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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  <a href="relat_1.html#V2">V5</a>  <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><a NAME="E1:4"/><i><font color="Green">E4</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="pcomps_1.html#K1">bool</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/>

<a NAME="E2:4"/><i><font color="Green">E5</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K3">Union</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="ref" href="card_3.html#D4">CARD_3:def 4</a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E3:4"/><b>then </b>
 <a href="card_3.html#K3">Union</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <span class="p2">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="topgen_3.html#E1:4"><i><font color="Green">E4</font></i></a>, <a class="ref" href="zfmisc_1.html#T95">ZFMISC_1:95</a></i>;<br/>
<a NAME="E4:4"/><b>then </b>
 <a href="card_3.html#K3">Union</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="pcomps_1.html#K1">bool</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T99">ZFMISC_1:99</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>3</sub></font> =  <a href="card_3.html#K3">Union</a> <font color="Maroon">c<sub>2</sub></font> as   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<b>assume </b><a NAME="E6:4"/><i><font color="Green">E6</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>2</sub></font>
 ;<br/>

<b>assume </b><a NAME="E7:4"/><i><font color="Green">E7</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> holds <br/>ex <font color="Olive">b<sub>4</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>3</sub></font> )
 ;<br/>

<b>assume </b><a NAME="E8:4"/><i><font color="Green">E8</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> holds <br/>ex <font color="Olive">b<sub>4</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>2</sub></font> <a href="xboole_0.html#K3">/\</a> <font color="Olive">b<sub>3</sub></font> )
 ;<br/>

<b>take </b>
<font color="Maroon">c<sub>3</sub></font>
;<br/>

<b>thus </b><a NAME="E9:4"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K3">Union</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="pre_topc.html#L1">TopStruct</a> ;<br/>

<b>assume </b><a NAME="E10:4"/><i><font color="Green">E9</font></i>: 
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp; the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="cantor_1.html#K1">UniCl</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>

<a NAME="E11:4"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="topgen_3.html#V1" target="_self">point-filtered</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="topgen_3.html#D1" target="_self">TOPGEN_3:def 1</a><br/>

<b>assume </b><a NAME="E1:4_1"/><i><font color="Green">E11</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>7</sub></font> )
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>8</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:4_1"/><i><font color="Green">E12</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="txt" href="topgen_3.html#E2:4"><i><font color="Green">E5</font></i></a>, <a class="ref" href="card_5.html#T10">CARD_5:10</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>9</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:4_1"/><i><font color="Green">E13</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> )
 <b>by </b><i><a class="txt" href="topgen_3.html#E1:4_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="topgen_3.html#E2:4"><i><font color="Green">E5</font></i></a>, <a class="ref" href="card_5.html#T10">CARD_5:10</a></i>;<br/>
<a NAME="E6:4_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="topgen_3.html#E1:4_1"><i><font color="Green">E11</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">c<sub>10</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:4_1"/><i><font color="Green">E14</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="topgen_3.html#E7:4"><i><font color="Green">E7</font></i></a>, <a class="txt" href="topgen_3.html#E3:4_1"><i><font color="Green">E12</font></i></a></i>;<br/>
<a NAME="E9:4_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="topgen_3.html#E1:4_1"><i><font color="Green">E11</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">c<sub>11</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E11:4_1"/><i><font color="Green">E15</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="topgen_3.html#E7:4"><i><font color="Green">E7</font></i></a>, <a class="txt" href="topgen_3.html#E5:4_1"><i><font color="Green">E13</font></i></a></i>;<br/>
<a NAME="E12:4_1"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="topgen_3.html#E2:4"><i><font color="Green">E5</font></i></a>, <a class="txt" href="topgen_3.html#E8:4_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="funct_1.html#D4">FUNCT_1:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>12</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E14:4_1"/><i><font color="Green">E17</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>10</sub></font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>11</sub></font> )
 <b>by </b><i><a class="txt" href="topgen_3.html#E8:4"><i><font color="Green">E8</font></i></a>, <a class="txt" href="topgen_3.html#E8:4_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="topgen_3.html#E11:4_1"><i><font color="Green">E15</font></i></a></i>;<br/>
<a NAME="E15:4_1"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="topgen_3.html#E2:4"><i><font color="Green">E5</font></i></a>, <a class="txt" href="topgen_3.html#E12:4_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="topgen_3.html#E14:4_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="card_5.html#T10">CARD_5:10</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>12</sub></font> as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<b>take </b>
<font color="Maroon">c<sub>13</sub></font>
;<br/>

