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

<b>let </b><font color="Maroon">M</font> be  non <a href="struct_0.html#V3">empty</a> <a href="metric_1.html#NM1">MetrSpace</a>;<br/><b>let </b><font color="Maroon">A</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="topmetr.html#M1" target="_self">SubSpace</a> of <font color="Maroon">M</font>;<br/>



<b>set </b><font color="Maroon">T</font> =  <a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">M</font>;<br/>
<b>set </b><font color="Maroon">R</font> =  <a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">A</font>;<br/>
<a NAME="E1:25"/><i><font color="Green">E28</font></i>: 
(  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font> &amp;  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">M</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E2:25"/>
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">A</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">M</font></span>)</span>
 <b>by </b><i/>; <i><font color="Red">:: according to </font></i><a class="ref" href="pre_topc.html#D9">PRE_TOPC:def 9</a><br/>

<b>let </b><font color="Maroon">P</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">A</font></span>)</span>;<br/>

<b>thus </b><a NAME="E3:25"/>
( <font color="Maroon">P</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> <font color="Maroon">A</font></span>)</span> implies  ex <font color="Olive">Q</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">M</font></span>)</span> st <br/>( <font color="Olive">Q</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> <font color="Maroon">M</font></span>)</span> &amp; <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <font color="Olive">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="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">A</font></span>)</span></span>)</span> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:25_1"/>
<font color="Maroon">P</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> <font color="Maroon">A</font></span>)</span>
 ;<br/>

<a NAME="E2:25_1"/><b>then </b><i><font color="Green">E32</font></i>: 
<font color="Maroon">P</font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K3">Family_open_set</a> <font color="Maroon">A</font>
 
<b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">QQ</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Olive">x</font>,<font color="Olive">r</font></span>)</span> where <font color="Olive">x</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">M</font>, <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <span class="p1">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Olive">x</font>,<font color="Olive">r</font></span>)</span> <a href="subset_1.html#K8">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font> ) </span>}</span> ;<br/>
<a NAME="E3:25_1"/>
for <font color="Olive">X</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Olive">x</font>,<font color="Olive">r</font></span>)</span> where <font color="Olive">x</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">M</font>, <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <span class="p1">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Olive">x</font>,<font color="Olive">r</font></span>)</span> <a href="subset_1.html#K8">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font> ) </span>}</span>  holds <br/><font color="Olive">X</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:25_1_1"/>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Olive">x</font>,<font color="Olive">r</font></span>)</span> where <font color="Olive">x</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">M</font>, <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <span class="p1">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Olive">x</font>,<font color="Olive">r</font></span>)</span> <a href="subset_1.html#K8">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font> ) </span>}</span> 
 ;<br/>

<a NAME="E2:25_1_1"/><b>then </b>
 ex <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">M</font> ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Olive">x</font>,<font color="Olive">r</font> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <span class="p1">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Olive">x</font>,<font color="Olive">r</font></span>)</span> <a href="subset_1.html#K8">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font> )
 
;<br/>
<b>hence </b><a NAME="E3:25_1_1"/>
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M</font>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">Q</font> =  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Olive">x</font>,<font color="Olive">r</font></span>)</span> where <font color="Olive">x</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">M</font>, <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <span class="p1">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Olive">x</font>,<font color="Olive">r</font></span>)</span> <a href="subset_1.html#K8">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font> ) </span>}</span>  as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">M</font> <b>by </b><i><a class="ref" href="zfmisc_1.html#T94">ZFMISC_1:94</a></i>;<br/>
<a NAME="E5:25_1"/>
for <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">M</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> holds <br/> ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Olive">x</font>,<font color="Olive">r</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Q</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">M</font>;<br/>

<b>assume </b><a NAME="E1:25_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font>
 ;<br/>

<b>then consider </b><font color="Maroon">Y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:25_1_2"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Olive">x</font>,<font color="Olive">r</font></span>)</span> where <font color="Olive">x</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">M</font>, <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <span class="p1">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Olive">x</font>,<font color="Olive">r</font></span>)</span> <a href="subset_1.html#K8">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font> ) </span>}</span>  )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">x'</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">M</font>, <font color="Maroon">r</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E5:25_1_2"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">x'</font>,<font color="Maroon">r</font>
 <b>and </b><br/><a NAME="E6:25_1_2"/>
( <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <span class="p1">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Maroon">x'</font>,<font color="Maroon">r</font></span>)</span> <a href="subset_1.html#K8">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font> )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">p</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E8:25_1_2"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">x</font>,<font color="Maroon">p</font> <a href="tarski.html#R1">c=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">x'</font>,<font color="Maroon">r</font> )
 <b>by </b><i>, , <a class="ref" href="pcomps_1.html#T30">PCOMPS_1:30</a></i>;<br/>
<b>take </b>
<font color="Maroon">p</font>
;<br/>

<b>thus </b><a NAME="E9:25_1_2"/>
<font color="Maroon">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i/>;<br/>

<a NAME="E10:25_1_2"/>
<font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Q</font>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<b>hence </b><a NAME="E11:25_1_2"/>
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">x</font>,<font color="Maroon">p</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Q</font>
 <b>by </b><i>, , <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:25_1"/><b>then </b><i><font color="Green">E52</font></i>: 
<font color="Maroon">Q</font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K3">Family_open_set</a> <font color="Maroon">M</font>
 
<b>by </b><i><a class="ref" href="pcomps_1.html#D5">PCOMPS_1:def 5</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 <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">M</font></span>)</span> <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">Q'</font>
;<br/>

<b>thus </b><a NAME="E8:25_1"/>
<font color="Maroon">Q'</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> <font color="Maroon">M</font></span>)</span>
 <b>by </b><i>, </i>;<br/>

