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

<b>let </b><font color="Maroon" title="c1">TS</font> be   <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/>

<b>assume </b><a NAME="E1:12"/><i><font color="Green" title="E12">A1</font></i>: 
<font color="Maroon" title="c1">TS</font> <a href="compts_1.html#NR2" title="COMPTS_1:NR.2">is_T2</a> 
 ;<br/>

<b>let </b><font color="Maroon" title="c2">F</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TS</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E2:12"/><i><font color="Green" title="E13">A2</font></i>: 
<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> 
 <b>and </b><br/><a NAME="E3:12"/><i><font color="Green" title="E14">A3</font></i>: 
<font color="Maroon" title="c2">F</font> is <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a>
 ;<br/>

<b>let </b><font color="Maroon" title="c3">a</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">TS</font>;<br/>

<b>assume </b><a NAME="E4:12"/><i><font color="Green" title="E15">A4</font></i>: 
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 ;<br/>

<a NAME="E5:12"/><i><font color="Green" title="E16">A5</font></i>: 
not <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font>
 
<b>by </b><i><a class="txt" href="compts_1.html#E4:12"><i><font color="Green" title="E15">A4</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">PS</font>, <font color="Olive" title="b2">QS</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TS</font> st <br/>( $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">PS</font> &amp; $3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">QS</font> &amp; <font color="Olive" title="b1">PS</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b2">QS</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">PS</font> &amp; $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">QS</font> &amp; <font color="Olive" title="b1">PS</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b2">QS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  );<br/>
<a NAME="E6:12"/><i><font color="Green" title="E17">A6</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> holds <br/> ex <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">y</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 <font color="Maroon" title="c1">TS</font> &amp; <font color="Olive" title="b3">z</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 <font color="Maroon" title="c1">TS</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>,<font color="Olive" title="b3">z</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:12_1"/><i><font color="Green" title="E18">A7</font></i>: 
<font color="Maroon" title="c4">x</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="c5">p</font> = <font color="Maroon" title="c4">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">TS</font> ;<br/>
<b>consider </b><font color="Maroon" title="c6">W</font>, <font color="Maroon" title="c7">V</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TS</font><b> such that </b><br/><a NAME="E4:12_1"/><i><font color="Green" title="E19">A8</font></i>: 
<font color="Maroon" title="c6">W</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>and </b><br/><a NAME="E5:12_1"/><i><font color="Green" title="E20">A9</font></i>: 
<font color="Maroon" title="c7">V</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>and </b><br/><a NAME="E6:12_1"/><i><font color="Green" title="E21">A10</font></i>: 
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">W</font>
 <b>and </b><br/><a NAME="E7:12_1"/><i><font color="Green" title="E22">A11</font></i>: 
<font color="Maroon" title="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">V</font>
 <b>and </b><br/><a NAME="E8:12_1"/><i><font color="Green" title="E23">A12</font></i>: 
<font color="Maroon" title="c6">W</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c7">V</font>
 <b>by </b><i><a class="txt" href="compts_1.html#E1:12"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="compts_1.html#E5:12"><i><font color="Green" title="E16">A5</font></i></a>, <a class="txt" href="compts_1.html#E1:12_1"><i><font color="Green" title="E18">A7</font></i></a>, <a class="ref" href="compts_1.html#D4" target="_self" title="COMPTS_1:def.4">Def4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">PS</font> = <font color="Maroon" title="c6">W</font>, <font color="Maroon" title="c9">QS</font> = <font color="Maroon" title="c7">V</font> as    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<b>take </b>
<font color="Maroon" title="c8">PS</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c9">QS</font>
;<br/>

<b>thus </b><a NAME="E10:12_1"/>
( <font color="Maroon" title="c8">PS</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 <font color="Maroon" title="c1">TS</font> &amp; <font color="Maroon" title="c9">QS</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 <font color="Maroon" title="c1">TS</font> )
 <b>by </b><i><a class="txt" href="compts_1.html#E4:12_1"><i><font color="Green" title="E19">A8</font></i></a>, <a class="txt" href="compts_1.html#E5:12_1"><i><font color="Green" title="E20">A9</font></i></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/>

