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

<b>set </b><font color="Maroon" title="c3">T</font> =  <a href="topgen_2.html#K5" title="TOPGEN_2:func.5">DiscrWithInfin</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">x0</font>;<br/>
<b>set </b><font color="Maroon" title="c4">O1</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">U</font> where <font color="Olive" title="b1">U</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> : not <font color="Maroon" title="c2">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">U</font> </span>}</span> ;<br/>
<b>set </b><font color="Maroon" title="c5">O2</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">F</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> where <font color="Olive" title="b1">F</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> : <font color="Olive" title="b1">F</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> </span>}</span> ;<br/>
<b>set </b><font color="Maroon" title="c6">O</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">U</font> where <font color="Olive" title="b1">U</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> : not <font color="Maroon" title="c2">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">U</font> </span>}</span>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">F</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> where <font color="Olive" title="b1">F</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> : <font color="Olive" title="b1">F</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> </span>}</span> ;<br/>
<a NAME="E1:35_1_1"/><i><font color="Green" title="E23">A1</font></i>: 
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="topgen_2.html#K5" title="TOPGEN_2:func.5">DiscrWithInfin</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">x0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> &amp; the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="topgen_2.html#K5" title="TOPGEN_2:func.5">DiscrWithInfin</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">x0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">U</font> where <font color="Olive" title="b1">U</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> : not <font color="Maroon" title="c2">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">U</font> </span>}</span>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">F</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> where <font color="Olive" title="b1">F</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> : <font color="Olive" title="b1">F</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> </span>}</span>  )
 
<b>by </b><i><a class="ref" href="topgen_2.html#D5" target="_self" title="TOPGEN_2:def.5">Def5</a></i>;<br/>
<a NAME="E2:35_1_1"/>
<span class="p1">(<span class="default"><a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font>
 
;<br/>
<a NAME="E3:35_1_1"/><b>then </b>
<font color="Maroon" title="c1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">F</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> where <font color="Olive" title="b1">F</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> : <font color="Olive" title="b1">F</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> </span>}</span> 
 
;<br/>
<b>hence </b><a NAME="E4:35_1_1"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="topgen_2.html#K5" title="TOPGEN_2:func.5">DiscrWithInfin</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">x0</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="topgen_2.html#K5" title="TOPGEN_2:func.5">DiscrWithInfin</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">x0</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:35_1_1"><i><font color="Green" title="E23">A1</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="pre_topc.html#D1" title="PRE_TOPC:def.1">PRE_TOPC:def 1</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">K10</a>(<a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">K10</a>(the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="topgen_2.html#K5" title="TOPGEN_2:func.5">DiscrWithInfin</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">x0</font></span>)</span>))  holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="topgen_2.html#K5" title="TOPGEN_2:func.5">DiscrWithInfin</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">x0</font></span>)</span> or  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="topgen_2.html#K5" title="TOPGEN_2:func.5">DiscrWithInfin</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">x0</font></span>)</span> ) ) &amp; ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">K10</a>(the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="topgen_2.html#K5" title="TOPGEN_2:func.5">DiscrWithInfin</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">x0</font></span>)</span>)  holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="topgen_2.html#K5" title="TOPGEN_2:func.5">DiscrWithInfin</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">x0</font></span>)</span> or  not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="topgen_2.html#K5" title="TOPGEN_2:func.5">DiscrWithInfin</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">x0</font></span>)</span> or <font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="topgen_2.html#K5" title="TOPGEN_2:func.5">DiscrWithInfin</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">x0</font></span>)</span> ) ) )</span><br/>

