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

<b>let </b><font color="Maroon" title="c1">X</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">FS</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="card_fil.html#K3" title="CARD_FIL:func.3">Filters</a> <font color="Maroon" title="c1">X</font></span>)</span>;<br/>



<b>assume </b><a NAME="E1:29"/><i><font color="Green" title="E22">A1</font></i>: 
<font color="Maroon" title="c2">FS</font> is <a href="ordinal1.html#V6" title="ORDINAL1:attr.6">c=-linear</a>
 ;<br/>

<a NAME="E2:29"/><i><font color="Green" title="E23">A2</font></i>: 
for <font color="Olive" title="b1">S</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">FS</font> holds <br/><font color="Olive" title="b1">S</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> <font color="Maroon" title="c1">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">S</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:29_1"/><i><font color="Green" title="E24">A3</font></i>: 
<font color="Maroon" title="c3">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">FS</font>
 ;<br/>

<a NAME="E2:29_1"/>
<font color="Maroon" title="c3">S</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="card_fil.html#K3" title="CARD_FIL:func.3">Filters</a> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="txt" href="card_fil.html#E1:29_1"><i><font color="Green" title="E24">A3</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:29_1"/>
<font color="Maroon" title="c3">S</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> <font color="Maroon" title="c1">X</font>
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c3">S</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:29"/><i><font color="Green" title="E24">A4</font></i>: 
<font color="Maroon" title="c3">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">FS</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E5:29"/>
<font color="Maroon" title="c3">S</font> is    <a href="card_fil.html#M1" title="CARD_FIL:mode.1">Filter</a> of <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="txt" href="card_fil.html#E4:29"><i><font color="Green" title="E24">A4</font></i></a>, <a class="ref" href="card_fil.html#T15" target="_self" title="CARD_FIL:th.15">Th15</a></i>;<br/>
<a NAME="E6:29"/><b>then </b>
<font color="Maroon" title="c1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">S</font>
 
<b>by </b><i><a class="ref" href="card_fil.html#T5" target="_self" title="CARD_FIL:th.5">Th5</a></i>;<br/>
<a NAME="E7:29"/><b>then </b><i><font color="Green" title="E25">A5</font></i>: 
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">FS</font> is  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="txt" href="card_fil.html#E2:29"><i><font color="Green" title="E23">A2</font></i></a>, <a class="txt" href="card_fil.html#E4:29"><i><font color="Green" title="E24">A4</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a>, <a class="ref" href="zfmisc_1.html#T94" title="ZFMISC_1:th.94">ZFMISC_1:94</a></i>;<br/>
<a NAME="E8:29"/><i><font color="Green" title="E26">A6</font></i>: 
not  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">FS</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:29_2"/>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">FS</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c4">S</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:29_2"/><i><font color="Green" title="E27">A7</font></i>: 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">S</font>
 <b>and </b><br/><a NAME="E4:29_2"/><i><font color="Green" title="E28">A8</font></i>: 
<font color="Maroon" title="c4">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">FS</font>
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<a NAME="E5:29_2"/>
<font color="Maroon" title="c4">S</font> is    <a href="card_fil.html#M1" title="CARD_FIL:mode.1">Filter</a> of <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="txt" href="card_fil.html#E4:29_2"><i><font color="Green" title="E28">A8</font></i></a>, <a class="ref" href="card_fil.html#T15" target="_self" title="CARD_FIL:th.15">Th15</a></i>;<br/>
<b>hence </b><a NAME="E6:29_2"/>
contradiction
 <b>by </b><i><a class="txt" href="card_fil.html#E3:29_2"><i><font color="Green" title="E27">A7</font></i></a>, <a class="ref" href="card_fil.html#D1" target="_self" title="CARD_FIL:def.1">Def1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:29"/>
for <font color="Olive" title="b1">Y1</font>, <font color="Olive" title="b2">Y2</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> holds <br/> ( ( <font color="Olive" title="b1">Y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">FS</font> &amp; <font color="Olive" title="b2">Y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">FS</font> implies <font color="Olive" title="b1">Y1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b2">Y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">FS</font> ) &amp; ( <font color="Olive" title="b1">Y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">FS</font> &amp; <font color="Olive" title="b1">Y1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">Y2</font> implies <font color="Olive" title="b2">Y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">FS</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">Y1</font>, <font color="Maroon" title="c5">Y2</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>;<br/>