<b>thus </b><a NAME="E9:25_1"/>
<font color="Maroon">P</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="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">A</font></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_3"><b>proof </b></a><div class="add">

<a NAME="E1:25_1_3"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">P</font> <a href="tarski.html#R1">c=</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="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">A</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:25_1_3_1"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 ;<br/>

<a NAME="E2:25_1_3_1"/><b>then </b>
<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="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">A</font></span>)</span>
 
;<br/>
<a NAME="E3:25_1_3_1"/><b>then </b><i><font color="Green">E75</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">A</font></span>)</span>
 
;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">a</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">A</font> <b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">P'</font> = <font color="Maroon">P</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">A</font> <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">r</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E7:25_1_3_1"/><i><font color="Green">E76</font></i>: 
( <font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">x</font>,<font color="Maroon">r</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P'</font> )
 <b>by </b><i>, , <a class="ref" href="pcomps_1.html#D5">PCOMPS_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">M</font> <b>by </b><i/>;<br/>
<a NAME="E9:25_1_3_1"/>
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">x</font>,<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Maroon">x'</font>,<font color="Maroon">r</font></span>)</span> <a href="subset_1.html#K8">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>
 
<b>by </b><i/>;<br/>
<a NAME="E10:25_1_3_1"/><b>then </b>
(  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">x'</font>,<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Olive">x</font>,<font color="Olive">r</font></span>)</span> where <font color="Olive">x</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">M</font>, <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <span class="p1">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Olive">x</font>,<font color="Olive">r</font></span>)</span> <a href="subset_1.html#K8">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font> ) </span>}</span>  &amp; <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">x'</font>,<font color="Maroon">r</font> )
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E11:25_1_3_1"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q'</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>hence </b><a NAME="E12:25_1_3_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</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="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">A</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="topmetr.html#T1" target="_self">Th1</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:25_1_3"/>
<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="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:25_1_3_2"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</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="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">A</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:25_1_3_2"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q'</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">Y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:25_1_3_2"/><i><font color="Green">E90</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Olive">x</font>,<font color="Olive">r</font></span>)</span> where <font color="Olive">x</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">M</font>, <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <span class="p1">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Olive">x</font>,<font color="Olive">r</font></span>)</span> <a href="subset_1.html#K8">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font> ) </span>}</span>  )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">x</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">M</font>, <font color="Maroon">r</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E6:25_1_3_2"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">x</font>,<font color="Maroon">r</font>
 <b>and </b><br/><a NAME="E7:25_1_3_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>and </b><br/><a NAME="E8:25_1_3_2"/>
<font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>and </b><br/><a NAME="E9:25_1_3_2"/><i><font color="Green">E92</font></i>: 
<span class="p1">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Maroon">x</font>,<font color="Maroon">r</font></span>)</span> <a href="subset_1.html#K8">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="topmetr.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E10:25_1_3_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E11:25_1_3_2"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Maroon">x</font>,<font color="Maroon">r</font></span>)</span> <a href="subset_1.html#K8">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="topmetr.html#T1" target="_self">Th1</a>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E12:25_1_3_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:25_1_3"/>
<font color="Maroon">P</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="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">A</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


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


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

<b>given </b><font color="Maroon">Q</font> being   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">M</font></span>)</span><b> such that </b><a NAME="E4:25"/><i><font color="Green">E93</font></i>: 
( <font color="Maroon">Q</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> <font color="Maroon">M</font></span>)</span> &amp; <font color="Maroon">P</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="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">A</font></span>)</span></span>)</span> )
 ;<br/>

<b>reconsider </b><font color="Maroon">Q'</font> = <font color="Maroon">Q</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">M</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">P'</font> = <font color="Maroon">P</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">A</font> <b>by </b><i/>;<br/>
<a NAME="E7:25"/>
for <font color="Olive">p</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">A</font>  st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P'</font> holds <br/> ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Olive">p</font>,<font color="Olive">r</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P'</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">A</font>;<br/>

<b>reconsider </b><font color="Maroon">p'</font> = <font color="Maroon">p</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">M</font> <b>by </b><i/>;<br/>
<b>assume </b><a NAME="E2:25_2"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P'</font>
 ;<br/>

<a NAME="E3:25_2"/><b>then </b>
( <font color="Maroon">p'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q'</font> &amp; <font color="Maroon">Q'</font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K3">Family_open_set</a> <font color="Maroon">M</font> )
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">r</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E5:25_2"/><i><font color="Green">E94</font></i>: 
( <font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p'</font>,<font color="Maroon">r</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Q'</font> )
 <b>by </b><i><a class="ref" href="pcomps_1.html#D5">PCOMPS_1:def 5</a></i>;<br/>
<a NAME="E6:25_2"/>
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p</font>,<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Maroon">p'</font>,<font color="Maroon">r</font></span>)</span> <a href="subset_1.html#K8">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>
 
<b>by </b><i/>;<br/>
<a NAME="E7:25_2"/><b>then </b>
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p</font>,<font color="Maroon">r</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Q</font> <a href="subset_1.html#K8">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T26">XBOOLE_1:26</a></i>;<br/>
<a NAME="E8:25_2"/><b>then </b>
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p</font>,<font color="Maroon">r</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Q</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="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">A</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E9:25_2"/><b>then </b>
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p</font>,<font color="Maroon">r</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P'</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E10:25_2"/>
 ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p</font>,<font color="Olive">r</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P'</font> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E8:25"/><b>then </b>
<font color="Maroon">P</font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K3">Family_open_set</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="pcomps_1.html#D5">PCOMPS_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E9:25"/>
<font color="Maroon">P</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> <font color="Maroon">A</font></span>)</span>
 <b>by </b><i/>;<br/>


</div>
