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

<b>let </b><font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c2">Y</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="pre_topc.html#V1" title="PRE_TOPC:attr.1">strict</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:38_1_2"/><i><font color="Green" title="E32">A5</font></i>: 
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  ex <font color="Olive" title="b1">f</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> ,<font color="Maroon" title="c1">X</font> st <br/>( <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="funcop_1.html#K2" title="FUNCOP_1:func.2">--&gt;</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span>)</span> &amp; ( for <font color="Olive" title="b2">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> holds <br/> ( <font color="Olive" title="b2">A</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> iff <font color="Olive" title="b1">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Olive" title="b2">A</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> ) ) ) )
 <b>and </b><br/><a NAME="E2:38_1_2"/><i><font color="Green" title="E33">A6</font></i>: 
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  ex <font color="Olive" title="b1">f</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> ,<font color="Maroon" title="c2">Y</font> st <br/>( <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="funcop_1.html#K2" title="FUNCOP_1:func.2">--&gt;</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span>)</span> &amp; ( for <font color="Olive" title="b2">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font> holds <br/> ( <font color="Olive" title="b2">A</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> iff <font color="Olive" title="b1">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Olive" title="b2">A</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> ) ) ) )
 ;<br/>

<b>consider </b><font color="Maroon" title="c3">f</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> ,<font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E4:38_1_2"/><i><font color="Green" title="E34">A7</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="funcop_1.html#K2" title="FUNCOP_1:func.2">--&gt;</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span>)</span>
 <b>and </b><br/><a NAME="E5:38_1_2"/><i><font color="Green" title="E35">A8</font></i>: 
for <font color="Olive" title="b1">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> holds <br/> ( <font color="Olive" title="b1">A</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> iff <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Olive" title="b1">A</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> )
 <b>by </b><i><a class="txt" href="frechet.html#E1:38_1_2"><i><font color="Green" title="E32">A5</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c4">g</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> ,<font color="Maroon" title="c2">Y</font><b> such that </b><br/><a NAME="E7:38_1_2"/><i><font color="Green" title="E36">A9</font></i>: 
<font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="funcop_1.html#K2" title="FUNCOP_1:func.2">--&gt;</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span>)</span>
 <b>and </b><br/><a NAME="E8:38_1_2"/><i><font color="Green" title="E37">A10</font></i>: 
for <font color="Olive" title="b1">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font> holds <br/> ( <font color="Olive" title="b1">A</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> iff <font color="Maroon" title="c4">g</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Olive" title="b1">A</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> )
 <b>by </b><i><a class="txt" href="frechet.html#E2:38_1_2"><i><font color="Green" title="E33">A6</font></i></a></i>;<br/>
<a NAME="E9:38_1_2"/>
the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">Y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_2_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:38_1_2_1"/>
the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">Y</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">V</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="E1:38_1_2_1_1"/><i><font color="Green" title="E38">A11</font></i>: 
<font color="Maroon" title="c5">V</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 <font color="Maroon" title="c1">X</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c6">V1</font> = <font color="Maroon" title="c5">V</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c7">V1</font> = <font color="Maroon" title="c6">V1</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> ;<br/>
<a NAME="E4:38_1_2_1_1"/>
<font color="Maroon" title="c7">V1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="txt" href="frechet.html#E1:38_1_2_1_1"><i><font color="Green" title="E38">A11</font></i></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E5:38_1_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c7">V1</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 
<b>by </b><i><a class="txt" href="frechet.html#E6"><i><font color="Green" title="E5">Lm3</font></i></a></i>;<br/>
<a NAME="E6:38_1_2_1_1"/><b>then </b><i><font color="Green" title="E39">A12</font></i>: 
<font color="Maroon" title="c4">g</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c7">V1</font></span>)</span> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 
<b>by </b><i><a class="txt" href="frechet.html#E4:38_1_2"><i><font color="Green" title="E34">A7</font></i></a>, <a class="txt" href="frechet.html#E5:38_1_2"><i><font color="Green" title="E35">A8</font></i></a>, <a class="txt" href="frechet.html#E7:38_1_2"><i><font color="Green" title="E36">A9</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">V2</font> = <font color="Maroon" title="c5">V</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font> <b>by </b><i><a class="txt" href="frechet.html#E1:38_1_2"><i><font color="Green" title="E32">A5</font></i></a>, <a class="txt" href="frechet.html#E2:38_1_2"><i><font color="Green" title="E33">A6</font></i></a>, <a class="txt" href="frechet.html#E1:38_1_2_1_1"><i><font color="Green" title="E38">A11</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">V2</font> = <font color="Maroon" title="c8">V2</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font> ;<br/>
<a NAME="E9:38_1_2_1_1"/>
<span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c9">V2</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 
<b>by </b><i><a class="txt" href="frechet.html#E1:38_1_2"><i><font color="Green" title="E32">A5</font></i></a>, <a class="txt" href="frechet.html#E2:38_1_2"><i><font color="Green" title="E33">A6</font></i></a>, <a class="txt" href="frechet.html#E8:38_1_2"><i><font color="Green" title="E37">A10</font></i></a>, <a class="txt" href="frechet.html#E6:38_1_2_1_1"><i><font color="Green" title="E39">A12</font></i></a></i>;<br/>
<a NAME="E10:38_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c9">V2</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="txt" href="frechet.html#E6"><i><font color="Green" title="E5">Lm3</font></i></a></i>;<br/>
<b>hence </b><a NAME="E11:38_1_2_1_1"/>
<font color="Maroon" title="c5">V</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 <font color="Maroon" title="c2">Y</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/>