<b>thus </b><a NAME="E17:4_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="topgen_3.html#E15:4_1"><i><font color="Green">E18</font></i></a></i>;<br/>

<b>thus </b><a NAME="E18:4_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="topgen_3.html#E6:4"><i><font color="Green">E6</font></i></a>, <a class="txt" href="topgen_3.html#E12:4_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="topgen_3.html#E14:4_1"><i><font color="Green">E17</font></i></a></i>;<br/>

<a NAME="E19:4_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>11</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="topgen_3.html#E8:4_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="topgen_3.html#E11:4_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="xboole_1.html#T27">XBOOLE_1:27</a></i>;<br/>
<b>hence </b><a NAME="E20:4_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="topgen_3.html#E14:4_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>;<br/><b>assume </b><a NAME="E2:4_2"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><a NAME="E3:4_2"/><b>then </b>
not <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i><a class="ref" href="pboole.html#D16">PBOOLE:def 16</a></i>;<br/><a NAME="E4:4_2"/><b>then </b>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="topgen_3.html#E6:4"><i><font color="Green">E6</font></i></a>, <a class="txt" href="topgen_3.html#E2:4_2"><i><font color="Green">E11</font></i></a>, <a class="txt" href="topgen_3.html#E2:4"><i><font color="Green">E5</font></i></a>, <a class="ref" href="card_5.html#T10">CARD_5:10</a></i>;<br/><b>hence </b><a NAME="E5:4_2"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> )
 ;<br/></div><b>end;</b></div>
<a NAME="E13:4"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> is <a href="abian.html#V2">covering</a>
 
<b>by </b><i><a class="ref" href="topgen_3.html#T1" target="_self">Th1</a></i>;<br/>
<b>hence </b><a NAME="E14:4"/>
<font color="Maroon">c<sub>4</sub></font> is   <a href="pre_topc.html#NM1">TopSpace</a>
 <b>by </b><i><a class="txt" href="topgen_3.html#E10:4"><i><font color="Green">E9</font></i></a>, <a class="txt" href="topgen_3.html#E11:4"><i><font color="Green">E10</font></i></a>, <a class="ref" href="topgen_3.html#T2" target="_self">Th2</a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>5</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/>

<b>assume </b><a NAME="E15:4"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>2</sub></font> as   <a href="struct_0.html#NM9">ManySortedSet</a> of <font color="Maroon">c<sub>5</sub></font> <b>by </b><i><a class="txt" href="topgen_3.html#E10:4"><i><font color="Green">E9</font></i></a></i>;<br/>
<a NAME="E17:4"/>
<font color="Maroon">c<sub>6</sub></font> is    <a href="topgen_2.html#M1">Neighborhood_System</a> of <font color="Maroon">c<sub>5</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>5</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="topgen_2.html#D3">TOPGEN_2:def 3</a><br/>

<a NAME="E1:4_3"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="topgen_3.html#E10:4"><i><font color="Green">E9</font></i></a>, <a class="txt" href="topgen_3.html#E2:4"><i><font color="Green">E5</font></i></a>, <a class="txt" href="topgen_3.html#E15:4"><i><font color="Green">E11</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>5</sub></font> <b>by </b><i><a class="txt" href="topgen_3.html#E10:4"><i><font color="Green">E9</font></i></a>, <a class="txt" href="topgen_3.html#E1:4"><i><font color="Green">E4</font></i></a>, <a class="txt" href="topgen_3.html#E15:4"><i><font color="Green">E11</font></i></a></i>;<br/>
<a NAME="E3:4_3"/>
<font color="Maroon">c<sub>8</sub></font> is    <a href="yellow_8.html#M1">Basis</a> of <font color="Maroon">c<sub>7</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_3_1"><b>proof </b></a><div class="add">

