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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be    <a href="pre_topc.html#L1">TopStruct</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:43"/><i><font color="Green">E12</font></i>: 
 <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font> #) <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>2</sub></font> #)
 <b>and </b><br/><a NAME="E2:43"/><i><font color="Green">E13</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="pre_topc.html#R1">is_a_cover_of</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="tops_2.html#V1">open</a> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="pre_topc.html#R1">is_a_cover_of</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> is <a href="finset_1.html#V1">finite</a> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="compts_1.html#D3">COMPTS_1:def 3</a><br/>

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>2</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="compts_1.html#D3">COMPTS_1:def 3</a><br/>

<b>assume </b><a NAME="E3:43"/><i><font color="Green">E14</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="pre_topc.html#R1">is_a_cover_of</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> is <a href="tops_2.html#V1">open</a> )
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="yellow14.html#E1:43"><i><font color="Green">E12</font></i></a></i>;<br/>
<a NAME="E5:43"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="pre_topc.html#R1">is_a_cover_of</a> <font color="Maroon">c<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1"><b>proof </b></a><div class="add">

<b>thus </b> <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font> = 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>

<br/>.= 
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="yellow14.html#E1:43"><i><font color="Green">E12</font></i></a></i>
<br/>.= 
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="yellow14.html#E3:43"><i><font color="Green">E14</font></i></a>, <a class="ref" href="pre_topc.html#D8">PRE_TOPC:def 8</a></i>
; <i><font color="Red">:: according to </font></i><a class="ref" href="pre_topc.html#D8">PRE_TOPC:def 8</a><br/>


</div><b>end;</b></div>
<a NAME="E6:43"/>
<font color="Maroon">c<sub>4</sub></font> is <a href="tops_2.html#V1">open</a>
 
<b>by </b><i><a class="txt" href="yellow14.html#E1:43"><i><font color="Green">E12</font></i></a>, <a class="txt" href="yellow14.html#E3:43"><i><font color="Green">E14</font></i></a>, <a class="ref" href="waybel_9.html#T19">WAYBEL_9:19</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E8:43"/><i><font color="Green">E16</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="pre_topc.html#R1">is_a_cover_of</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> is <a href="finset_1.html#V1">finite</a> )
 <b>by </b><i><a class="txt" href="yellow14.html#E2:43"><i><font color="Green">E13</font></i></a>, <a class="txt" href="yellow14.html#E5:43"><i><font color="Green">E15</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="yellow14.html#E1:43"><i><font color="Green">E12</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>6</sub></font>
;<br/>

<a NAME="E10:43"/>
<font color="Maroon">c<sub>6</sub></font> <a href="pre_topc.html#R1">is_a_cover_of</a> <font color="Maroon">c<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_2"><b>proof </b></a><div class="add">

<b>thus </b> <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font> = 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>

<br/>.= 
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="yellow14.html#E1:43"><i><font color="Green">E12</font></i></a></i>
<br/>.= 
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="txt" href="yellow14.html#E8:43"><i><font color="Green">E16</font></i></a>, <a class="ref" href="pre_topc.html#D8">PRE_TOPC:def 8</a></i>
; <i><font color="Red">:: according to </font></i><a class="ref" href="pre_topc.html#D8">PRE_TOPC:def 8</a><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E11:43"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="pre_topc.html#R1">is_a_cover_of</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> is <a href="finset_1.html#V1">finite</a> )
 <b>by </b><i><a class="txt" href="yellow14.html#E8:43"><i><font color="Green">E16</font></i></a></i>;<br/>


</div>