<b>thus </b><a NAME="E1:29_3"/>
( <font color="Maroon" title="c4">Y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">FS</font> &amp; <font color="Maroon" title="c5">Y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">FS</font> implies <font color="Maroon" title="c4">Y1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">Y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">FS</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:29_3_1"/><i><font color="Green" title="E27">A9</font></i>: 
( <font color="Maroon" title="c4">Y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">FS</font> &amp; <font color="Maroon" title="c5">Y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">FS</font> )
 ;<br/>

<b>then consider </b><font color="Maroon" title="c6">S1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:29_3_1"/><i><font color="Green" title="E28">A10</font></i>: 
<font color="Maroon" title="c4">Y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">S1</font>
 <b>and </b><br/><a NAME="E4:29_3_1"/><i><font color="Green" title="E29">A11</font></i>: 
<font color="Maroon" title="c6">S1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">FS</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="c7">S2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:29_3_1"/><i><font color="Green" title="E30">A12</font></i>: 
<font color="Maroon" title="c5">Y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">S2</font>
 <b>and </b><br/><a NAME="E7:29_3_1"/><i><font color="Green" title="E31">A13</font></i>: 
<font color="Maroon" title="c7">S2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">FS</font>
 <b>by </b><i><a class="txt" href="card_fil.html#E1:29_3_1"><i><font color="Green" title="E27">A9</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<a NAME="E8:29_3_1"/><i><font color="Green" title="E32">A14</font></i>: 
( <font color="Maroon" title="c6">S1</font> is    <a href="card_fil.html#M1" title="CARD_FIL:mode.1">Filter</a> of <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c7">S2</font> is    <a href="card_fil.html#M1" title="CARD_FIL:mode.1">Filter</a> of <font color="Maroon" title="c1">X</font> )
 
<b>by </b><i><a class="txt" href="card_fil.html#E4:29_3_1"><i><font color="Green" title="E29">A11</font></i></a>, <a class="txt" href="card_fil.html#E7:29_3_1"><i><font color="Green" title="E31">A13</font></i></a>, <a class="ref" href="card_fil.html#T15" target="_self" title="CARD_FIL:th.15">Th15</a></i>;<br/>
<a NAME="E9:29_3_1"/><i><font color="Green" title="E33">A15</font></i>: 
<font color="Maroon" title="c6">S1</font>,<font color="Maroon" title="c7">S2</font> <a href="xboole_0.html#R3" title="XBOOLE_0:pred.3">are_c=-comparable</a> 
 
