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

<b>let </b><font color="Maroon" title="c1">T</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">F</font>, <font color="Olive" title="b2">G</font> being  <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b1">F</font> is <a href="tops_2.html#V2" title="TOPS_2:attr.2">closed</a> &amp; <font color="Olive" title="b2">G</font> is <a href="tops_2.html#V2" title="TOPS_2:attr.2">closed</a> holds <br/> <a href="topgen_4.html#K2" title="TOPGEN_4:func.2">INTERSECTION</a> <font color="Olive" title="b1">F</font>,<font color="Olive" title="b2">G</font> is <a href="tops_2.html#V2" title="TOPS_2:attr.2">closed</a></span><br/><b>let </b><font color="Maroon" title="c2">F</font>, <font color="Maroon" title="c3">G</font> be   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">F</font> is <a href="tops_2.html#V2" title="TOPS_2:attr.2">closed</a> &amp; <font color="Maroon" title="c3">G</font> is <a href="tops_2.html#V2" title="TOPS_2:attr.2">closed</a> implies  <a href="topgen_4.html#K2" title="TOPGEN_4:func.2">INTERSECTION</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">G</font> is <a href="tops_2.html#V2" title="TOPS_2:attr.2">closed</a> )</span><br/>





<b>assume </b><b>that </b><br/><a NAME="E1:42"/><i><font color="Green" title="E21">A1</font></i>: 
<font color="Maroon" title="c2">F</font> is <a href="tops_2.html#V2" title="TOPS_2:attr.2">closed</a>
 <b>and </b><br/><a NAME="E2:42"/><i><font color="Green" title="E22">A2</font></i>: 
<font color="Maroon" title="c3">G</font> is <a href="tops_2.html#V2" title="TOPS_2:attr.2">closed</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="topgen_4.html#K2" title="TOPGEN_4:func.2">INTERSECTION</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">G</font> is <a href="tops_2.html#V2" title="TOPS_2:attr.2">closed</a></span><br/>

<a NAME="E3:42"/>
for <font color="Olive" title="b1">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topgen_4.html#K2" title="TOPGEN_4:func.2">INTERSECTION</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">G</font> holds <br/><font color="Olive" title="b1">A</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topgen_4.html#K2" title="TOPGEN_4:func.2">INTERSECTION</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">G</font> implies <font color="Maroon" title="c4">A</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> )</span><br/>

<b>assume </b><a NAME="E1:42_1"/>
<font color="Maroon" title="c4">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topgen_4.html#K2" title="TOPGEN_4:func.2">INTERSECTION</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">A</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a></span><br/>

<b>then consider </b><font color="Maroon" title="c5">X</font>, <font color="Maroon" title="c6">Y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:42_1"/><i><font color="Green" title="E23">A3</font></i>: 
( <font color="Maroon" title="c5">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> &amp; <font color="Maroon" title="c6">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">G</font> &amp; <font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c6">Y</font> )
 <b>by </b><i><a class="ref" href="setfam_1.html#D5" title="SETFAM_1:def.5">SETFAM_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">X</font> = <font color="Maroon" title="c5">X</font>, <font color="Maroon" title="c8">Y</font> = <font color="Maroon" title="c6">Y</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> <b>by </b><i><a class="txt" href="#E3:42_1"><i><font color="Green" title="E23">A3</font></i></a></i>;<br/>
<a NAME="E5:42_1"/>
( <font color="Maroon" title="c7">X</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> &amp; <font color="Maroon" title="c8">Y</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> )
 
<b>by </b><i><a class="txt" href="#E1:42"><i><font color="Green" title="E21">A1</font></i></a>, <a class="txt" href="#E2:42"><i><font color="Green" title="E22">A2</font></i></a>, <a class="txt" href="#E3:42_1"><i><font color="Green" title="E23">A3</font></i></a>, <a class="ref" href="tops_2.html#D2" title="TOPS_2:def.2">TOPS_2:def 2</a></i>;<br/>
<b>hence </b><a NAME="E6:42_1"/>
<font color="Maroon" title="c4">A</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 <b>by </b><i><a class="txt" href="#E3:42_1"><i><font color="Green" title="E23">A3</font></i></a>, <a class="ref" href="tops_1.html#T35" title="TOPS_1:th.35">TOPS_1:35</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:42"/>
 <a href="topgen_4.html#K2" title="TOPGEN_4:func.2">INTERSECTION</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">G</font> is <a href="tops_2.html#V2" title="TOPS_2:attr.2">closed</a>
 <b>by </b><i><a class="ref" href="tops_2.html#D2" title="TOPS_2:def.2">TOPS_2:def 2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
