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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="pre_topc.html#L1">TopStruct</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>





<a NAME="E1:8"/>
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D9">PRE_TOPC:def 9</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> =  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<b>assume </b><a NAME="E3:8"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E4:8"/><b>then </b><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>
<b>thus </b><a NAME="E5:8"/>
( <font color="Maroon">c<sub>3</sub></font> is <a href="compts_1.html#V6" target="_self">compact</a> implies for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> is <a href="compts_1.html#V6" target="_self">compact</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:8_1"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="compts_1.html#V6" target="_self">compact</a>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>

<b>assume </b><a NAME="E2:8_1"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>thus </b><a NAME="E3:8_1"/>
<font color="Maroon">c<sub>5</sub></font> is <a href="compts_1.html#V6" target="_self">compact</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</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#D7" target="_self">COMPTS_1:def 7</a><br/>

<b>set </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font>;<br/>
<b>assume </b><b>that </b><br/><a NAME="E1:8_1_1"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="compts_1.html#R1" target="_self">is_a_cover_of</a> <font color="Maroon">c<sub>5</sub></font>
 <b>and </b><br/><a NAME="E2:8_1_1"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>6</sub></font> is <a href="tops_2.html#V1">open</a>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>8</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="E4:8_1_1"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>8</sub></font> is <a href="tops_2.html#V1">open</a>
 <b>and </b><br/><a NAME="E5:8_1_1"/><i><font color="Green">E16</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="tops_2.html#K1">|</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="compts_1.html#E2:8_1_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="tops_2.html#T49">TOPS_2:49</a></i>;<br/>
<a NAME="E6:8_1_1"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>6</sub></font> &amp;  <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="tops_2.html#K1">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>8</sub></font> )
 
<b>by </b><i><a class="txt" href="compts_1.html#E1:8_1_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="compts_1.html#D1" target="_self">Def1</a>, <a class="ref" href="tops_2.html#T44">TOPS_2:44</a></i>;<br/>
<a NAME="E7:8_1_1"/><b>then </b>
( <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>6</sub></font> &amp;  <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>8</sub></font> )
 
<b>by </b><i><a class="txt" href="compts_1.html#E2:8_1"><i><font color="Green">E12</font></i></a>, <a class="txt" href="compts_1.html#E5:8_1_1"><i><font color="Green">E16</font></i></a></i>;<br/>
<a NAME="E8:8_1_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E9:8_1_1"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="compts_1.html#R1" target="_self">is_a_cover_of</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="compts_1.html#D1" target="_self">Def1</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>9</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="E11:8_1_1"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>and </b><br/><a NAME="E12:8_1_1"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="compts_1.html#R1" target="_self">is_a_cover_of</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E13:8_1_1"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>9</sub></font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="txt" href="compts_1.html#E1:8_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="compts_1.html#E4:8_1_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="compts_1.html#D7" target="_self">Def7</a></i>;<br/>
<a NAME="E14:8_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="tops_2.html#K1">|</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
 
;<br/>
<a NAME="E15:8_1_1"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="tops_2.html#K1">|</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/>
<a NAME="E16:8_1_1"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="tops_2.html#K1">|</a> <font color="Maroon">c<sub>4</sub></font> <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>2</sub></font>
 
;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</sub></font> <a href="tops_2.html#K1">|</a> <font color="Maroon">c<sub>4</sub></font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>2</sub></font> ;<br/>
<b>take </b>
<font color="Maroon">c<sub>10</sub></font>
;<br/>

<a NAME="E18:8_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="tops_2.html#K1">|</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>8</sub></font> <a href="tops_2.html#K1">|</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="compts_1.html#E11:8_1_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="tops_2.html#T40">TOPS_2:40</a></i>;<br/>
<b>hence </b><a NAME="E19:8_1_1"/>
<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="compts_1.html#E5:8_1_1"><i><font color="Green">E16</font></i></a></i>;<br/>