<div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">K10</a>(the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="topgen_2.html#K5" title="TOPGEN_2:func.5">DiscrWithInfin</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">x0</font></span>)</span>)  holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="topgen_2.html#K5" title="TOPGEN_2:func.5">DiscrWithInfin</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">x0</font></span>)</span> or  not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="topgen_2.html#K5" title="TOPGEN_2:func.5">DiscrWithInfin</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">x0</font></span>)</span> or <font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="topgen_2.html#K5" title="TOPGEN_2:func.5">DiscrWithInfin</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">x0</font></span>)</span> )</span>
<div class="add"><b>let </b><font color="Maroon" title="c7">a</font> be   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <span class="p1">(<span class="default"><a href="topgen_2.html#K5" title="TOPGEN_2:func.5">DiscrWithInfin</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">x0</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="topgen_2.html#K5" title="TOPGEN_2:func.5">DiscrWithInfin</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">x0</font></span>)</span> implies  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="topgen_2.html#K5" title="TOPGEN_2:func.5">DiscrWithInfin</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">x0</font></span>)</span> )</span><br/><b>assume </b><a NAME="E1:35_1_1_1"/><i><font color="Green" title="E24">A2</font></i>: 
<font color="Maroon" title="c7">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="topgen_2.html#K5" title="TOPGEN_2:func.5">DiscrWithInfin</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">x0</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="topgen_2.html#K5" title="TOPGEN_2:func.5">DiscrWithInfin</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">x0</font></span>)</span></span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:35_1_1_1_1"/>
(  not <font color="Maroon" title="c7">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">U</font> where <font color="Olive" title="b1">U</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> : not <font color="Maroon" title="c2">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">U</font> </span>}</span>  or <font color="Maroon" title="c7">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">U</font> where <font color="Olive" title="b1">U</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> : not <font color="Maroon" title="c2">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">U</font> </span>}</span>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:35_1_1_1_1_1"/>
not <font color="Maroon" title="c7">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">U</font> where <font color="Olive" title="b1">U</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> : not <font color="Maroon" title="c2">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">U</font> </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="topgen_2.html#K5" title="TOPGEN_2:func.5">DiscrWithInfin</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">x0</font></span>)</span></span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c8">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:35_1_1_1_1_1"/><i><font color="Green" title="E25">A3</font></i>: 
( <font color="Maroon" title="c8">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">a</font> &amp; not <font color="Maroon" title="c8">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">U</font> where <font color="Olive" title="b1">U</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> : not <font color="Maroon" title="c2">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">U</font> </span>}</span>  )
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><a NAME="E4:35_1_1_1_1_1"/>
<font color="Maroon" title="c8">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">F</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> where <font color="Olive" title="b1">F</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> : <font color="Olive" title="b1">F</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> </span>}</span> 
 <b>by </b><i><a class="txt" href="#E1:35_1_1"><i><font color="Green" title="E23">A1</font></i></a>, <a class="txt" href="#E1:35_1_1_1"><i><font color="Green" title="E24">A2</font></i></a>, <a class="txt" href="#E3:35_1_1_1_1_1"><i><font color="Green" title="E25">A3</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>then consider </b><font color="Maroon" title="c9">F</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E6:35_1_1_1_1_1"/><i><font color="Green" title="E26">A4</font></i>: 
( <font color="Maroon" title="c8">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">F</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &amp; <font color="Maroon" title="c9">F</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> )
