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

<b>let </b><font color="Maroon">TS</font> be   <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">K</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">TS</font>;<br/><b>let </b><font color="Maroon">L</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">TS</font>;<br/>





<a NAME="E1:55"/><i><font color="Green">E21</font></i>: 
<span class="p1">(<span class="default"><a href="tops_1.html#K2" target="_self">Fr</a> <font color="Maroon">K</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="tops_1.html#K2" target="_self">Fr</a> <font color="Maroon">L</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tops_1.html#K2" target="_self">Fr</a> <span class="p3">(<span class="default"><font color="Maroon">K</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">L</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="tops_1.html#K2" target="_self">Fr</a> <span class="p3">(<span class="default"><font color="Maroon">K</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">L</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tops_1.html#K2" target="_self">Fr</a> <font color="Maroon">K</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="tops_1.html#K2" target="_self">Fr</a> <font color="Maroon">L</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1"><b>proof </b></a><div class="add">

<a NAME="E1:55_1"/><i><font color="Green">E24</font></i>: 
 <a href="tops_1.html#K2" target="_self">Fr</a> <font color="Maroon">K</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tops_1.html#K2" target="_self">Fr</a> <span class="p3">(<span class="default"><font color="Maroon">K</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">L</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="tops_1.html#K2" target="_self">Fr</a> <span class="p3">(<span class="default"><font color="Maroon">K</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">L</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tops_1.html#K2" target="_self">Fr</a> <font color="Maroon">K</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="tops_1.html#K2" target="_self">Fr</a> <font color="Maroon">L</font></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E2:55_1"/>
 <a href="tops_1.html#K2" target="_self">Fr</a> <font color="Maroon">L</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tops_1.html#K2" target="_self">Fr</a> <span class="p3">(<span class="default"><font color="Maroon">K</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">L</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="tops_1.html#K2" target="_self">Fr</a> <span class="p3">(<span class="default"><font color="Maroon">K</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">L</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tops_1.html#K2" target="_self">Fr</a> <font color="Maroon">K</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="tops_1.html#K2" target="_self">Fr</a> <font color="Maroon">L</font></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:55_1"/>
<span class="p1">(<span class="default"><a href="tops_1.html#K2" target="_self">Fr</a> <font color="Maroon">K</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="tops_1.html#K2" target="_self">Fr</a> <font color="Maroon">L</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tops_1.html#K2" target="_self">Fr</a> <span class="p3">(<span class="default"><font color="Maroon">K</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">L</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="tops_1.html#K2" target="_self">Fr</a> <span class="p3">(<span class="default"><font color="Maroon">K</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">L</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tops_1.html#K2" target="_self">Fr</a> <font color="Maroon">K</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="tops_1.html#K2" target="_self">Fr</a> <font color="Maroon">L</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:55"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tops_1.html#K2" target="_self">Fr</a> <span class="p3">(<span class="default"><font color="Maroon">K</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">L</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="tops_1.html#K2" target="_self">Fr</a> <span class="p3">(<span class="default"><font color="Maroon">K</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">L</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tops_1.html#K2" target="_self">Fr</a> <font color="Maroon">K</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="tops_1.html#K2" target="_self">Fr</a> <font color="Maroon">L</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="tops_1.html#K2" target="_self">Fr</a> <font color="Maroon">K</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="tops_1.html#K2" target="_self">Fr</a> <font color="Maroon">L</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:55_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tops_1.html#K2" target="_self">Fr</a> <span class="p3">(<span class="default"><font color="Maroon">K</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">L</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="tops_1.html#K2" target="_self">Fr</a> <span class="p3">(<span class="default"><font color="Maroon">K</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">L</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tops_1.html#K2" target="_self">Fr</a> <font color="Maroon">K</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="tops_1.html#K2" target="_self">Fr</a> <font color="Maroon">L</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:55_2"/><b>then </b><i><font color="Green">E36</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="tops_1.html#K2" target="_self">Fr</a> <span class="p2">(<span class="default"><font color="Maroon">K</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">L</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="tops_1.html#K2" target="_self">Fr</a> <span class="p2">(<span class="default"><font color="Maroon">K</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">L</font></span>)</span></span>)</span> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="tops_1.html#K2" target="_self">Fr</a> <font color="Maroon">K</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="tops_1.html#K2" target="_self">Fr</a> <font color="Maroon">L</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<div><i><font color="Green">E37</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="55_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:55_2_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K2" target="_self">Fr</a> <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">L</font></span>)</span>
 ;<br/><a NAME="E2:55_2_1"/>
 <a href="tops_1.html#K2" target="_self">Fr</a> <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">L</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="tops_1.html#K2" target="_self">Fr</a> <font color="Maroon">K</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="tops_1.html#K2" target="_self">Fr</a> <font color="Maroon">L</font></span>)</span>
 <b>by </b><i><a class="txt" href="tops_1.html#E40"><i><font color="Green">Lemma34</font></i></a></i>;<br/><b>hence </b><a NAME="E3:55_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="tops_1.html#K2" target="_self">Fr</a> <font color="Maroon">K</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="tops_1.html#K2" target="_self">Fr</a> <font color="Maroon">L</font></span>)</span>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E39</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="55_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:55_2_2"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K2" target="_self">Fr</a> <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">L</font></span>)</span>
 ;<br/><a NAME="E2:55_2_2"/>
 <a href="tops_1.html#K2" target="_self">Fr</a> <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">L</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="tops_1.html#K2" target="_self">Fr</a> <font color="Maroon">K</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="tops_1.html#K2" target="_self">Fr</a> <font color="Maroon">L</font></span>)</span>
 <b>by </b><i><a class="ref" href="tops_1.html#T38" target="_self">Th38</a></i>;<br/><b>hence </b><a NAME="E3:55_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="tops_1.html#K2" target="_self">Fr</a> <font color="Maroon">K</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="tops_1.html#K2" target="_self">Fr</a> <font color="Maroon">L</font></span>)</span>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_2_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:55_2_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="tops_1.html#K2" target="_self">Fr</a> <font color="Maroon">K</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="tops_1.html#K2" target="_self">Fr</a> <font color="Maroon">L</font></span>)</span>
 ;<br/><a NAME="E2:55_2_3"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K2" target="_self">Fr</a> <font color="Maroon">K</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K2" target="_self">Fr</a> <font color="Maroon">L</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E3:55_2_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="tops_1.html#K2" target="_self">Fr</a> <font color="Maroon">K</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="tops_1.html#K2" target="_self">Fr</a> <font color="Maroon">L</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:55_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="tops_1.html#K2" target="_self">Fr</a> <font color="Maroon">K</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="tops_1.html#K2" target="_self">Fr</a> <font color="Maroon">L</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="tops_1.html#E4:55_2"><i><font color="Green">E39</font></i></a>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:55"/>
<span class="p1">(<span class="default"><a href="tops_1.html#K2" target="_self">Fr</a> <font color="Maroon">K</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="tops_1.html#K2" target="_self">Fr</a> <font color="Maroon">L</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tops_1.html#K2" target="_self">Fr</a> <span class="p3">(<span class="default"><font color="Maroon">K</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">L</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="tops_1.html#K2" target="_self">Fr</a> <span class="p3">(<span class="default"><font color="Maroon">K</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">L</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tops_1.html#K2" target="_self">Fr</a> <font color="Maroon">K</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="tops_1.html#K2" target="_self">Fr</a> <font color="Maroon">L</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div>
