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

<b>reconsider </b><font color="Maroon">c<sub>1</sub></font> = <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> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<b>set </b><font color="Maroon">c<sub>2</sub></font> = <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>;<br/>
<a NAME="E2:38_1_1"/>
<a href="numbers.html#K1">REAL</a>  <a href="xboole_0.html#K2">\/</a> <a href="numbers.html#K5">NAT</a>  <a href="hidden.html#R1">=</a>  <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>;<br/>
<a NAME="E3:38_1_1"/><b>then </b><i><font color="Green">E33</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<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="p2">(<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></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K3">R^1</a> 
 
<b>by </b><i><a class="ref" href="frechet.html#T15" target="_self">Th15</a>, <a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<a NAME="E4:38_1_1"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<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="p2">(<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></span>)</span> <a href="tarski.html#R1">c=</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>
 
<b>by </b><i><a class="ref" href="frechet.html#T16" target="_self">Th16</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>3</sub></font> = <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> as   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K3">R^1</a> ,<font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="frechet.html#E3:38_1_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K6">\</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K11">"</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> is <a href="pre_topc.html#V4">closed</a> ) </span>}</span> ;<br/>
<a NAME="E6:38_1_1"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K6">\</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K11">"</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> is <a href="pre_topc.html#V4">closed</a> ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="pcomps_1.html#K1">bool</a> <font color="Maroon">c<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></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_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K6">\</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K11">"</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> is <a href="pre_topc.html#V4">closed</a> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:38_1_1_1"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>6</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V4">closed</a> ) )
 ;<br/>
<b>thus </b><a NAME="E4:38_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K1">bool</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="frechet.html#E3:38_1_1_1"><i><font color="Green">E34</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K6">\</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K11">"</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> is <a href="pre_topc.html#V4">closed</a> ) </span>}</span>  as   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> =  <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font> #);<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="struct_0.html#NM7">Function</a> of <a href="topmetr.html#K3">R^1</a> ,<a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font> #) ;<br/>
<a NAME="E9:38_1_1"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font> #) holds <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V4">closed</a> iff <font color="Maroon">c<sub>7</sub></font> <a href="relset_1.html#K11">"</a> <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V4">closed</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font> #);<br/>

<b>thus </b><a NAME="E1:38_1_1_2"/>
( <font color="Maroon">c<sub>8</sub></font> is <a href="pre_topc.html#V4">closed</a> implies <font color="Maroon">c<sub>7</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>8</sub></font> is <a href="pre_topc.html#V4">closed</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:38_1_1_2_1"/>
<font color="Maroon">c<sub>8</sub></font> is <a href="pre_topc.html#V4">closed</a>
 ;<br/>

<a NAME="E2:38_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font> #)</span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>8</sub></font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D6">PRE_TOPC:def 6</a></i>;<br/>
<a NAME="E3:38_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font> #)</span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font> #)
 
<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">c<sub>9</sub></font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E5:38_1_1_2_1"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font> #)</span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>8</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V4">closed</a> ) )
 ;<br/>
<a NAME="E6:38_1_1_2_1"/>
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font> #)</span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font> #)</span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="frechet.html#E5:38_1_1_2_1"><i><font color="Green">E35</font></i></a></i>;<br/>
<a NAME="E7:38_1_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font> #)</span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font> #)</span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T22">PRE_TOPC:22</a></i>;<br/>
<b>hence </b><a NAME="E8:38_1_1_2_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>8</sub></font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i><a class="txt" href="frechet.html#E5:38_1_1_2_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="pre_topc.html#T22">PRE_TOPC:22</a></i>;<br/>


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

<b>assume </b><a NAME="E2:38_1_1_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>8</sub></font> is <a href="pre_topc.html#V4">closed</a>
 ;<br/>

<a NAME="E3:38_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 
;<br/>
<a NAME="E4:38_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font> #)</span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 
;<br/>
<a NAME="E5:38_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font> #)</span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>8</sub></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>hence </b><a NAME="E6:38_1_1_2"/>
<font color="Maroon">c<sub>8</sub></font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i><a class="ref" href="pre_topc.html#D6">PRE_TOPC:def 6</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> =  <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font> #) as  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> <b>by </b><i><a class="ref" href="frechet.html#T6" target="_self">Th6</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>8</sub></font>
;<br/>

<b>thus </b><a NAME="E11:38_1_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>8</sub></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>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="struct_0.html#NM7">Function</a> of <a href="topmetr.html#K3">R^1</a> ,<font color="Maroon">c<sub>8</sub></font> ;<br/>
<b>take </b>
<font color="Maroon">c<sub>9</sub></font>
;<br/>

<b>thus </b><a NAME="E13:38_1_1"/>
( <font color="Maroon">c<sub>9</sub></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">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>8</sub></font> holds <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V4">closed</a> iff <font color="Maroon">c<sub>9</sub></font> <a href="relset_1.html#K11">"</a> <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V4">closed</a> ) ) )
 <b>by </b><i><a class="txt" href="frechet.html#E9:38_1_1"><i><font color="Green">E34</font></i></a></i>;<br/>


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