<b>by </b><i><a class="txt" href="card_fil.html#E1:29"><i><font color="Green" title="E22">A1</font></i></a>, <a class="txt" href="card_fil.html#E4:29_3_1"><i><font color="Green" title="E29">A11</font></i></a>, <a class="txt" href="card_fil.html#E7:29_3_1"><i><font color="Green" title="E31">A13</font></i></a>, <a class="ref" href="ordinal1.html#D9" title="ORDINAL1:def.9">ORDINAL1:def 9</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:29_3_1_1"/>
( <font color="Maroon" title="c6">S1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">S2</font> or <font color="Maroon" title="c7">S2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">S1</font> )
 </span><b>by </b><i><a class="txt" href="card_fil.html#E9:29_3_1"><i><font color="Green" title="E33">A15</font></i></a>, <a class="ref" href="xboole_0.html#D9" title="XBOOLE_0:def.9">XBOOLE_0:def 9</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_3_1_1_1"><b>suppose </b></a><a NAME="E1:29_3_1_1_1"/>
<font color="Maroon" title="c6">S1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">S2</font>
 ;<br/><div class="add"><a NAME="E2:29_3_1_1_1"/><b>then </b>
<font color="Maroon" title="c4">Y1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">Y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">S2</font>
 <b>by </b><i><a class="txt" href="card_fil.html#E3:29_3_1"><i><font color="Green" title="E28">A10</font></i></a>, <a class="txt" href="card_fil.html#E6:29_3_1"><i><font color="Green" title="E30">A12</font></i></a>, <a class="txt" href="card_fil.html#E8:29_3_1"><i><font color="Green" title="E32">A14</font></i></a>, <a class="ref" href="card_fil.html#D1" target="_self" title="CARD_FIL:def.1">Def1</a></i>;<br/><b>hence </b><a NAME="E3:29_3_1_1_1"/>
<font color="Maroon" title="c4">Y1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">Y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">FS</font>
 <b>by </b><i><a class="txt" href="card_fil.html#E7:29_3_1"><i><font color="Green" title="E31">A13</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_3_1_1_2"><b>suppose </b></a><a NAME="E1:29_3_1_1_2"/>
<font color="Maroon" title="c7">S2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">S1</font>
 ;<br/><div class="add"><a NAME="E2:29_3_1_1_2"/><b>then </b>
<font color="Maroon" title="c4">Y1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">Y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">S1</font>
 <b>by </b><i><a class="txt" href="card_fil.html#E3:29_3_1"><i><font color="Green" title="E28">A10</font></i></a>, <a class="txt" href="card_fil.html#E6:29_3_1"><i><font color="Green" title="E30">A12</font></i></a>, <a class="txt" href="card_fil.html#E8:29_3_1"><i><font color="Green" title="E32">A14</font></i></a>, <a class="ref" href="card_fil.html#D1" target="_self" title="CARD_FIL:def.1">Def1</a></i>;<br/><b>hence </b><a NAME="E3:29_3_1_1_2"/>
<font color="Maroon" title="c4">Y1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">Y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">FS</font>
 <b>by </b><i><a class="txt" href="card_fil.html#E4:29_3_1"><i><font color="Green" title="E29">A11</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></div><b>end;</b></div>

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

<b>assume </b><a NAME="E2:29_3"/><i><font color="Green" title="E27">A16</font></i>: 
( <font color="Maroon" title="c4">Y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">FS</font> &amp; <font color="Maroon" title="c4">Y1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">Y2</font> )
 ;<br/>

<b>then consider </b><font color="Maroon" title="c6">S1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:29_3"/><i><font color="Green" title="E28">A17</font></i>: 
<font color="Maroon" title="c4">Y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">S1</font>
 <b>and </b><br/><a NAME="E5:29_3"/><i><font color="Green" title="E29">A18</font></i>: 
<font color="Maroon" title="c6">S1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">FS</font>
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<a NAME="E6:29_3"/>
<font color="Maroon" title="c6">S1</font> is    <a href="card_fil.html#M1" title="CARD_FIL:mode.1">Filter</a> of <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="txt" href="card_fil.html#E5:29_3"><i><font color="Green" title="E29">A18</font></i></a>, <a class="ref" href="card_fil.html#T15" target="_self" title="CARD_FIL:th.15">Th15</a></i>;<br/>
<a NAME="E7:29_3"/><b>then </b>
<font color="Maroon" title="c5">Y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">S1</font>
 
<b>by </b><i><a class="txt" href="card_fil.html#E2:29_3"><i><font color="Green" title="E27">A16</font></i></a>, <a class="txt" href="card_fil.html#E4:29_3"><i><font color="Green" title="E28">A17</font></i></a>, <a class="ref" href="card_fil.html#D1" target="_self" title="CARD_FIL:def.1">Def1</a></i>;<br/>
<b>hence </b><a NAME="E8:29_3"/>
<font color="Maroon" title="c5">Y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">FS</font>
 <b>by </b><i><a class="txt" href="card_fil.html#E5:29_3"><i><font color="Green" title="E29">A18</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>
<b>hence </b><a NAME="E10:29"/>
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">FS</font> is    <a href="card_fil.html#M1" title="CARD_FIL:mode.1">Filter</a> of <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="card_fil.html#E7:29"><i><font color="Green" title="E25">A5</font></i></a>, <a class="txt" href="card_fil.html#E8:29"><i><font color="Green" title="E26">A6</font></i></a>, <a class="ref" href="card_fil.html#D1" target="_self" title="CARD_FIL:def.1">Def1</a></i>;<br/>


</div>