<b>take </b>
<font color="Maroon" title="c6">W</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c7">V</font>
;<br/>

<b>thus </b><a NAME="E11:12_1"/>
( <font color="Maroon" title="c8">PS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">W</font> &amp; <font color="Maroon" title="c9">QS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">V</font> &amp; <font color="Maroon" title="c6">W</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c7">V</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">W</font> &amp; <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">V</font> &amp; <font color="Maroon" title="c6">W</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 <b>by </b><i><a class="txt" href="compts_1.html#E4:12_1"><i><font color="Green" title="E19">A8</font></i></a>, <a class="txt" href="compts_1.html#E5:12_1"><i><font color="Green" title="E20">A9</font></i></a>, <a class="txt" href="compts_1.html#E6:12_1"><i><font color="Green" title="E21">A10</font></i></a>, <a class="txt" href="compts_1.html#E7:12_1"><i><font color="Green" title="E22">A11</font></i></a>, <a class="txt" href="compts_1.html#E8:12_1"><i><font color="Green" title="E23">A12</font></i></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c4">f</font>, <font color="Maroon" title="c5">g</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E8:12"/><i><font color="Green" title="E18">A13</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> )
 <b>and </b><br/><a NAME="E9:12"/><i><font color="Green" title="E19">A14</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>,<font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>]
 <b>from </b><i><a class="ref" href="wellord2.html#S2" title="WELLORD2:sch.2">WELLORD2:sch 2</a>(<a class="txt" href="compts_1.html#E6:12"><i><font color="Green" title="E17">A6</font></i></a>);<br/></i>
<a NAME="E10:12"/>
<font color="Maroon" title="c5">g</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c2">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">TS</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:12_2"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">g</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c2">F</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:12_2"/>
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">g</font>
 <b>and </b><br/><a NAME="E4:12_2"/><i><font color="Green" title="E20">A15</font></i>: 
<font color="Maroon" title="c7">y</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="E5:12_2"/><i><font color="Green" title="E21">A16</font></i>: 
<font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E6:12_2"/>
 ex <font color="Olive" title="b1">PS</font>, <font color="Olive" title="b2">QS</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TS</font> st <br/>( <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">PS</font> &amp; <font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">QS</font> &amp; <font color="Olive" title="b1">PS</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b2">QS</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">PS</font> &amp; <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">QS</font> &amp; <font color="Olive" title="b1">PS</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b2">QS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<b>by </b><i><a class="txt" href="compts_1.html#E9:12"><i><font color="Green" title="E19">A14</font></i></a>, <a class="txt" href="compts_1.html#E4:12_2"><i><font color="Green" title="E20">A15</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:12_2"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">TS</font>
 <b>by </b><i><a class="txt" href="compts_1.html#E5:12_2"><i><font color="Green" title="E21">A16</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c6">C</font> = <font color="Maroon" title="c5">g</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c2">F</font> as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">TS</font> ;<br/>
<a NAME="E12:12"/>
<font color="Maroon" title="c2">F</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="c6">C</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:12_3"/><i><font color="Green" title="E20">A17</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c8">PS</font>, <font color="Maroon" title="c9">QS</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TS</font><b> such that </b><br/><a NAME="E3:12_3"/>
<font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">PS</font>
 <b>and </b><br/><a NAME="E4:12_3"/><i><font color="Green" title="E21">A18</font></i>: 
<font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">QS</font>
 <b>and </b><br/><a NAME="E5:12_3"/>
<font color="Maroon" title="c8">PS</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>and </b><br/><a NAME="E6:12_3"/>
<font color="Maroon" title="c9">QS</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>and </b><br/><a NAME="E7:12_3"/>
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">PS</font>
 <b>and </b><br/><a NAME="E8:12_3"/><i><font color="Green" title="E22">A19</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">QS</font>
 <b>and </b><br/><a NAME="E9:12_3"/>
<font color="Maroon" title="c8">PS</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c9">QS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="compts_1.html#E9:12"><i><font color="Green" title="E19">A14</font></i></a></i>;<br/>
<a NAME="E10:12_3"/>
<font color="Maroon" title="c9">QS</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">C</font>
 
