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<div class="add">

<b>let </b><font color="Maroon">X</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#V1">strict</a> <a href="pre_topc.html#NM1">TopSpace</a>, <font color="Maroon">Y</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#V1">strict</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:38_1_2"/><i><font color="Green">E40</font></i>: 
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="numbers.html#K1">REAL</a>  <a href="subset_1.html#K6">\</a> <a href="numbers.html#K5">NAT</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="numbers.html#K1">REAL</a> </span><a href="tarski.html#K1">}</a></span> &amp;  ex <font color="Olive">f</font> being  <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K3">R^1</a> ,<font color="Maroon">X</font> st <br/>( <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <a href="numbers.html#K1">REAL</a> </span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><a href="numbers.html#K5">NAT</a>  <a href="funcop_1.html#K2">--&gt;</a> <a href="numbers.html#K1">REAL</a> </span>)</span> &amp; ( for <font color="Olive">A</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> holds <br/> ( <font color="Olive">A</font> is <a href="pre_topc.html#V4">closed</a> iff <font color="Olive">f</font> <a href="relset_1.html#K11">"</a> <font color="Olive">A</font> is <a href="pre_topc.html#V4">closed</a> ) ) ) )
 <b>and </b><br/><a NAME="E2:38_1_2"/><i><font color="Green">E43</font></i>: 
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="numbers.html#K1">REAL</a>  <a href="subset_1.html#K6">\</a> <a href="numbers.html#K5">NAT</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="numbers.html#K1">REAL</a> </span><a href="tarski.html#K1">}</a></span> &amp;  ex <font color="Olive">f</font> being  <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K3">R^1</a> ,<font color="Maroon">Y</font> st <br/>( <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <a href="numbers.html#K1">REAL</a> </span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><a href="numbers.html#K5">NAT</a>  <a href="funcop_1.html#K2">--&gt;</a> <a href="numbers.html#K1">REAL</a> </span>)</span> &amp; ( for <font color="Olive">A</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font> holds <br/> ( <font color="Olive">A</font> is <a href="pre_topc.html#V4">closed</a> iff <font color="Olive">f</font> <a href="relset_1.html#K11">"</a> <font color="Olive">A</font> is <a href="pre_topc.html#V4">closed</a> ) ) ) )
 ;<br/>

<a NAME="E3:38_1_2"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">Y</font>
 
<b>by </b><i><a class="txt" href="frechet.html#E6"><i><font color="Green">Lemma33</font></i></a>, </i>;<br/>
<a NAME="E4:38_1_2"/><b>then </b><i><font color="Green">E44</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">Y</font>
 
;<br/>
<b>consider </b><font color="Maroon">f</font> being   <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K3">R^1</a> ,<font color="Maroon">X</font><b> such that </b><br/><a NAME="E6:38_1_2"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <a href="numbers.html#K1">REAL</a> </span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><a href="numbers.html#K5">NAT</a>  <a href="funcop_1.html#K2">--&gt;</a> <a href="numbers.html#K1">REAL</a> </span>)</span>
 <b>and </b><br/><a NAME="E7:38_1_2"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">A</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> holds <br/> ( <font color="Olive">A</font> is <a href="pre_topc.html#V4">closed</a> iff <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Olive">A</font> is <a href="pre_topc.html#V4">closed</a> )
 <b>by </b><i><a class="txt" href="frechet.html#E6"><i><font color="Green">Lemma33</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">g</font> being   <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K3">R^1</a> ,<font color="Maroon">Y</font><b> such that </b><br/><a NAME="E9:38_1_2"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <a href="numbers.html#K1">REAL</a> </span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><a href="numbers.html#K5">NAT</a>  <a href="funcop_1.html#K2">--&gt;</a> <a href="numbers.html#K1">REAL</a> </span>)</span>
 <b>and </b><br/><a NAME="E10:38_1_2"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">A</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font> holds <br/> ( <font color="Olive">A</font> is <a href="pre_topc.html#V4">closed</a> iff <font color="Maroon">g</font> <a href="relset_1.html#K11">"</a> <font color="Olive">A</font> is <a href="pre_topc.html#V4">closed</a> )
 <b>by </b><i/>;<br/>
<a NAME="E11:38_1_2"/>
the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">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">topology</a> of <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">Y</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">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">V</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="E1:38_1_2_1_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">V</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">V1</font> = <font color="Maroon">V</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> ;<br/>
<b>reconsider </b><font color="Maroon">V1</font> = <font color="Maroon">V1</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> ;<br/>
<a NAME="E4:38_1_2_1_1"/>
<font color="Maroon">V1</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="txt" href="frechet.html#E1:38_1_2"><i><font color="Green">E40</font></i></a>, <a class="ref" href="pre_topc.html#D5">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="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">V1</font> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i/>;<br/>
<a NAME="E6:38_1_2_1_1"/><b>then </b><i><font color="Green">E54</font></i>: 
<font color="Maroon">g</font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">V1</font></span>)</span> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i><a class="ref" href="frechet.html#T5" target="_self">Th5</a>, , </i>;<br/>
<b>reconsider </b><font color="Maroon">V2</font> = <font color="Maroon">V</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font> <b>by </b><i><a class="txt" href="frechet.html#E6"><i><font color="Green">Lemma33</font></i></a>, , <a class="txt" href="frechet.html#E1:38_1_2"><i><font color="Green">E40</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">V2</font> = <font color="Maroon">V2</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font> ;<br/>
<a NAME="E9:38_1_2_1_1"/>
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">Y</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">V2</font> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i><a class="ref" href="frechet.html#T4" target="_self">Th4</a>, , </i>;<br/>
<a NAME="E10:38_1_2_1_1"/><b>then </b>
<font color="Maroon">V2</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E11:38_1_2_1_1"/>
<font color="Maroon">V</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#D5">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">topology</a> of <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">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">V</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="E1:38_1_2_1_2"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">V</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">Y</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">V1</font> = <font color="Maroon">V</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font> ;<br/>
<b>reconsider </b><font color="Maroon">V1</font> = <font color="Maroon">V1</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font> ;<br/>
<a NAME="E4:38_1_2_1_2"/>
<font color="Maroon">V1</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="txt" href="frechet.html#E1:38_1_2"><i><font color="Green">E40</font></i></a>, <a class="ref" href="pre_topc.html#D5">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="pre_topc.html#K2">[#]</a> <font color="Maroon">Y</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">V1</font> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i/>;<br/>
<a NAME="E6:38_1_2_1_2"/><b>then </b><i><font color="Green">E56</font></i>: 
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">Y</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">V1</font></span>)</span> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i><a class="ref" href="frechet.html#T5" target="_self">Th5</a>, , </i>;<br/>
<b>reconsider </b><font color="Maroon">V2</font> = <font color="Maroon">V</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> <b>by </b><i><a class="txt" href="frechet.html#E6"><i><font color="Green">Lemma33</font></i></a>, , <a class="txt" href="frechet.html#E1:38_1_2"><i><font color="Green">E40</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">V2</font> = <font color="Maroon">V2</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> ;<br/>
<a NAME="E9:38_1_2_1_2"/>
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">V2</font> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i><a class="ref" href="frechet.html#T4" target="_self">Th4</a>, , </i>;<br/>
<a NAME="E10:38_1_2_1_2"/><b>then </b>
<font color="Maroon">V2</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E11:38_1_2_1_2"/>
<font color="Maroon">V</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>


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


</div><b>end;</b></div>
<b>hence </b><a NAME="E12:38_1_2"/>
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="txt" href="frechet.html#E6"><i><font color="Green">Lemma33</font></i></a>, </i>;<br/>


</div>