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

<b>thus </b><a NAME="E2:38_1_2_1"/>
the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_2_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">V</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="E1:38_1_2_1_2"/><i><font color="Green" title="E38">A13</font></i>: 
<font color="Maroon" title="c5">V</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 <font color="Maroon" title="c2">Y</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c6">V1</font> = <font color="Maroon" title="c5">V</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c7">V1</font> = <font color="Maroon" title="c6">V1</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font> ;<br/>
<a NAME="E4:38_1_2_1_2"/>
<font color="Maroon" title="c7">V1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="txt" href="frechet.html#E1:38_1_2_1_2"><i><font color="Green" title="E38">A13</font></i></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E5:38_1_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c7">V1</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 
<b>by </b><i><a class="txt" href="frechet.html#E6"><i><font color="Green" title="E5">Lm3</font></i></a></i>;<br/>
<a NAME="E6:38_1_2_1_2"/><b>then </b><i><font color="Green" title="E39">A14</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c7">V1</font></span>)</span> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 
<b>by </b><i><a class="txt" href="frechet.html#E4:38_1_2"><i><font color="Green" title="E34">A7</font></i></a>, <a class="txt" href="frechet.html#E7:38_1_2"><i><font color="Green" title="E36">A9</font></i></a>, <a class="txt" href="frechet.html#E8:38_1_2"><i><font color="Green" title="E37">A10</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">V2</font> = <font color="Maroon" title="c5">V</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="txt" href="frechet.html#E1:38_1_2"><i><font color="Green" title="E32">A5</font></i></a>, <a class="txt" href="frechet.html#E2:38_1_2"><i><font color="Green" title="E33">A6</font></i></a>, <a class="txt" href="frechet.html#E1:38_1_2_1_2"><i><font color="Green" title="E38">A13</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">V2</font> = <font color="Maroon" title="c8">V2</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> ;<br/>
<a NAME="E9:38_1_2_1_2"/>
<span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c9">V2</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 
<b>by </b><i><a class="txt" href="frechet.html#E1:38_1_2"><i><font color="Green" title="E32">A5</font></i></a>, <a class="txt" href="frechet.html#E2:38_1_2"><i><font color="Green" title="E33">A6</font></i></a>, <a class="txt" href="frechet.html#E5:38_1_2"><i><font color="Green" title="E35">A8</font></i></a>, <a class="txt" href="frechet.html#E6:38_1_2_1_2"><i><font color="Green" title="E39">A14</font></i></a></i>;<br/>
<a NAME="E10:38_1_2_1_2"/><b>then </b>
<font color="Maroon" title="c9">V2</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="txt" href="frechet.html#E6"><i><font color="Green" title="E5">Lm3</font></i></a></i>;<br/>
<b>hence </b><a NAME="E11:38_1_2_1_2"/>
<font color="Maroon" title="c5">V</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 <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/>


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


</div><b>end;</b></div>
<b>hence </b><a NAME="E10:38_1_2"/>
<font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">Y</font>
 <b>by </b><i><a class="txt" href="frechet.html#E1:38_1_2"><i><font color="Green" title="E32">A5</font></i></a>, <a class="txt" href="frechet.html#E2:38_1_2"><i><font color="Green" title="E33">A6</font></i></a></i>;<br/>


</div>
