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

<b>let </b><font color="Maroon">c<sub>1</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="E1:39"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="compts_1.html#V2">compact</a>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>set </b><font color="Maroon">c<sub>3</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is    <a href="yellow19.html#M1" target="_self">Subset</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> : verum </span>}</span> ;<br/>
<a NAME="E2:39"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is    <a href="yellow19.html#M1" target="_self">Subset</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> : verum </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</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="E1:39_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is    <a href="yellow19.html#M1" target="_self">Subset</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> : verum </span>}</span> 
 ;<br/>

<a NAME="E2:39_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="yellow19.html#M1" target="_self">Subset</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> st <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Olive">b<sub>1</sub></font>
 
;<br/>
<b>hence </b><a NAME="E3:39_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is    <a href="yellow19.html#M1" target="_self">Subset</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> : verum </span>}</span>  as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<b>consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">c<sub>4</sub></font>;<br/>
<a NAME="E5:39"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>4</sub></font> is <a href="tops_2.html#V2">closed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="tops_2.html#D2">TOPS_2:def 2</a><br/>

<b>assume </b><a NAME="E1:39_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E2:39_2"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="yellow19.html#M1" target="_self">Subset</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> st <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Olive">b<sub>1</sub></font>
 
;<br/>
<b>hence </b><a NAME="E3:39_2"/>
<font color="Maroon">c<sub>6</sub></font> is <a href="pre_topc.html#V4">closed</a>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E6:39"/>
<font color="Maroon">c<sub>4</sub></font> is <a href="compts_1.html#V1">centered</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_3"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="yellow19.html#M1" target="_self">Subset</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>
<a NAME="E2:39_3"/>
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 
;<br/>
<b>hence </b><a NAME="E3:39_3"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="compts_1.html#D2">COMPTS_1:def 2</a><br/>

<b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E4:39_3"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> is <a href="finset_1.html#V1">finite</a> )
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] means ex <font color="Olive">b<sub>1</sub></font> being   <a href="yellow19.html#M1" target="_self">Subset</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>ex <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel_9.html#K6">|</a> <font color="Olive">b<sub>2</sub></font></span>)</span> );<br/>
<div><i><font color="Green">E28</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="39_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:39_3_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/><a NAME="E2:39_3_1"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="yellow19.html#E4:39_3"><i><font color="Green">E27</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>9</sub></font> being    <a href="yellow19.html#M1" target="_self">Subset</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E4:39_3_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/><b>consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E6:39_3_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel_9.html#K6">|</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow19.html#D2" target="_self">Def2</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font> as    <a href="hidden.html#M1">set</a>  ;<br/><b>take </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>11</sub></font>;<br/><b>thus </b><a NAME="E8:39_3_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 ;<br/><b>thus </b><a NAME="E9:39_3_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>12</sub></font>]
 <b>by </b><i><a class="txt" href="yellow19.html#E4:39_3_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="yellow19.html#E6:39_3_1"><i><font color="Green">E30</font></i></a></i>;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E7:39_3"/><i><font color="Green">E29</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> )
 <b>and </b><br/><a NAME="E8:39_3"/><i><font color="Green">E30</font></i>: 
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>7</sub></font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="wellord2.html#S1">WELLORD2:sch 1</a>(<a class="txt" href="yellow19.html#E5:39_3"><i><font color="Green">E28</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> =  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>8</sub></font> as  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="yellow19.html#E4:39_3"><i><font color="Green">E27</font></i></a>, <a class="txt" href="yellow19.html#E7:39_3"><i><font color="Green">E29</font></i></a>, <a class="ref" href="finset_1.html#T26">FINSET_1:26</a></i>;<br/>
<a NAME="E10:39_3"/>
(  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font> is <a href="waybel_0.html#V1">directed</a> &amp;  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a>, <a class="ref" href="waybel_0.html#D6">WAYBEL_0:def 6</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E12:39_3"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E13:39_3"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="ref" href="waybel_0.html#T1">WAYBEL_0:1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_3_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>11</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:39_3_2"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>12</sub></font> being    <a href="yellow19.html#M1" target="_self">Subset</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>13</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E3:39_3_2"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel_9.html#K6">|</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="yellow19.html#E8:39_3"><i><font color="Green">E30</font></i></a></i>;<br/><a NAME="E4:39_3_2"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="yellow19.html#E7:39_3"><i><font color="Green">E29</font></i></a>, <a class="txt" href="yellow19.html#E1:39_3_2"><i><font color="Green">E32</font></i></a>, <a class="txt" href="yellow19.html#E3:39_3_2"><i><font color="Green">E33</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E5:39_3_2"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="yellow19.html#E13:39_3"><i><font color="Green">E31</font></i></a>, <a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a></i>;<br/><a NAME="E6:39_3_2"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel_9.html#K6">|</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="waybel_9.html#D7">WAYBEL_9:def 7</a></i>;<br/><a NAME="E7:39_3_2"/><b>then </b>
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel_9.html#K6">|</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> &amp; the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel_9.html#K6">|</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel_9.html#K6">|</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> )
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="waybel_9.html#D7">WAYBEL_9:def 7</a></i>;<br/><a NAME="E8:39_3_2"/><b>then </b>
( the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel_9.html#K6">|</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font> &amp; the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel_9.html#K6">|</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="txt" href="yellow19.html#E3:39_3_2"><i><font color="Green">E33</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E9:39_3_2"/><b>then </b>
( <font color="Maroon">c<sub>2</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</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="yellow19.html#E3:39_3_2"><i><font color="Green">E33</font></i></a>, <a class="ref" href="pre_topc.html#T48">PRE_TOPC:48</a></i>;<br/><b>hence </b><a NAME="E10:39_3_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E15:39_3"/>
 <a href="setfam_1.html#K1">meet</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="yellow19.html#E4:39_3"><i><font color="Green">E27</font></i></a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:39"/><b>then </b><i><font color="Green">E27</font></i>: 
 <a href="setfam_1.html#K6">meet</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="yellow19.html#E1:39"><i><font color="Green">E25</font></i></a>, <a class="txt" href="yellow19.html#E5:39"><i><font color="Green">E26</font></i></a>, <a class="ref" href="compts_1.html#T13">COMPTS_1:13</a></i>;<br/>
<a NAME="E8:39"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K6">meet</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>5</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<b>take </b>
<font color="Maroon">c<sub>6</sub></font>
;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="yellow19.html#M1" target="_self">Subset</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/><a NAME="E1:39_4"/>
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><b>hence </b><a NAME="E2:39_4"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="yellow19.html#E7:39"><i><font color="Green">E27</font></i></a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E11:39"/>
<font color="Maroon">c<sub>6</sub></font> <a href="waybel_9.html#R3">is_a_cluster_point_of</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="yellow19.html#T31" target="_self">Th31</a></i>;<br/>


</div>
