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

<b>let </b><font color="Maroon" title="c1">Y</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a>  <a href="pre_topc.html#L1" title="PRE_TOPC:struct.1">TopStruct</a> ;<br/><b>let </b><font color="Maroon" title="c2">F</font> be   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">Y</font>;<br/>



<b>assume </b><a NAME="E1:20"/><i><font color="Green" title="E13">A1</font></i>: 
<font color="Maroon" title="c2">F</font> is <a href="tex_4.html#V2" title="TEX_4:attr.2">anti-discrete-set-family</a>
 ;<br/>

<b>assume </b><a NAME="E2:20"/><i><font color="Green" title="E14">A2</font></i>: 
 <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c2">F</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/>

<a NAME="E3:20"/>
for <font color="Olive" title="b1">G</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font>  holds <br/>(  not <font color="Olive" title="b1">G</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> or  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c2">F</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b1">G</font> or  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c2">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">G</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">G</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font>;<br/>

<b>assume </b><a NAME="E1:20_1"/><i><font color="Green" title="E15">A3</font></i>: 
<font color="Maroon" title="c3">G</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 ;<br/>

<b>assume </b><a NAME="E2:20_1"/>
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c2">F</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">G</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c4">A0</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:20_1"/><i><font color="Green" title="E16">A4</font></i>: 
<font color="Maroon" title="c4">A0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font>
 <b>and </b><br/><a NAME="E5:20_1"/><i><font color="Green" title="E17">A5</font></i>: 
<font color="Maroon" title="c4">A0</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">G</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T98" title="ZFMISC_1:th.98">ZFMISC_1:98</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">A0</font> = <font color="Maroon" title="c4">A0</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font> <b>by </b><i><a class="txt" href="tex_4.html#E4:20_1"><i><font color="Green" title="E16">A4</font></i></a></i>;<br/>
<a NAME="E7:20_1"/>
<font color="Maroon" title="c5">A0</font> is <a href="tex_4.html#V1" title="TEX_4:attr.1">anti-discrete</a>
 
<b>by </b><i><a class="txt" href="tex_4.html#E1:20"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="tex_4.html#E4:20_1"><i><font color="Green" title="E16">A4</font></i></a>, <a class="ref" href="tex_4.html#D5" target="_self" title="TEX_4:def.5">Def5</a></i>;<br/>
<a NAME="E8:20_1"/><b>then </b><i><font color="Green" title="E18">A6</font></i>: 
<font color="Maroon" title="c5">A0</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">G</font>
 
<b>by </b><i><a class="txt" href="tex_4.html#E1:20_1"><i><font color="Green" title="E15">A3</font></i></a>, <a class="txt" href="tex_4.html#E5:20_1"><i><font color="Green" title="E17">A5</font></i></a>, <a class="ref" href="tex_4.html#D3" target="_self" title="TEX_4:def.3">Def3</a></i>;<br/>
<a NAME="E9:20_1"/>
 <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c2">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">A0</font>
 
<b>by </b><i><a class="txt" href="tex_4.html#E4:20_1"><i><font color="Green" title="E16">A4</font></i></a>, <a class="ref" href="setfam_1.html#T4" title="SETFAM_1:th.4">SETFAM_1:4</a></i>;<br/>
<a NAME="E10:20_1"/><b>then </b><i><font color="Green" title="E19">A7</font></i>: 
 <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c2">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">G</font>
 
<b>by </b><i><a class="txt" href="tex_4.html#E8:20_1"><i><font color="Green" title="E18">A6</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E11:20_1"/>
for <font color="Olive" title="b1">B</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> holds <br/><font color="Olive" title="b1">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">G</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">B</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:20_1_1"/><i><font color="Green" title="E20">A8</font></i>: 
<font color="Maroon" title="c6">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c7">B0</font> = <font color="Maroon" title="c6">B</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font> ;<br/>
<a NAME="E3:20_1_1"/>
 <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c2">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">B0</font>
 
<b>by </b><i><a class="txt" href="tex_4.html#E1:20_1_1"><i><font color="Green" title="E20">A8</font></i></a>, <a class="ref" href="setfam_1.html#T4" title="SETFAM_1:th.4">SETFAM_1:4</a></i>;<br/>
<a NAME="E4:20_1_1"/><b>then </b>
 <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c2">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">B0</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">G</font>
 
<b>by </b><i><a class="txt" href="tex_4.html#E10:20_1"><i><font color="Green" title="E19">A7</font></i></a>, <a class="ref" href="xboole_1.html#T19" title="XBOOLE_1:th.19">XBOOLE_1:19</a></i>;<br/>
<a NAME="E5:20_1_1"/><b>then </b>
<font color="Maroon" title="c7">B0</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">G</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="tex_4.html#E2:20"><i><font color="Green" title="E14">A2</font></i></a>, <a class="ref" href="xboole_1.html#T3" title="XBOOLE_1:th.3">XBOOLE_1:3</a></i>;<br/>
<a NAME="E6:20_1_1"/><b>then </b><i><font color="Green" title="E21">A9</font></i>: 
<font color="Maroon" title="c7">B0</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">G</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E7:20_1_1"/>
<font color="Maroon" title="c7">B0</font> is <a href="tex_4.html#V1" title="TEX_4:attr.1">anti-discrete</a>
 
<b>by </b><i><a class="txt" href="tex_4.html#E1:20"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="tex_4.html#E1:20_1_1"><i><font color="Green" title="E20">A8</font></i></a>, <a class="ref" href="tex_4.html#D5" target="_self" title="TEX_4:def.5">Def5</a></i>;<br/>
<b>hence </b><a NAME="E8:20_1_1"/>
<font color="Maroon" title="c6">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">G</font>
 <b>by </b><i><a class="txt" href="tex_4.html#E1:20_1"><i><font color="Green" title="E15">A3</font></i></a>, <a class="txt" href="tex_4.html#E6:20_1_1"><i><font color="Green" title="E21">A9</font></i></a>, <a class="ref" href="tex_4.html#D3" target="_self" title="TEX_4:def.3">Def3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E12:20_1"/>
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c2">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">G</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T94" title="ZFMISC_1:th.94">ZFMISC_1:94</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:20"/>
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c2">F</font> is <a href="tex_4.html#V1" title="TEX_4:attr.1">anti-discrete</a>
 <b>by </b><i><a class="ref" href="tex_4.html#D3" target="_self" title="TEX_4:def.3">Def3</a></i>;<br/>


</div>
