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

<b>let </b><font color="Maroon" title="c1">L</font> be   <a href="lattices.html#NM2" title="LATTICES:NM.2">D_Lattice</a>;<br/><b>let </b><font color="Maroon" title="c2">b</font>, <font color="Maroon" title="c3">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>;<br/><b>let </b><font color="Maroon" title="c4">Z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>







<b>assume </b><a NAME="E1:31"/><i><font color="Green" title="E22">A1</font></i>: 
( <font color="Maroon" title="c4">Z</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>  &amp; <font color="Maroon" title="c4">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="openlatt.html#K10" title="OPENLATT:func.10">SF_have</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="openlatt.html#K10" title="OPENLATT:func.10">SF_have</a> <font color="Maroon" title="c3">a</font></span>)</span> &amp; <font color="Maroon" title="c4">Z</font> is <a href="ordinal1.html#V6" title="ORDINAL1:attr.6">c=-linear</a> )
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c5">Z</font> = <font color="Maroon" title="c4">Z</font> as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>take </b><font color="Maroon" title="c6">Y</font> =  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c5">Z</font>;<br/>

<a NAME="E3:31"/>
<font color="Maroon" title="c6">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="openlatt.html#K10" title="OPENLATT:func.10">SF_have</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="openlatt.html#K10" title="OPENLATT:func.10">SF_have</a> <font color="Maroon" title="c3">a</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1"><b>proof </b></a><div class="add">

<a NAME="E1:31_1"/><i><font color="Green" title="E23">A2</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="c6">Y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:31_1_1"/>
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Y</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">X</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:31_1_1"/><i><font color="Green" title="E24">A3</font></i>: 
( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">X</font> &amp; <font color="Maroon" title="c7">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Z</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>thus </b><a NAME="E4:31_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="openlatt.html#E1:31"><i><font color="Green" title="E22">A1</font></i></a>, <a class="txt" href="openlatt.html#E3:31_1_1"><i><font color="Green" title="E24">A3</font></i></a>, <a class="ref" href="openlatt.html#T19" target="_self" title="OPENLATT:th.19">Th19</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c7">X</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">Z</font>;<br/>
<a NAME="E3:31_1"/>
<font color="Maroon" title="c7">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="openlatt.html#K10" title="OPENLATT:func.10">SF_have</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="openlatt.html#K10" title="OPENLATT:func.10">SF_have</a> <font color="Maroon" title="c3">a</font></span>)</span>
 
<b>by </b><i><a class="txt" href="openlatt.html#E1:31"><i><font color="Green" title="E22">A1</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E4:31_1"/><b>then </b><i><font color="Green" title="E24">A4</font></i>: 
<font color="Maroon" title="c2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">X</font>
 
<b>by </b><i><a class="ref" href="openlatt.html#T19" target="_self" title="OPENLATT:th.19">Th19</a></i>;<br/>
<a NAME="E5:31_1"/><b>then </b><i><font color="Green" title="E25">A5</font></i>: 
<font color="Maroon" title="c2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Y</font>
 
<b>by </b><i><a class="txt" href="openlatt.html#E1:31"><i><font color="Green" title="E22">A1</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">Y</font> = <font color="Maroon" title="c6">Y</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="txt" href="openlatt.html#E1:31"><i><font color="Green" title="E22">A1</font></i></a>, <a class="txt" href="openlatt.html#E4:31_1"><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></i>;<br/>
<a NAME="E7:31_1"/>
<font color="Maroon" title="c8">Y</font> is    <a href="filter_0.html#M1" title="FILTER_0:mode.1">Filter</a> of <font color="Maroon" title="c1">L</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">p</font>, <font color="Maroon" title="c10">q</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="filter_0.html#D1" title="FILTER_0:def.1">FILTER_0:def 1</a><br/>

<b>thus </b><a NAME="E1:31_1_2"/>
( <font color="Maroon" title="c9">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Y</font> &amp; <font color="Maroon" title="c10">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Y</font> implies <font color="Maroon" title="c9">p</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c10">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Y</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:31_1_2_1"/>
<font color="Maroon" title="c9">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Y</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c11">X1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:31_1_2_1"/><i><font color="Green" title="E26">A6</font></i>: 
( <font color="Maroon" title="c9">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">X1</font> &amp; <font color="Maroon" title="c11">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Z</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>assume </b><a NAME="E4:31_1_2_1"/>
<font color="Maroon" title="c10">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Y</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c12">X2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:31_1_2_1"/><i><font color="Green" title="E27">A7</font></i>: 
( <font color="Maroon" title="c10">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">X2</font> &amp; <font color="Maroon" title="c12">X2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Z</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<a NAME="E7:31_1_2_1"/><i><font color="Green" title="E28">A8</font></i>: 
( <font color="Maroon" title="c11">X1</font>,<font color="Maroon" title="c12">X2</font> <a href="xboole_0.html#R3" title="XBOOLE_0:pred.3">are_c=-comparable</a>  &amp; <font color="Maroon" title="c11">X1</font> is    <a href="filter_0.html#M1" title="FILTER_0:mode.1">Filter</a> of <font color="Maroon" title="c1">L</font> &amp; <font color="Maroon" title="c12">X2</font> is    <a href="filter_0.html#M1" title="FILTER_0:mode.1">Filter</a> of <font color="Maroon" title="c1">L</font> )
 
