b>let </b><font color="Maroon">a</font> be   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">b</font> be   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">c</font> be   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">d</font> be   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">x1</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span>,<span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span>;<br/><b>let </b><font color="Maroon">g</font> be   <a href="partfun1.html#NM1">PartFunc</a> of  <a href="numbers.html#K1">REAL</a> , <a href="numbers.html#K1">REAL</a> ;<br/>







<b>assume </b><a NAME="E1:21"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">b</font> &amp; <font color="Maroon">c</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">d</font> &amp; <font color="Maroon">f</font> <a href="tmap_1.html#R1">is_continuous_at</a> <font color="Maroon">x</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> &amp; <font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a> &amp; <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> )
 ;<br/>

<a NAME="E2:21"/><b>then </b>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E3:21"/><b>then </b>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i>, <a class="ref" href="topmetr.html#T25">TOPMETR:25</a></i>;<br/>
<a NAME="E4:21"/><b>then </b><i><font color="Green">E38</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K1">.]</a></span>
 
;<br/>
<a NAME="E5:21"/>
for <font color="Olive">c</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>   st <font color="Olive">c</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> holds <br/><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">c</font>
 
;<br/>
<a NAME="E6:21"/><b>then </b><i><font color="Green">E39</font></i>: 
<font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i>, <a class="ref" href="partfun2.html#T32">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">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> or <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> or ( <font color="Maroon">x1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font> &amp; <font color="Maroon">x1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">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">E46</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 ;<br/><div class="add"><a NAME="E2:21_1_1"/><i><font color="Green">E48</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E3:21_1_1"/>
for <font color="Olive">N1</font> being   <a href="rcomp_1.html#M1">Neighbourhood</a> of <font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font>  ex <font color="Olive">N</font> being   <a href="rcomp_1.html#M1">Neighbourhood</a> of <font color="Maroon">x1</font> st <font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> <font color="Olive">N</font> <a href="tarski.html#R1">c=</a> <font color="Olive">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">N1</font> be    <a href="rcomp_1.html#M1">Neighbourhood</a> of <font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font>;<br/>

<b>reconsider </b><font color="Maroon">N2</font> = <font color="Maroon">N1</font> as   <a href="struct_0.html#NM2">Subset</a> of <a href="metric_1.html#K8">RealSpace</a>  <b>by </b><i><a class="ref" href="metric_1.html#D14">METRIC_1:def 14</a></i>;<br/>
<a NAME="E2:21_1_1_1"/>
<font color="Maroon">N2</font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K3">Family_open_set</a> <a href="metric_1.html#K8">RealSpace</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E3:21_1_1_1"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">N2</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <a href="metric_1.html#K8">RealSpace</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="topmetr.html#T16">TOPMETR:16</a></i>;<br/>
<a NAME="E4:21_1_1_1"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:21_1_1_1"/><b>then </b><i><font color="Green">E51</font></i>: 
( <font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N1</font> &amp; <font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font></span><a href="rcomp_1.html#K1">.]</a></span> )
 
<b>by </b><i>, <a class="ref" href="jordan5a.html#T1" target="_self">Th1</a>, <a class="ref" href="rcomp_1.html#T37">RCOMP_1:37</a>, <a class="ref" href="topmetr.html#T25">TOPMETR:25</a></i>;<br/>
<b>set </b><font color="Maroon">NN1</font> = <font color="Maroon">N1</font> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font></span><a href="rcomp_1.html#K1">.]</a></span>;<br/>
<a NAME="E6:21_1_1_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N1</font> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E7:21_1_1_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">N1</font> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font></span><a href="rcomp_1.html#K1">.]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">N1</font> <a href="subset_1.html#K8">/\</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="topmetr.html#T25">TOPMETR:25</a></i>;<br/>
<a NAME="E8:21_1_1_1"/><b>then </b><i><font color="Green">E55</font></i>: 
<font color="Maroon">N1</font> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font></span><a href="rcomp_1.html#K1">.]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">N1</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span></span>)</span>
 
;<br/>
<b>reconsider </b><font color="Maroon">NN</font> = <font color="Maroon">N1</font> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font></span><a href="rcomp_1.html#K1">.]</a></span> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span> <b>by </b><i>, <a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E10:21_1_1_1"/>
<font color="Maroon">NN</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span>
 