<b>by </b><i><a class="txt" href="compts_1.html#E8:12"><i><font color="Green" title="E18">A13</font></i></a>, <a class="txt" href="compts_1.html#E1:12_3"><i><font color="Green" title="E20">A17</font></i></a>, <a class="txt" href="compts_1.html#E4:12_3"><i><font color="Green" title="E21">A18</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<b>hence </b><a NAME="E11:12_3"/>
<font color="Maroon" title="c7">x</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="c6">C</font>
 <b>by </b><i><a class="txt" href="compts_1.html#E8:12_3"><i><font color="Green" title="E22">A19</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:12"/><b>then </b><i><font color="Green" title="E20">A20</font></i>: 
<font color="Maroon" title="c6">C</font> <a href="compts_1.html#R1" title="COMPTS_1:pred.1">is_a_cover_of</a> <font color="Maroon" title="c2">F</font>
 
<b>by </b><i><a class="ref" href="compts_1.html#D1" target="_self" title="COMPTS_1:def.1">Def1</a></i>;<br/>
<a NAME="E14:12"/><i><font color="Green" title="E21">A21</font></i>: 
<font color="Maroon" title="c6">C</font> is <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">G</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TS</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="tops_2.html#D1" title="TOPS_2:def.1">TOPS_2:def 1</a><br/>

<b>assume </b><a NAME="E1:12_4"/>
<font color="Maroon" title="c7">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">C</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c8">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:12_4"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">g</font>
 <b>and </b><br/><a NAME="E4:12_4"/><i><font color="Green" title="E22">A22</font></i>: 
<font color="Maroon" title="c8">x</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="E5:12_4"/><i><font color="Green" title="E23">A23</font></i>: 
<font color="Maroon" title="c7">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E6:12_4"/>
 ex <font color="Olive" title="b1">PS</font>, <font color="Olive" title="b2">QS</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TS</font> st <br/>( <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">PS</font> &amp; <font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">QS</font> &amp; <font color="Olive" title="b1">PS</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b2">QS</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">PS</font> &amp; <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">QS</font> &amp; <font color="Olive" title="b1">PS</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b2">QS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<b>by </b><i><a class="txt" href="compts_1.html#E9:12"><i><font color="Green" title="E19">A14</font></i></a>, <a class="txt" href="compts_1.html#E4:12_4"><i><font color="Green" title="E22">A22</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:12_4"/>
<font color="Maroon" title="c7">G</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="txt" href="compts_1.html#E5:12_4"><i><font color="Green" title="E23">A23</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon" title="c7">C'</font> being   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">TS</font><b> such that </b><br/><a NAME="E16:12"/><i><font color="Green" title="E22">A24</font></i>: 
<font color="Maroon" title="c7">C'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">C</font>
 <b>and </b><br/><a NAME="E17:12"/><i><font color="Green" title="E23">A25</font></i>: 
<font color="Maroon" title="c7">C'</font> <a href="compts_1.html#R1" title="COMPTS_1:pred.1">is_a_cover_of</a> <font color="Maroon" title="c2">F</font>
 <b>and </b><br/><a NAME="E18:12"/><i><font color="Green" title="E24">A26</font></i>: 
<font color="Maroon" title="c7">C'</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>by </b><i><a class="txt" href="compts_1.html#E3:12"><i><font color="Green" title="E14">A3</font></i></a>, <a class="txt" href="compts_1.html#E13:12"><i><font color="Green" title="E20">A20</font></i></a>, <a class="ref" href="compts_1.html#D7" target="_self" title="COMPTS_1:def.7">Def7</a></i>;<br/>
<a NAME="E19:12"/>
<font color="Maroon" title="c7">C'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">g</font>
 