<b>by </b><i><a class="txt" href="openlatt.html#E1:31"><i><font color="Green" title="E22">A1</font></i></a>, <a class="txt" href="openlatt.html#E3:31_1_2_1"><i><font color="Green" title="E26">A6</font></i></a>, <a class="txt" href="openlatt.html#E6:31_1_2_1"><i><font color="Green" title="E27">A7</font></i></a>, <a class="ref" href="openlatt.html#T19" target="_self" title="OPENLATT:th.19">Th19</a>, <a class="ref" href="ordinal1.html#D9" title="ORDINAL1:def.9">ORDINAL1:def 9</a></i>;<br/>
<a NAME="E8:31_1_2_1"/><b>then </b>
( <font color="Maroon" title="c11">X1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">X2</font> or <font color="Maroon" title="c12">X2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">X1</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D9" title="XBOOLE_0:def.9">XBOOLE_0:def 9</a></i>;<br/>
<a NAME="E9:31_1_2_1"/><b>then </b>
( <font color="Maroon" title="c9">p</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c10">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">X1</font> or <font color="Maroon" title="c9">p</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c10">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">X2</font> )
 
<b>by </b><i><a class="txt" href="openlatt.html#E3:31_1_2_1"><i><font color="Green" title="E26">A6</font></i></a>, <a class="txt" href="openlatt.html#E6:31_1_2_1"><i><font color="Green" title="E27">A7</font></i></a>, <a class="txt" href="openlatt.html#E7:31_1_2_1"><i><font color="Green" title="E28">A8</font></i></a>, <a class="ref" href="filter_0.html#D1" title="FILTER_0:def.1">FILTER_0:def 1</a></i>;<br/>
<b>hence </b><a NAME="E10:31_1_2_1"/>
<font color="Maroon" title="c9">p</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c10">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Y</font>
 <b>by </b><i><a class="txt" href="openlatt.html#E3:31_1_2_1"><i><font color="Green" title="E26">A6</font></i></a>, <a class="txt" href="openlatt.html#E6:31_1_2_1"><i><font color="Green" title="E27">A7</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>assume </b><a NAME="E2:31_1_2"/>
<font color="Maroon" title="c9">p</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c10">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Y</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c11">X1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:31_1_2"/><i><font color="Green" title="E26">A9</font></i>: 
( <font color="Maroon" title="c9">p</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c10">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">X1</font> &amp; <font color="Maroon" title="c11">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Z</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<a NAME="E5:31_1_2"/>
<font color="Maroon" title="c11">X1</font> is    <a href="filter_0.html#M1" title="FILTER_0:mode.1">Filter</a> of <font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="txt" href="openlatt.html#E1:31"><i><font color="Green" title="E22">A1</font></i></a>, <a class="txt" href="openlatt.html#E4:31_1_2"><i><font color="Green" title="E26">A9</font></i></a>, <a class="ref" href="openlatt.html#T19" target="_self" title="OPENLATT:th.19">Th19</a></i>;<br/>
<a NAME="E6:31_1_2"/><b>then </b>
( <font color="Maroon" title="c9">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">X1</font> &amp; <font color="Maroon" title="c10">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">X1</font> )
 
<b>by </b><i><a class="txt" href="openlatt.html#E4:31_1_2"><i><font color="Green" title="E26">A9</font></i></a>, <a class="ref" href="filter_0.html#D1" title="FILTER_0:def.1">FILTER_0:def 1</a></i>;<br/>
<b>hence </b><a NAME="E7:31_1_2"/>
( <font color="Maroon" title="c9">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Y</font> &amp; <font color="Maroon" title="c10">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Y</font> )
 <b>by </b><i><a class="txt" href="openlatt.html#E4:31_1_2"><i><font color="Green" title="E26">A9</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="E8:31_1"/>
<font color="Maroon" title="c6">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="openlatt.html#K10" title="OPENLATT:func.10">SF_have</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="openlatt.html#K10" title="OPENLATT:func.10">SF_have</a> <font color="Maroon" title="c3">a</font></span>)</span>
 <b>by </b><i><a class="txt" href="openlatt.html#E1:31_1"><i><font color="Green" title="E23">A2</font></i></a>, <a class="txt" href="openlatt.html#E5:31_1"><i><font color="Green" title="E25">A5</font></i></a>, <a class="txt" href="openlatt.html#E27"><i><font color="Green" title="E19">Lm1</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:31"/>
( <font color="Maroon" title="c6">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="openlatt.html#K10" title="OPENLATT:func.10">SF_have</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="openlatt.html#K10" title="OPENLATT:func.10">SF_have</a> <font color="Maroon" title="c3">a</font></span>)</span> &amp; ( for <font color="Olive" title="b1">X1</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Z</font> holds <br/><font color="Olive" title="b1">X1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">Y</font> ) )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></i>;<br/>


</div>