<b>by </b><i>, <a class="txt" href="jordan5a.html#E1:21"><i><font color="Green">E28</font></i></a>, <a class="ref" href="pre_topc.html#D9">PRE_TOPC:def 9</a>, <a class="ref" href="topmetr.html#D7">TOPMETR:def 7</a></i>;<br/>
<a NAME="E11:21_1_1_1"/><b>then </b><i><font color="Green">E56</font></i>: 
<font color="Maroon">NN</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">f0</font> = <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span> <b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">N1'</font> = <font color="Maroon">NN</font> as    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">f0</font> <b>by </b><i>, <a class="ref" href="jordan5a.html#T1" target="_self">Th1</a>, , , <a class="ref" href="connsp_2.html#T5">CONNSP_2:5</a></i>;<br/>
<b>consider </b><font color="Maroon">H</font> being    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">x</font><b> such that </b><br/><a NAME="E15:21_1_1_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">H</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">N1'</font>
 <b>by </b><i>, <a class="ref" href="jordan5a.html#T1" target="_self">Th1</a>, <a class="ref" href="tmap_1.html#D2">TMAP_1:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon">H1</font> being   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span><b> such that </b><br/><a NAME="E17:21_1_1_1"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">H1</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">H1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">H</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H1</font> )
 <b>by </b><i><a class="ref" href="connsp_2.html#T8">CONNSP_2:8</a></i>;<br/>
<a NAME="E18:21_1_1_1"/>
<font color="Maroon">H1</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span>
 
<b>by </b><i><a class="ref" href="jordan5a.html#T2" target="_self">Th2</a>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon">Q</font> being   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a> <b> such that </b><br/><a NAME="E20:21_1_1_1"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">Q</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <a href="topmetr.html#K3">R^1</a>  &amp; <font color="Maroon">H1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Q</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="pre_topc.html#D9">PRE_TOPC:def 9</a></i>;<br/>
<b>reconsider </b><font color="Maroon">Q'</font> = <font color="Maroon">Q</font> as   <a href="struct_0.html#NM2">Subset</a> of <a href="metric_1.html#K8">RealSpace</a>  <b>by </b><i><a class="ref" href="topmetr.html#T16">TOPMETR:16</a>, <a class="ref" href="topmetr.html#D7">TOPMETR:def 7</a></i>;<br/>
<a NAME="E22:21_1_1_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">Q'</font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K3">Family_open_set</a> <a href="metric_1.html#K8">RealSpace</a> 
 
<b>by </b><i>, <a class="ref" href="topmetr.html#T16">TOPMETR:16</a>, <a class="ref" href="topmetr.html#D7">TOPMETR:def 7</a></i>;<br/>
<b>reconsider </b><font color="Maroon">Q1</font> = <font color="Maroon">Q'</font> as   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="metric_1.html#D14">METRIC_1:def 14</a></i>;<br/>
<a NAME="E24:21_1_1_1"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">Q1</font> is <a href="rcomp_1.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="jordan5a.html#T3" target="_self">Th3</a>, </i>;<br/>
<a NAME="E25:21_1_1_1"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q1</font>
 
<b>by </b><i>, <a class="ref" href="jordan5a.html#T2" target="_self">Th2</a>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">N</font> being    <a href="rcomp_1.html#M1">Neighbourhood</a> of <font color="Maroon">x1</font><b> such that </b><br/><a NAME="E27:21_1_1_1"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">N</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Q1</font>
 <b>by </b><i>, <a class="ref" href="rcomp_1.html#T39">RCOMP_1:39</a></i>;<br/>
<b>take </b>
<font color="Maroon">N</font>
;<br/>

<a NAME="E28:21_1_1_1"/>
<font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">N</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">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">aa</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:21_1_1_1_1"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">aa</font> <a href="hidden.html#R2">in</a> <font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">N</font>
 ;<br/>

<a NAME="E2:21_1_1_1_1"/>
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">N</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span>
 
;<br/>
<a NAME="E3:21_1_1_1_1"/><b>then </b>
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">N</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i>, <a class="ref" href="topmetr.html#T25">TOPMETR:25</a></i>;<br/>
<a NAME="E4:21_1_1_1_1"/><b>then </b>
<font color="Maroon">aa</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i>, </i>;<br/>
<b>then reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">aa</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  ;<br/>
<b>consider </b><font color="Maroon">cc</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E7:21_1_1_1_1"/><i><font color="Green">E74</font></i>: 
( <font color="Maroon">cc</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> &amp; <font color="Maroon">cc</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> &amp; <font color="Maroon">a'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">cc</font> )
 <b>by </b><i>, <a class="ref" href="partfun2.html#T78">PARTFUN2:78</a></i>;<br/>
<a NAME="E8:21_1_1_1_1"/>
<font color="Maroon">cc</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E9:21_1_1_1_1"/><b>then </b>
<font color="Maroon">cc</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span>
 
;<br/>
<a NAME="E10:21_1_1_1_1"/><b>then </b>
<font color="Maroon">cc</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H1</font>
 