<b>by </b><i><a class="txt" href="compts_1.html#E8:12"><i><font color="Green" title="E18">A13</font></i></a>, <a class="txt" href="compts_1.html#E16:12"><i><font color="Green" title="E22">A24</font></i></a>, <a class="ref" href="relat_1.html#T146" title="RELAT_1:th.146">RELAT_1:146</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c8">H'</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E21:12"/><i><font color="Green" title="E25">A27</font></i>: 
<font color="Maroon" title="c8">H'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">g</font>
 <b>and </b><br/><a NAME="E22:12"/><i><font color="Green" title="E26">A28</font></i>: 
<font color="Maroon" title="c8">H'</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>and </b><br/><a NAME="E23:12"/><i><font color="Green" title="E27">A29</font></i>: 
<font color="Maroon" title="c5">g</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c8">H'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">C'</font>
 <b>by </b><i><a class="txt" href="compts_1.html#E18:12"><i><font color="Green" title="E24">A26</font></i></a>, <a class="ref" href="orders_1.html#T195" title="ORDERS_1:th.195">ORDERS_1:195</a></i>;<br/>
<a NAME="E24:12"/>
<font color="Maroon" title="c4">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c8">H'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">TS</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:12_5"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c8">H'</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c10">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:12_5"/>
<font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">f</font>
 <b>and </b><br/><a NAME="E4:12_5"/><i><font color="Green" title="E28">A30</font></i>: 
<font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">H'</font>
 <b>and </b><br/><a NAME="E5:12_5"/><i><font color="Green" title="E29">A31</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E6:12_5"/>
 ex <font color="Olive" title="b1">PS</font>, <font color="Olive" title="b2">QS</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TS</font> st <br/>( <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">PS</font> &amp; <font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">QS</font> &amp; <font color="Olive" title="b1">PS</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b2">QS</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">PS</font> &amp; <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">QS</font> &amp; <font color="Olive" title="b1">PS</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b2">QS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<b>by </b><i><a class="txt" href="compts_1.html#E8:12"><i><font color="Green" title="E18">A13</font></i></a>, <a class="txt" href="compts_1.html#E9:12"><i><font color="Green" title="E19">A14</font></i></a>, <a class="txt" href="compts_1.html#E21:12"><i><font color="Green" title="E25">A27</font></i></a>, <a class="txt" href="compts_1.html#E4:12_5"><i><font color="Green" title="E28">A30</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:12_5"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">TS</font>
 <b>by </b><i><a class="txt" href="compts_1.html#E5:12_5"><i><font color="Green" title="E29">A31</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c9">B</font> = <font color="Maroon" title="c4">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c8">H'</font> as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">TS</font> ;<br/>
<b>take </b><font color="Maroon" title="c10">G0</font> =  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c9">B</font>;<br/>

<b>take </b><font color="Maroon" title="c11">G1</font> =  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c7">C'</font>;<br/>

<a NAME="E26:12"/><i><font color="Green" title="E28">A32</font></i>: 
<font color="Maroon" title="c9">B</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 
<b>by </b><i><a class="txt" href="compts_1.html#E22:12"><i><font color="Green" title="E26">A28</font></i></a>, <a class="ref" href="finset_1.html#T17" title="FINSET_1:th.17">FINSET_1:17</a></i>;<br/>
<a NAME="E27:12"/>
<font color="Maroon" title="c9">B</font> is <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">G</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TS</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="tops_2.html#D1" title="TOPS_2:def.1">TOPS_2:def 1</a><br/>

