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

<b>let </b><font color="Maroon">T</font> be   <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">F</font> be   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font>, <font color="Maroon">G</font> be   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font>;<br/>



<b>thus </b><a NAME="E1:13"/>
 <a href="topgen_1.html#K1" target="_self">Fr</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">G</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="topgen_1.html#K1" target="_self">Fr</a> <font color="Maroon">F</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topgen_1.html#K1" target="_self">Fr</a> <font color="Maroon">G</font></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1"><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="E1:13_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topgen_1.html#K1" target="_self">Fr</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">G</font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">A</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> ;<br/>
<b>consider </b><font color="Maroon">B</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E4:13_1"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="tops_1.html#K2">Fr</a> <font color="Maroon">B</font> &amp; <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">G</font> )
 <b>by </b><i>, </i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:13_1_1"/>
( <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> or <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> )
 </span><b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_1_1_1"><b>suppose </b></a><a NAME="E1:13_1_1_1"/>
<font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<br/><div class="add"><a NAME="E2:13_1_1_1"/><b>then </b>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="topgen_1.html#K1" target="_self">Fr</a> <font color="Maroon">F</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:13_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topgen_1.html#K1" target="_self">Fr</a> <font color="Maroon">F</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topgen_1.html#K1" target="_self">Fr</a> <font color="Maroon">G</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_1_1_2"><b>suppose </b></a><a NAME="E1:13_1_1_2"/>
<font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 ;<br/><div class="add"><a NAME="E2:13_1_1_2"/><b>then </b>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="topgen_1.html#K1" target="_self">Fr</a> <font color="Maroon">G</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:13_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topgen_1.html#K1" target="_self">Fr</a> <font color="Maroon">F</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topgen_1.html#K1" target="_self">Fr</a> <font color="Maroon">G</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></div><b>end;</b></div>

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

<a NAME="E2:13"/>
<font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E3:13"/><b>then </b><i><font color="Green">E42</font></i>: 
 <a href="topgen_1.html#K1" target="_self">Fr</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a>  <a href="topgen_1.html#K1" target="_self">Fr</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">G</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E4:13"/>
<font color="Maroon">G</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E5:13"/><b>then </b>
 <a href="topgen_1.html#K1" target="_self">Fr</a> <font color="Maroon">G</font> <a href="tarski.html#R1">c=</a>  <a href="topgen_1.html#K1" target="_self">Fr</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">G</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E6:13"/>
<span class="p1">(<span class="default"><a href="topgen_1.html#K1" target="_self">Fr</a> <font color="Maroon">F</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topgen_1.html#K1" target="_self">Fr</a> <font color="Maroon">G</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="topgen_1.html#K1" target="_self">Fr</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">G</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>


</div>