<a NAME="E20:8_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="compts_1.html#E12:8_1_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="compts_1.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E21:8_1_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</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="compts_1.html#E4:8"><i><font color="Green">E10</font></i></a>, <a class="txt" href="compts_1.html#E2:8_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="tops_2.html#T42">TOPS_2:42</a></i>;<br/>
<b>hence </b><a NAME="E22:8_1_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="compts_1.html#R1" target="_self">is_a_cover_of</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="compts_1.html#D1" target="_self">Def1</a></i>;<br/>

<b>thus </b><a NAME="E23:8_1_1"/>
<font color="Maroon">c<sub>10</sub></font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="txt" href="compts_1.html#E13:8_1_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="tops_2.html#T46">TOPS_2:46</a></i>;<br/>


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


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

<b>thus </b><a NAME="E6:8"/>
( ( for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> is <a href="compts_1.html#V6" target="_self">compact</a> ) implies <font color="Maroon">c<sub>3</sub></font> is <a href="compts_1.html#V6" target="_self">compact</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:8_2"/><i><font color="Green">E11</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> is <a href="compts_1.html#V6" target="_self">compact</a>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <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="compts_1.html#E3:8"><i><font color="Green">E9</font></i></a></i>;<br/>
<a NAME="E3:8_2"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>5</sub></font> is <a href="compts_1.html#V6" target="_self">compact</a>
 
<b>by </b><i><a class="txt" href="compts_1.html#E1:8_2"><i><font color="Green">E11</font></i></a></i>;<br/>
<b>thus </b><a NAME="E4:8_2"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="compts_1.html#V6" target="_self">compact</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="compts_1.html#D7" target="_self">COMPTS_1:def 7</a><br/>

<b>set </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font>;<br/>
<b>assume </b><b>that </b><br/><a NAME="E1:8_2_1"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="compts_1.html#R1" target="_self">is_a_cover_of</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E2:8_2_1"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>6</sub></font> is <a href="tops_2.html#V1">open</a>
 ;<br/>

<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="E4:8_2_1"/><i><font color="Green">E15</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>6</sub></font>
 <b>and </b><br/><a NAME="E5:8_2_1"/><i><font color="Green">E16</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="tops_2.html#K1">|</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E6:8_2_1"/><i><font color="Green">E17</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>6</sub></font> holds <br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="tops_2.html#T50">TOPS_2:50</a></i>;<br/>
<a NAME="E7:8_2_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="tops_2.html#K1">|</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
 
;<br/>
<a NAME="E8:8_2_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="tops_2.html#K1">|</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/>
<a NAME="E9:8_2_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="tops_2.html#K1">|</a> <font color="Maroon">c<sub>4</sub></font> <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>2</sub></font>
 
;<br/>
<b>then 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="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="compts_1.html#E5:8_2_1"><i><font color="Green">E16</font></i></a></i>;<br/>
<a NAME="E11:8_2_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="compts_1.html#E1:8_2_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="compts_1.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E12:8_2_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="compts_1.html#E4:8"><i><font color="Green">E10</font></i></a>, <a class="txt" href="compts_1.html#E5:8_2_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="tops_2.html#T42">TOPS_2:42</a></i>;<br/>
<a NAME="E13:8_2_1"/><b>then </b><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="compts_1.html#R1" target="_self">is_a_cover_of</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="compts_1.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E14:8_2_1"/>
<font color="Maroon">c<sub>9</sub></font> is <a href="tops_2.html#V1">open</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>10</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="tops_2.html#D1">TOPS_2:def 1</a><br/>