<b>assume </b><a NAME="E1:12_6"/>
<font color="Maroon" title="c12">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">B</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c13">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:12_6"/><i><font color="Green" title="E29">A33</font></i>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">f</font>
 <b>and </b><br/><a NAME="E4:12_6"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">H'</font>
 <b>and </b><br/><a NAME="E5:12_6"/><i><font color="Green" title="E30">A34</font></i>: 
<font color="Maroon" title="c12">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E6:12_6"/>
 ex <font color="Olive" title="b1">PS</font>, <font color="Olive" title="b2">QS</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TS</font> st <br/>( <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">PS</font> &amp; <font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">QS</font> &amp; <font color="Olive" title="b1">PS</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b2">QS</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">PS</font> &amp; <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">QS</font> &amp; <font color="Olive" title="b1">PS</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b2">QS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<b>by </b><i><a class="txt" href="compts_1.html#E8:12"><i><font color="Green" title="E18">A13</font></i></a>, <a class="txt" href="compts_1.html#E9:12"><i><font color="Green" title="E19">A14</font></i></a>, <a class="txt" href="compts_1.html#E3:12_6"><i><font color="Green" title="E29">A33</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:12_6"/>
<font color="Maroon" title="c12">G</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="txt" href="compts_1.html#E5:12_6"><i><font color="Green" title="E30">A34</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E28:12"/>
<font color="Maroon" title="c10">G0</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="txt" href="compts_1.html#E26:12"><i><font color="Green" title="E28">A32</font></i></a>, <a class="ref" href="tops_2.html#T27" title="TOPS_2:th.27">TOPS_2:27</a></i>;<br/>

<b>thus </b><a NAME="E29:12"/>
<font color="Maroon" title="c11">G1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="txt" href="compts_1.html#E14:12"><i><font color="Green" title="E21">A21</font></i></a>, <a class="txt" href="compts_1.html#E16:12"><i><font color="Green" title="E22">A24</font></i></a>, <a class="ref" href="tops_2.html#T18" title="TOPS_2:th.18">TOPS_2:18</a>, <a class="ref" href="tops_2.html#T26" title="TOPS_2:th.26">TOPS_2:26</a></i>;<br/>

<b>consider </b><font color="Maroon" title="c12">z</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">F</font>;<br/>
<a NAME="E31:12"/>
<font color="Maroon" title="c2">F</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">C'</font>
 
<b>by </b><i><a class="txt" href="compts_1.html#E17:12"><i><font color="Green" title="E23">A25</font></i></a>, <a class="ref" href="compts_1.html#D1" target="_self" title="COMPTS_1:def.1">Def1</a></i>;<br/>
<a NAME="E32:12"/><b>then </b>
<font color="Maroon" title="c12">z</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">C'</font>
 
