<?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>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>thus </b><a NAME="E1:74"/>
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#R2">in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><span class="p1">(<span class="default"><a href="setfam_1.html#K6">meet</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p3">(<span class="default"><a href="setfam_1.html#K6">meet</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1"><b>proof </b></a><div class="add">

<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/>

<b>assume </b><a NAME="E1:74_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E2:74_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="setfam_1.html#K6">meet</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p3">(<span class="default"><a href="setfam_1.html#K6">meet</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="ref" href="setfam_1.html#T4">SETFAM_1:4</a>, <a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<b>hence </b><a NAME="E3:74_1"/>
<span class="p1">(<span class="default"><a href="setfam_1.html#K6">meet</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p3">(<span class="default"><a href="setfam_1.html#K6">meet</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>


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

<b>thus </b><a NAME="E2:74"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or 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> is <a href="tops_1.html#V4">condensed</a> &amp; ( 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#R2">in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>2</sub></font> ) holds <br/><font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K6">meet</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p3">(<span class="default"><a href="setfam_1.html#K6">meet</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:74_2"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<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/>

<b>assume </b><a NAME="E2:74_2"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="tops_1.html#V4">condensed</a>
 ;<br/>

<b>assume </b><a NAME="E3:74_2"/>
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#R2">in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font>
 ;<br/>

<a NAME="E4:74_2"/><b>then </b>
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>2</sub></font> holds <br/><font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font>
 
;<br/>
<a NAME="E5:74_2"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="tdlat_2.html#E1:74_2"><i><font color="Green">E40</font></i></a>, <a class="ref" href="setfam_1.html#T6">SETFAM_1:6</a></i>;<br/>
<a NAME="E6:74_2"/><b>then </b>
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K6">meet</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="tops_1.html#T48">TOPS_1:48</a></i>;<br/>
<a NAME="E7:74_2"/><b>then </b>
( <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp;  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p2">(<span class="default"><a href="setfam_1.html#K6">meet</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="tdlat_2.html#E2:74_2"><i><font color="Green">E41</font></i></a>, <a class="ref" href="pre_topc.html#T49">PRE_TOPC:49</a>, <a class="ref" href="tops_1.html#D6">TOPS_1:def 6</a></i>;<br/>
<a NAME="E8:74_2"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p2">(<span class="default"><a href="setfam_1.html#K6">meet</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>hence </b><a NAME="E9:74_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K6">meet</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p3">(<span class="default"><a href="setfam_1.html#K6">meet</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="tdlat_2.html#E5:74_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="xboole_1.html#T19">XBOOLE_1:19</a></i>;<br/>


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


</div>