 ;<br/><a NAME="E7:35_1_1_1_1_1"/>
<font color="Maroon" title="c8">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c7">a</font>
 <b>by </b><i><a class="txt" href="#E3:35_1_1_1_1_1"><i><font color="Green" title="E25">A3</font></i></a>, <a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></i>;<br/><a NAME="E8:35_1_1_1_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c7">a</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c7">a</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c7">a</font> )
 <b>by </b><i><a class="txt" href="#E1:35_1_1"><i><font color="Green" title="E23">A1</font></i></a>, <a class="txt" href="#E6:35_1_1_1_1_1"><i><font color="Green" title="E26">A4</font></i></a>, <a class="ref" href="finset_1.html#T13" title="FINSET_1:th.13">FINSET_1:13</a>, <a class="ref" href="subset_1.html#T36" title="SUBSET_1:th.36">SUBSET_1:36</a></i>;<br/><a NAME="E9:35_1_1_1_1_1"/><b>then </b>
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">F</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> where <font color="Olive" title="b1">F</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> : <font color="Olive" title="b1">F</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> </span>}</span> 
 <b>by </b><i><a class="txt" href="#E1:35_1_1"><i><font color="Green" title="E23">A1</font></i></a></i>;<br/><b>hence </b><a NAME="E10:35_1_1_1_1_1"/>
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="topgen_2.html#K5" title="TOPGEN_2:func.5">DiscrWithInfin</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">x0</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:35_1_1"><i><font color="Green" title="E23">A1</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:35_1_1_1_1_2"/><i><font color="Green" title="E25">A5</font></i>: 
<font color="Maroon" title="c7">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">U</font> where <font color="Olive" title="b1">U</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> : not <font color="Maroon" title="c2">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">U</font> </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="topgen_2.html#K5" title="TOPGEN_2:func.5">DiscrWithInfin</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">x0</font></span>)</span></span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:35_1_1_1_1_2_1"/>
<font color="Maroon" title="c2">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c7">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><b>then consider </b><font color="Maroon" title="c8">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:35_1_1_1_1_2_1"/><i><font color="Green" title="E26">A6</font></i>: 
( <font color="Maroon" title="c2">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">b</font> &amp; <font color="Maroon" title="c8">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">a</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/><a NAME="E4:35_1_1_1_1_2_1"/>
<font color="Maroon" title="c8">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">U</font> where <font color="Olive" title="b1">U</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> : not <font color="Maroon" title="c2">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">U</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="#E1:35_1_1_1_1_2"><i><font color="Green" title="E25">A5</font></i></a>, <a class="txt" href="#E3:35_1_1_1_1_2_1"><i><font color="Green" title="E26">A6</font></i></a></i>;<br/><a NAME="E5:35_1_1_1_1_2_1"/><b>then </b>
 ex <font color="Olive" title="b1">U</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Maroon" title="c8">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">U</font> &amp; not <font color="Maroon" title="c2">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">U</font> )
 ;<br/><b>hence </b><a NAME="E6:35_1_1_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E3:35_1_1_1_1_2_1"><i><font color="Green" title="E26">A6</font></i></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><a NAME="E3:35_1_1_1_1_2"/><b>then </b>
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">U</font> where <font color="Olive" title="b1">U</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> : not <font color="Maroon" title="c2">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">U</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="#E1:35_1_1"><i><font color="Green" title="E23">A1</font></i></a></i>;<br/><b>hence </b><a NAME="E4:35_1_1_1_1_2"/>
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="topgen_2.html#K5" title="TOPGEN_2:func.5">DiscrWithInfin</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">x0</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:35_1_1"><i><font color="Green" title="E23">A1</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</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></div><b>end;</b></div></div>
<b>end;</b></div>