<b>by </b><i><a class="txt" href="compts_1.html#E2:12"><i><font color="Green" title="E13">A2</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c13">D</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E34:12"/>
<font color="Maroon" title="c12">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">D</font>
 <b>and </b><br/><a NAME="E35:12"/><i><font color="Green" title="E29">A35</font></i>: 
<font color="Maroon" title="c13">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">C'</font>
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">D'</font> = <font color="Maroon" title="c13">D</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TS</font> <b>by </b><i><a class="txt" href="compts_1.html#E35:12"><i><font color="Green" title="E29">A35</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c15">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E38:12"/><i><font color="Green" title="E30">A36</font></i>: 
<font color="Maroon" title="c15">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">g</font>
 <b>and </b><br/><a NAME="E39:12"/><i><font color="Green" title="E31">A37</font></i>: 
<font color="Maroon" title="c15">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">H'</font>
 <b>and </b><br/><a NAME="E40:12"/>
<font color="Maroon" title="c14">D'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">y</font>
 <b>by </b><i><a class="txt" href="compts_1.html#E23:12"><i><font color="Green" title="E27">A29</font></i></a>, <a class="txt" href="compts_1.html#E35:12"><i><font color="Green" title="E29">A35</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c16">PS</font>, <font color="Maroon" title="c17">QS</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TS</font><b> such that </b><br/><a NAME="E42:12"/><i><font color="Green" title="E32">A38</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">PS</font>
 <b>and </b><br/><a NAME="E43:12"/>
<font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">QS</font>
 <b>and </b><br/><a NAME="E44:12"/>
<font color="Maroon" title="c16">PS</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>and </b><br/><a NAME="E45:12"/>
<font color="Maroon" title="c17">QS</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>and </b><br/><a NAME="E46:12"/>
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">PS</font>
 <b>and </b><br/><a NAME="E47:12"/>
<font color="Maroon" title="c15">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">QS</font>
 <b>and </b><br/><a NAME="E48:12"/>
<font color="Maroon" title="c16">PS</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c17">QS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="compts_1.html#E8:12"><i><font color="Green" title="E18">A13</font></i></a>, <a class="txt" href="compts_1.html#E9:12"><i><font color="Green" title="E19">A14</font></i></a>, <a class="txt" href="compts_1.html#E38:12"><i><font color="Green" title="E30">A36</font></i></a></i>;<br/>
<a NAME="E49:12"/><i><font color="Green" title="E33">A39</font></i>: 
<font color="Maroon" title="c9">B</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="compts_1.html#E8:12"><i><font color="Green" title="E18">A13</font></i></a>, <a class="txt" href="compts_1.html#E38:12"><i><font color="Green" title="E30">A36</font></i></a>, <a class="txt" href="compts_1.html#E39:12"><i><font color="Green" title="E31">A37</font></i></a>, <a class="txt" href="compts_1.html#E42:12"><i><font color="Green" title="E32">A38</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E50:12"/>
for <font color="Olive" title="b1">G</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">B</font> holds <br/><font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">G</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c18">G</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:12_7"/><i><font color="Green" title="E34">A40</font></i>: 
<font color="Maroon" title="c18">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">B</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c19">G'</font> = <font color="Maroon" title="c18">G</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TS</font> ;<br/>
<b>consider </b><font color="Maroon" title="c20">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:12_7"/><i><font color="Green" title="E35">A41</font></i>: 
<font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">f</font>
 <b>and </b><br/><a NAME="E5:12_7"/>
<font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">H'</font>
 <b>and </b><br/><a NAME="E6:12_7"/><i><font color="Green" title="E36">A42</font></i>: 
<font color="Maroon" title="c19">G'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x</font>
 <b>by </b><i><a class="txt" href="compts_1.html#E1:12_7"><i><font color="Green" title="E34">A40</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E7:12_7"/>
 ex <font color="Olive" title="b1">PS</font>, <font color="Olive" title="b2">QS</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TS</font> st <br/>( <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">PS</font> &amp; <font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">QS</font> &amp; <font color="Olive" title="b1">PS</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b2">QS</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">PS</font> &amp; <font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">QS</font> &amp; <font color="Olive" title="b1">PS</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b2">QS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<b>by </b><i><a class="txt" href="compts_1.html#E8:12"><i><font color="Green" title="E18">A13</font></i></a>, <a class="txt" href="compts_1.html#E9:12"><i><font color="Green" title="E19">A14</font></i></a>, <a class="txt" href="compts_1.html#E4:12_7"><i><font color="Green" title="E35">A41</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:12_7"/>
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">G</font>
 <b>by </b><i><a class="txt" href="compts_1.html#E6:12_7"><i><font color="Green" title="E36">A42</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E51:12"/>
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">G0</font>
 <b>by </b><i><a class="txt" href="compts_1.html#E49:12"><i><font color="Green" title="E33">A39</font></i></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></i>;<br/>

<b>thus </b><a NAME="E52:12"/>
<font color="Maroon" title="c2">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">G1</font>
 <b>by </b><i><a class="txt" href="compts_1.html#E17:12"><i><font color="Green" title="E23">A25</font></i></a>, <a class="ref" href="compts_1.html#D1" target="_self" title="COMPTS_1:def.1">Def1</a></i>;<br/>

<b>thus </b><a NAME="E53:12"/>
<font color="Maroon" title="c10">G0</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c11">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_8"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:12_8"/><i><font color="Green" title="E34">A43</font></i>: 
<font color="Maroon" title="c10">G0</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c11">G1</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/>