<b>assume </b><a NAME="E1:8_2_1_1"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <b>by </b><i><a class="txt" href="compts_1.html#E5:8_2_1"><i><font color="Green">E16</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>12</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:8_2_1_1"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> )
 <b>by </b><i><a class="txt" href="compts_1.html#E5:8_2_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="compts_1.html#E1:8_2_1_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="tops_2.html#D3">TOPS_2:def 3</a></i>;<br/>
<a NAME="E5:8_2_1_1"/>
<font color="Maroon">c<sub>12</sub></font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="txt" href="compts_1.html#E2:8_2_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="compts_1.html#E4:8_2_1_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="tops_2.html#D1">TOPS_2:def 1</a></i>;<br/>
<a NAME="E6:8_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E7:8_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="compts_1.html#E4:8_2_1_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="pre_topc.html#D9">PRE_TOPC:def 9</a></i>;<br/>
<b>hence </b><a NAME="E8:8_2_1_1"/>
<font color="Maroon">c<sub>10</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/>


</div><b>end;</b></div>
<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>2</sub></font><b> such that </b><br/><a NAME="E16:8_2_1"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>9</sub></font>
 <b>and </b><br/><a NAME="E17:8_2_1"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="compts_1.html#R1" target="_self">is_a_cover_of</a> <font color="Maroon">c<sub>5</sub></font>
 <b>and </b><br/><a NAME="E18:8_2_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>10</sub></font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="txt" href="compts_1.html#E3:8_2"><i><font color="Green">E12</font></i></a>, <a class="txt" href="compts_1.html#E13:8_2_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="compts_1.html#D7" target="_self">Def7</a></i>;<br/>
<b>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="E20:8_2_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>8</sub></font>
 <b>and </b><br/><a NAME="E21:8_2_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>11</sub></font> is <a href="finset_1.html#V1">finite</a>
 <b>and </b><br/><a NAME="E22:8_2_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="compts_1.html#E16:8_2_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="compts_1.html#E18:8_2_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="orders_1.html#T195">ORDERS_1:195</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>11</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="compts_1.html#E4:8_2_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="compts_1.html#E20:8_2_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="tops_2.html#T3">TOPS_2:3</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>12</sub></font>
;<br/>

<a NAME="E24:8_2_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>12</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_2_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>13</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="E1:8_2_1_2"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E2:8_2_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</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="compts_1.html#E17:8_2_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="compts_1.html#D1" target="_self">Def1</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>14</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:8_2_1_2"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font>
 <b>and </b><br/><a NAME="E5:8_2_1_2"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>14</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="compts_1.html#E1:8_2_1_2"><i><font color="Green">E25</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>15</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:8_2_1_2"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>8</sub></font>
 <b>and </b><br/><a NAME="E8:8_2_1_2"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font>
 <b>and </b><br/><a NAME="E9:8_2_1_2"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="compts_1.html#E22:8_2_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="compts_1.html#E5:8_2_1_2"><i><font color="Green">E27</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>15</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="compts_1.html#E4:8_2_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="compts_1.html#E7:8_2_1_2"><i><font color="Green">E28</font></i></a></i>;<br/>
<a NAME="E11:8_2_1_2"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="compts_1.html#E4:8_2_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="compts_1.html#E6:8_2_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="compts_1.html#E7:8_2_1_2"><i><font color="Green">E28</font></i></a>, <a class="txt" href="compts_1.html#E9:8_2_1_2"><i><font color="Green">E30</font></i></a></i>;<br/>
<a NAME="E12:8_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>16</sub></font>
 
<b>by </b><i><a class="txt" href="compts_1.html#E4:8_2_1_2"><i><font color="Green">E26</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E13:8_2_1_2"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="compts_1.html#E8:8_2_1_2"><i><font color="Green">E29</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E25:8_2_1"/>
( <font color="Maroon">c<sub>12</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="compts_1.html#R1" target="_self">is_a_cover_of</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> is <a href="finset_1.html#V1">finite</a> )
 <b>by </b><i><a class="txt" href="compts_1.html#E4:8_2_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="compts_1.html#E20:8_2_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="compts_1.html#E21:8_2_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="compts_1.html#D1" target="_self">Def1</a></i>;<br/>


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


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


</div>