<b>by </b><i>, , , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E11:21_1_1_1_1"/><b>then </b>
<font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">cc</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">H</font>
 
<b>by </b><i>, <a class="ref" href="jordan5a.html#T2" target="_self">Th2</a>, <a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/>
<b>hence </b><a NAME="E12:21_1_1_1_1"/>
<font color="Maroon">aa</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N1</font>
 <b>by </b><i><a class="txt" href="jordan5a.html#E2"><i><font color="Green">Lemma30</font></i></a>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E29:21_1_1_1"/>
<font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">N</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">N1</font>
 ;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E4:21_1_1"/>
<font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K1">.]</a></span> <a href="fcont_1.html#R1">is_continuous_in</a> <font color="Maroon">x1</font>
 <b>by </b><i>, , , <a class="ref" href="fcont_1.html#T5">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">E77</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 ;<br/><div class="add"><a NAME="E2:21_1_2"/><i><font color="Green">E78</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E3:21_1_2"/>
for <font color="Olive">N1</font> being   <a href="rcomp_1.html#M1">Neighbourhood</a> of <font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font>  ex <font color="Olive">N</font> being   <a href="rcomp_1.html#M1">Neighbourhood</a> of <font color="Maroon">x1</font> st <font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> <font color="Olive">N</font> <a href="tarski.html#R1">c=</a> <font color="Olive">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">N1</font> be    <a href="rcomp_1.html#M1">Neighbourhood</a> of <font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font>;<br/>

<b>reconsider </b><font color="Maroon">N2</font> = <font color="Maroon">N1</font> as   <a href="struct_0.html#NM2">Subset</a> of <a href="metric_1.html#K8">RealSpace</a>  <b>by </b><i><a class="ref" href="metric_1.html#D14">METRIC_1:def 14</a></i>;<br/>
<a NAME="E2:21_1_2_1"/>
<font color="Maroon">N2</font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K3">Family_open_set</a> <a href="metric_1.html#K8">RealSpace</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E3:21_1_2_1"/><b>then </b><i><font color="Green">E79</font></i>: 
<font color="Maroon">N2</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <a href="metric_1.html#K8">RealSpace</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="topmetr.html#T16">TOPMETR:16</a></i>;<br/>
<a NAME="E4:21_1_2_1"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:21_1_2_1"/><b>then </b><i><font color="Green">E82</font></i>: 
( <font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N1</font> &amp; <font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="jordan5a.html#T1" target="_self">Th1</a>, <a class="ref" href="rcomp_1.html#T37">RCOMP_1:37</a></i>;<br/>
<b>set </b><font color="Maroon">NN1</font> = <font color="Maroon">N1</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span></span>)</span>;<br/>
<a NAME="E6:21_1_2_1"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N1</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">NN</font> = <font color="Maroon">N1</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span></span>)</span> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span> ;<br/>
<a NAME="E8:21_1_2_1"/>
<font color="Maroon">NN</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="pre_topc.html#D9">PRE_TOPC:def 9</a>, <a class="ref" href="topmetr.html#D7">TOPMETR:def 7</a></i>;<br/>
<a NAME="E9:21_1_2_1"/><b>then </b><i><font color="Green">E93</font></i>: 
<font color="Maroon">NN</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">f0</font> = <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span> <b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">N1'</font> = <font color="Maroon">NN</font> as    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">f0</font> <b>by </b><i>, <a class="ref" href="jordan5a.html#T1" target="_self">Th1</a>, , , <a class="ref" href="connsp_2.html#T5">CONNSP_2:5</a></i>;<br/>
<b>consider </b><font color="Maroon">H</font> being    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">x</font><b> such that </b><br/><a NAME="E13:21_1_2_1"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">H</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">N1'</font>
 <b>by </b><i>, <a class="ref" href="jordan5a.html#T1" target="_self">Th1</a>, <a class="ref" href="tmap_1.html#D2">TMAP_1:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon">H1</font> being   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span><b> such that </b><br/><a NAME="E15:21_1_2_1"/><i><font color="Green">E97</font></i>: 
( <font color="Maroon">H1</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">H1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">H</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H1</font> )
 <b>by </b><i><a class="ref" href="connsp_2.html#T8">CONNSP_2:8</a></i>;<br/>
<a NAME="E16:21_1_2_1"/>
<font color="Maroon">H1</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon">Q</font> being   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a> <b> such that </b><br/><a NAME="E18:21_1_2_1"/><i><font color="Green">E98</font></i>: 
( <font color="Maroon">Q</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <a href="topmetr.html#K3">R^1</a>  &amp; <font color="Maroon">H1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Q</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="pre_topc.html#D9">PRE_TOPC:def 9</a></i>;<br/>
<b>reconsider </b><font color="Maroon">Q'</font> = <font color="Maroon">Q</font> as   <a href="struct_0.html#NM2">Subset</a> of <a href="metric_1.html#K8">RealSpace</a>  <b>by </b><i><a class="ref" href="topmetr.html#T16">TOPMETR:16</a>, <a class="ref" href="topmetr.html#D7">TOPMETR:def 7</a></i>;<br/>
<a NAME="E20:21_1_2_1"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">Q'</font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K3">Family_open_set</a> <a href="metric_1.html#K8">RealSpace</a> 
 