<b>consider </b><font color="Maroon" title="c18">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c9">B</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c7">C'</font></span>)</span>;<br/>
<a NAME="E3:12_8"/><i><font color="Green" title="E35">A44</font></i>: 
( <font color="Maroon" title="c18">x</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">C'</font> &amp; <font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c9">B</font> )
 
<b>by </b><i><a class="txt" href="compts_1.html#E1:12_8"><i><font color="Green" title="E34">A43</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="c19">A</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:12_8"/><i><font color="Green" title="E36">A45</font></i>: 
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">A</font>
 <b>and </b><br/><a NAME="E6:12_8"/><i><font color="Green" title="E37">A46</font></i>: 
<font color="Maroon" title="c19">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">C'</font>
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c20">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E8:12_8"/><i><font color="Green" title="E38">A47</font></i>: 
<font color="Maroon" title="c20">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">g</font>
 <b>and </b><br/><a NAME="E9:12_8"/><i><font color="Green" title="E39">A48</font></i>: 
<font color="Maroon" title="c20">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">H'</font>
 <b>and </b><br/><a NAME="E10:12_8"/><i><font color="Green" title="E40">A49</font></i>: 
<font color="Maroon" title="c19">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">z</font>
 <b>by </b><i><a class="txt" href="compts_1.html#E23:12"><i><font color="Green" title="E27">A29</font></i></a>, <a class="txt" href="compts_1.html#E6:12_8"><i><font color="Green" title="E37">A46</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c21">PS</font>, <font color="Maroon" title="c22">QS</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TS</font><b> such that </b><br/><a NAME="E12:12_8"/><i><font color="Green" title="E41">A50</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">PS</font>
 <b>and </b><br/><a NAME="E13:12_8"/><i><font color="Green" title="E42">A51</font></i>: 
<font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">QS</font>
 <b>and </b><br/><a NAME="E14:12_8"/>
<font color="Maroon" title="c21">PS</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>and </b><br/><a NAME="E15:12_8"/>
<font color="Maroon" title="c22">QS</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>and </b><br/><a NAME="E16:12_8"/>
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">PS</font>
 <b>and </b><br/><a NAME="E17:12_8"/>
<font color="Maroon" title="c20">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">QS</font>
 <b>and </b><br/><a NAME="E18:12_8"/><i><font color="Green" title="E43">A52</font></i>: 
<font color="Maroon" title="c21">PS</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c22">QS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="compts_1.html#E8:12"><i><font color="Green" title="E18">A13</font></i></a>, <a class="txt" href="compts_1.html#E9:12"><i><font color="Green" title="E19">A14</font></i></a>, <a class="txt" href="compts_1.html#E8:12_8"><i><font color="Green" title="E38">A47</font></i></a></i>;<br/>
<a NAME="E19:12_8"/>
<font color="Maroon" title="c21">PS</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">B</font>
 
<b>by </b><i><a class="txt" href="compts_1.html#E8:12"><i><font color="Green" title="E18">A13</font></i></a>, <a class="txt" href="compts_1.html#E8:12_8"><i><font color="Green" title="E38">A47</font></i></a>, <a class="txt" href="compts_1.html#E9:12_8"><i><font color="Green" title="E39">A48</font></i></a>, <a class="txt" href="compts_1.html#E12:12_8"><i><font color="Green" title="E41">A50</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E20:12_8"/><b>then </b>
( <font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">QS</font> &amp; <font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">PS</font> )
 
<b>by </b><i><a class="txt" href="compts_1.html#E3:12_8"><i><font color="Green" title="E35">A44</font></i></a>, <a class="txt" href="compts_1.html#E5:12_8"><i><font color="Green" title="E36">A45</font></i></a>, <a class="txt" href="compts_1.html#E10:12_8"><i><font color="Green" title="E40">A49</font></i></a>, <a class="txt" href="compts_1.html#E13:12_8"><i><font color="Green" title="E42">A51</font></i></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E21:12_8"/>
contradiction
 <b>by </b><i><a class="txt" href="compts_1.html#E18:12_8"><i><font color="Green" title="E43">A52</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>


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


</div>