<a NAME="E1:4_3_1"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="cantor_1.html#K1">UniCl</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="topgen_3.html#E2:4"><i><font color="Green">E5</font></i></a>, <a class="txt" href="topgen_3.html#E1:4_3"><i><font color="Green">E12</font></i></a>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a>, <a class="ref" href="cantor_1.html#T1">CANTOR_1:1</a></i>;<br/>
<b>hence </b><a NAME="E2:4_3_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="topgen_3.html#E10:4"><i><font color="Green">E9</font></i></a>, <a class="txt" href="topgen_3.html#E15:4"><i><font color="Green">E11</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_8.html#D2">YELLOW_8:def 2</a><br/>

<a NAME="E3:4_3_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> holds <br/><font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="topgen_3.html#E6:4"><i><font color="Green">E6</font></i></a>, <a class="txt" href="topgen_3.html#E15:4"><i><font color="Green">E11</font></i></a>, <a class="txt" href="topgen_3.html#E10:4"><i><font color="Green">E9</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:4_3_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="ref" href="setfam_1.html#T58">SETFAM_1:58</a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>5</sub></font>;<br/>

<b>assume </b><a NAME="E5:4_3_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>5</sub></font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a><br/>

<b>then consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>5</sub></font><b> such that </b><br/><a NAME="E7:4_3_1"/><i><font color="Green">E13</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="txt" href="topgen_3.html#E10:4"><i><font color="Green">E9</font></i></a>, <a class="txt" href="topgen_3.html#E15:4"><i><font color="Green">E11</font></i></a>, <a class="ref" href="cantor_1.html#D1">CANTOR_1:def 1</a></i>;<br/>
<b>assume </b><a NAME="E8:4_3_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>11</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E10:4_3_1"/><i><font color="Green">E14</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="txt" href="topgen_3.html#E7:4_3_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E11:4_3_1"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="topgen_3.html#E7:4_3_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="topgen_3.html#E10:4_3_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="card_5.html#T10">CARD_5:10</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>12</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E13:4_3_1"/><i><font color="Green">E15</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>11</sub></font> )
 <b>by </b><i><a class="txt" href="topgen_3.html#E10:4_3_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="topgen_3.html#E7:4"><i><font color="Green">E7</font></i></a>, <a class="txt" href="topgen_3.html#E2:4"><i><font color="Green">E5</font></i></a></i>;<br/>
<a NAME="E14:4_3_1"/>
( <font color="Maroon">c<sub>11</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="cantor_1.html#K1">UniCl</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="topgen_3.html#E10:4"><i><font color="Green">E9</font></i></a>, <a class="txt" href="topgen_3.html#E2:4"><i><font color="Green">E5</font></i></a>, <a class="txt" href="topgen_3.html#E15:4"><i><font color="Green">E11</font></i></a>, <a class="txt" href="topgen_3.html#E7:4_3_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="topgen_3.html#E10:4_3_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="topgen_3.html#E13:4_3_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a>, <a class="ref" href="cantor_1.html#T1">CANTOR_1:1</a>, <a class="ref" href="card_5.html#T10">CARD_5:10</a></i>;<br/>
<a NAME="E15:4_3_1"/><b>then </b>
( <font color="Maroon">c<sub>12</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="cantor_1.html#K1">UniCl</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="topgen_3.html#E13:4_3_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>hence </b><a NAME="E16:4_3_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>5</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>9</sub></font> )
 <b>by </b><i><a class="txt" href="topgen_3.html#E13:4_3_1"><i><font color="Green">E15</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:4_3"/>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> is    <a href="yellow_8.html#M1">Basis</a> of <font color="Maroon">c<sub>7</sub></font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E18:4"/>
<font color="Maroon">c<sub>2</sub></font> is    <a href="topgen_2.html#M1">Neighborhood_System</a> of <font color="Maroon">c<sub>5</sub></font>
 ;<br/>


</div>