<b>by </b><i><a class="txt" href="jordan5a.html#E2"><i><font color="Green">Lemma30</font></i></a>, <a class="ref" href="topmetr.html#T16">TOPMETR:16</a>, <a class="ref" href="topmetr.html#D7">TOPMETR:def 7</a></i>;<br/>
<b>reconsider </b><font color="Maroon">Q1</font> = <font color="Maroon">Q'</font> as   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="metric_1.html#D14">METRIC_1:def 14</a></i>;<br/>
<a NAME="E22:21_1_2_1"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">Q1</font> is <a href="rcomp_1.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="jordan5a.html#T2" target="_self">Th2</a>, </i>;<br/>
<a NAME="E23:21_1_2_1"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q1</font>
 
<b>by </b><i>, , <a class="txt" href="jordan5a.html#E2"><i><font color="Green">Lemma30</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">N</font> being    <a href="rcomp_1.html#M1">Neighbourhood</a> of <font color="Maroon">x1</font><b> such that </b><br/><a NAME="E25:21_1_2_1"/><i><font color="Green">E101</font></i>: 
<font color="Maroon">N</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Q1</font>
 <b>by </b><i>, <a class="ref" href="rcomp_1.html#T39">RCOMP_1:39</a></i>;<br/>
<b>take </b>
<font color="Maroon">N</font>
;<br/>

<a NAME="E26:21_1_2_1"/>
<font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">N</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">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">aa</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:21_1_2_1_1"/><i><font color="Green">E154</font></i>: 
<font color="Maroon">aa</font> <a href="hidden.html#R2">in</a> <font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">N</font>
 ;<br/>

<a NAME="E2:21_1_2_1_1"/>
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">N</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span>
 
;<br/>
<a NAME="E3:21_1_2_1_1"/><b>then </b>
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">N</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i>, <a class="ref" href="topmetr.html#T25">TOPMETR:25</a></i>;<br/>
<a NAME="E4:21_1_2_1_1"/><b>then </b>
<font color="Maroon">aa</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i>, </i>;<br/>
<b>then reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">aa</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  ;<br/>
<b>consider </b><font color="Maroon">cc</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E7:21_1_2_1_1"/><i><font color="Green">E155</font></i>: 
( <font color="Maroon">cc</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> &amp; <font color="Maroon">cc</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> &amp; <font color="Maroon">a'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">cc</font> )
 <b>by </b><i>, <a class="ref" href="partfun2.html#T78">PARTFUN2:78</a></i>;<br/>
<a NAME="E8:21_1_2_1_1"/>
<font color="Maroon">cc</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E9:21_1_2_1_1"/><b>then </b>
<font color="Maroon">cc</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span>
 
;<br/>
<a NAME="E10:21_1_2_1_1"/><b>then </b>
<font color="Maroon">cc</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H1</font>
 
<b>by </b><i><a class="txt" href="jordan5a.html#E2"><i><font color="Green">Lemma30</font></i></a>, <a class="ref" href="jordan5a.html#T3" target="_self">Th3</a>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E11:21_1_2_1_1"/><b>then </b>
<font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">cc</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">H</font>
 
<b>by </b><i>, , <a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/>
<b>hence </b><a NAME="E12:21_1_2_1_1"/>
<font color="Maroon">aa</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N1</font>
 <b>by </b><i><a class="txt" href="jordan5a.html#E1:21"><i><font color="Green">E28</font></i></a>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E27:21_1_2_1"/>
<font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">N</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">N1</font>
 ;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E4:21_1_2"/>
<font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K1">.]</a></span> <a href="fcont_1.html#R1">is_continuous_in</a> <font color="Maroon">x1</font>
 <b>by </b><i>, , , <a class="ref" href="fcont_1.html#T5">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">x1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font> &amp; <font color="Maroon">x1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:21_1_3"/>
<font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K1">.]</a></span> <a href="fcont_1.html#R1">is_continuous_in</a> <font color="Maroon">x1</font>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