<b>let </b><font color="Maroon" title="c7">a</font>, <font color="Maroon" title="c8">b</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="topgen_2.html#K5" title="TOPGEN_2:func.5">DiscrWithInfin</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">x0</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="topgen_2.html#K5" title="TOPGEN_2:func.5">DiscrWithInfin</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">x0</font></span>)</span> or  not <font color="Maroon" title="c8">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="topgen_2.html#K5" title="TOPGEN_2:func.5">DiscrWithInfin</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">x0</font></span>)</span> or <font color="Maroon" title="c7">a</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c8">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="topgen_2.html#K5" title="TOPGEN_2:func.5">DiscrWithInfin</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">x0</font></span>)</span> )</span><br/>

<b>assume </b><a NAME="E6:35_1_1"/><i><font color="Green" title="E24">A7</font></i>: 
( <font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="topgen_2.html#K5" title="TOPGEN_2:func.5">DiscrWithInfin</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">x0</font></span>)</span> &amp; <font color="Maroon" title="c8">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="topgen_2.html#K5" title="TOPGEN_2:func.5">DiscrWithInfin</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">x0</font></span>)</span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">a</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c8">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="topgen_2.html#K5" title="TOPGEN_2:func.5">DiscrWithInfin</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">x0</font></span>)</span></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:35_1_1_2"/>
( ( <font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">F</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> where <font color="Olive" title="b1">F</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> : <font color="Olive" title="b1">F</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> </span>}</span>  &amp; <font color="Maroon" title="c8">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">F</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> where <font color="Olive" title="b1">F</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> : <font color="Olive" title="b1">F</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> </span>}</span>  ) or <font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">U</font> where <font color="Olive" title="b1">U</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> : not <font color="Maroon" title="c2">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">U</font> </span>}</span>  or <font color="Maroon" title="c8">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">U</font> where <font color="Olive" title="b1">U</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> : not <font color="Maroon" title="c2">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">U</font> </span>}</span>  )
 </span><b>by </b><i><a class="txt" href="#E1:35_1_1"><i><font color="Green" title="E23">A1</font></i></a>, <a class="txt" href="#E6:35_1_1"><i><font color="Green" title="E24">A7</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_2_1"><b>suppose </b></a><a NAME="E1:35_1_1_2_1"/><i><font color="Green" title="E25">A8</font></i>: 
( <font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">F</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> where <font color="Olive" title="b1">F</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> : <font color="Olive" title="b1">F</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> </span>}</span>  &amp; <font color="Maroon" title="c8">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">F</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> where <font color="Olive" title="b1">F</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> : <font color="Olive" title="b1">F</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> </span>}</span>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">a</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c8">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="topgen_2.html#K5" title="TOPGEN_2:func.5">DiscrWithInfin</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">x0</font></span>)</span></span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c9">F1</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E3:35_1_1_2_1"/><i><font color="Green" title="E26">A9</font></i>: 
( <font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">F1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &amp; <font color="Maroon" title="c9">F1</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> )
 ;<br/><b>consider </b><font color="Maroon" title="c10">F2</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E5:35_1_1_2_1"/><i><font color="Green" title="E27">A10</font></i>: 
( <font color="Maroon" title="c8">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">F2</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &amp; <font color="Maroon" title="c10">F2</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> )
 <b>by </b><i><a class="txt" href="#E1:35_1_1_2_1"><i><font color="Green" title="E25">A8</font></i></a></i>;<br/><a NAME="E6:35_1_1_2_1"/>
( <font color="Maroon" title="c7">a</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c8">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c10">F2</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &amp; <font color="Maroon" title="c9">F1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c10">F2</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> )
 <b>by </b><i><a class="txt" href="#E3:35_1_1_2_1"><i><font color="Green" title="E26">A9</font></i></a>, <a class="txt" href="#E5:35_1_1_2_1"><i><font color="Green" title="E27">A10</font></i></a>, <a class="ref" href="xboole_1.html#T53" title="XBOOLE_1:th.53">XBOOLE_1:53</a></i>;<br/><a NAME="E7:35_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c7">a</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c8">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">F</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> where <font color="Olive" title="b1">F</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> : <font color="Olive" title="b1">F</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> </span>}</span> 
 ;<br/><b>hence </b><a NAME="E8:35_1_1_2_1"/>
<font color="Maroon" title="c7">a</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c8">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="topgen_2.html#K5" title="TOPGEN_2:func.5">DiscrWithInfin</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">x0</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:35_1_1"><i><font color="Green" title="E23">A1</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_2_2"><b>suppose </b></a><a NAME="E1:35_1_1_2_2"/>
( <font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">U</font> where <font color="Olive" title="b1">U</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> : not <font color="Maroon" title="c2">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">U</font> </span>}</span>  or <font color="Maroon" title="c8">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">U</font> where <font color="Olive" title="b1">U</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> : not <font color="Maroon" title="c2">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">U</font> </span>}</span>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">a</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c8">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="topgen_2.html#K5" title="TOPGEN_2:func.5">DiscrWithInfin</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">x0</font></span>)</span></span><br/><div class="add"><a NAME="E2:35_1_1_2_2"/><b>then </b>
(  ex <font color="Olive" title="b1">U</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">U</font> &amp; not <font color="Maroon" title="c2">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">U</font> ) or  ex <font color="Olive" title="b1">U</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Maroon" title="c8">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">U</font> &amp; not <font color="Maroon" title="c2">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">U</font> ) )
 ;<br/><a NAME="E3:35_1_1_2_2"/><b>then </b>
not <font color="Maroon" title="c2">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">a</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c8">b</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E4:35_1_1_2_2"/><b>then </b>
<font color="Maroon" title="c7">a</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c8">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">U</font> where <font color="Olive" title="b1">U</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> : not <font color="Maroon" title="c2">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">U</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="#E1:35_1_1"><i><font color="Green" title="E23">A1</font></i></a></i>;<br/><b>hence </b><a NAME="E5:35_1_1_2_2"/>
<font color="Maroon" title="c7">a</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c8">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="topgen_2.html#K5" title="TOPGEN_2:func.5">DiscrWithInfin</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">x0</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:35_1_1"><i><font color="Green" title="E23">A1</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</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></div><b>end;</b></div>

</div>
