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

<b>let </b><font color="Maroon" title="c1">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">BL</font> be  non <a href="realset2.html#V3" title="REALSET2:attr.3">trivial</a> <a href="lattices.html#NM8" title="LATTICES:NM.8">B_Lattice</a>;<br/>



<b>assume </b><a NAME="E1:40"/><i><font color="Green" title="E34">A1</font></i>: 
(  <a href="lopclset.html#K7" title="LOPCLSET:func.7">ultraset</a> <font color="Maroon" title="c2">BL</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c1">X</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="lopclset.html#K10" title="LOPCLSET:func.10">StoneR</a> <font color="Maroon" title="c2">BL</font></span>)</span> )
 ;<br/>

<b>assume </b><a NAME="E2:40"/><i><font color="Green" title="E35">A2</font></i>: 
for <font color="Olive" title="b1">Y</font> being  <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of <font color="Maroon" title="c1">X</font>  holds not  <a href="lopclset.html#K7" title="LOPCLSET:func.7">ultraset</a> <font color="Maroon" title="c2">BL</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Olive" title="b1">Y</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c3">Y</font> being   <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E4:40"/><i><font color="Green" title="E36">A3</font></i>: 
not <font color="Maroon" title="c3">Y</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 <b>by </b><i><a class="txt" href="lopclset.html#E1:40"><i><font color="Green" title="E34">A1</font></i></a>, <a class="ref" href="lopclset.html#T32" target="_self" title="LOPCLSET:th.32">Th32</a>, <a class="ref" href="zfmisc_1.html#T2" title="ZFMISC_1:th.2">ZFMISC_1:2</a></i>;<br/>
<a NAME="E5:40"/><i><font color="Green" title="E37">A4</font></i>: 
<font color="Maroon" title="c3">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="finsub_1.html#D5" title="FINSUB_1:def.5">FINSUB_1:def 5</a></i>;<br/>
<a NAME="E6:40"/><b>then </b><i><font color="Green" title="E38">A5</font></i>: 
<font color="Maroon" title="c3">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="lopclset.html#K10" title="LOPCLSET:func.10">StoneR</a> <font color="Maroon" title="c2">BL</font>
 
<b>by </b><i><a class="txt" href="lopclset.html#E1:40"><i><font color="Green" title="E34">A1</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c4">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">Y</font>;<br/>
<a NAME="E8:40"/><i><font color="Green" title="E39">A6</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="txt" href="lopclset.html#E4:40"><i><font color="Green" title="E36">A3</font></i></a>, <a class="txt" href="lopclset.html#E5:40"><i><font color="Green" title="E37">A4</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E9:40"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lopclset.html#K10" title="LOPCLSET:func.10">StoneR</a> <font color="Maroon" title="c2">BL</font>
 
<b>by </b><i><a class="txt" href="lopclset.html#E4:40"><i><font color="Green" title="E36">A3</font></i></a>, <a class="txt" href="lopclset.html#E6:40"><i><font color="Green" title="E38">A5</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="c5">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">BL</font><b> such that </b><br/><a NAME="E11:40"/><i><font color="Green" title="E40">A7</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="lopclset.html#K9" title="LOPCLSET:func.9">UFilter</a> <font color="Maroon" title="c2">BL</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">b</font>
 <b>by </b><i><a class="ref" href="lopclset.html#T26" target="_self" title="LOPCLSET:th.26">Th26</a></i>;<br/>
<b>set </b><font color="Maroon" title="c6">CY</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">a</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a> </span>)</span> where <font color="Olive" title="b1">a</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">BL</font> : <span class="p1">(<span class="default"><a href="lopclset.html#K9" title="LOPCLSET:func.9">UFilter</a> <font color="Maroon" title="c2">BL</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> </span>}</span> ;<br/>
<b>consider </b><font color="Maroon" title="c7">c</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">BL</font><b> such that </b><br/><a NAME="E13:40"/><i><font color="Green" title="E41">A8</font></i>: 
<font color="Maroon" title="c7">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a> 
 ;<br/>
<a NAME="E14:40"/><i><font color="Green" title="E42">A9</font></i>: 
<font color="Maroon" title="c7">c</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">a</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a> </span>)</span> where <font color="Olive" title="b1">a</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">BL</font> : <span class="p1">(<span class="default"><a href="lopclset.html#K9" title="LOPCLSET:func.9">UFilter</a> <font color="Maroon" title="c2">BL</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="lopclset.html#E8:40"><i><font color="Green" title="E39">A6</font></i></a>, <a class="txt" href="lopclset.html#E11:40"><i><font color="Green" title="E40">A7</font></i></a>, <a class="txt" href="lopclset.html#E13:40"><i><font color="Green" title="E41">A8</font></i></a></i>;<br/>
<a NAME="E15:40"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">a</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a> </span>)</span> where <font color="Olive" title="b1">a</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">BL</font> : <span class="p1">(<span class="default"><a href="lopclset.html#K9" title="LOPCLSET:func.9">UFilter</a> <font color="Maroon" title="c2">BL</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">BL</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">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="E1:40_1"/>
<font color="Maroon" title="c8">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">a</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a> </span>)</span> where <font color="Olive" title="b1">a</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">BL</font> : <span class="p1">(<span class="default"><a href="lopclset.html#K9" title="LOPCLSET:func.9">UFilter</a> <font color="Maroon" title="c2">BL</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c9">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">BL</font><b> such that </b><br/><a NAME="E3:40_1"/><i><font color="Green" title="E43">A10</font></i>: 
( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a>  &amp; <span class="p1">(<span class="default"><a href="lopclset.html#K9" title="LOPCLSET:func.9">UFilter</a> <font color="Maroon" title="c2">BL</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> )
 ;<br/>
<b>thus </b><a NAME="E4:40_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">BL</font>
 <b>by </b><i><a class="txt" href="lopclset.html#E3:40_1"><i><font color="Green" title="E43">A10</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c8">CY</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">a</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a> </span>)</span> where <font color="Olive" title="b1">a</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">BL</font> : <span class="p1">(<span class="default"><a href="lopclset.html#K9" title="LOPCLSET:func.9">UFilter</a> <font color="Maroon" title="c2">BL</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> </span>}</span>  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="c2">BL</font> <b>by </b><i><a class="txt" href="lopclset.html#E14:40"><i><font color="Green" title="E42">A9</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c9">H</font> = <span class="p1"><a href="filter_0.html#K3" title="FILTER_0:func.3">&lt;.</a><span class="default"><font color="Maroon" title="c8">CY</font></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span>;<br/>
<a NAME="E17:40"/>
for <font color="Olive" title="b1">B</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">BL</font>  st <font color="Olive" title="b1">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">CY</font> holds <br/> <a href="lattice4.html#K2" title="LATTICE4:func.2">FinMeet</a> <font color="Olive" title="b1">B</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c2">BL</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">B</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">BL</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:40_2"/><i><font color="Green" title="E43">A11</font></i>: 
<font color="Maroon" title="c10">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">CY</font>
 <b>and </b><br/><a NAME="E2:40_2"/><i><font color="Green" title="E44">A12</font></i>: 
 <a href="lattice4.html#K2" title="LATTICE4:func.2">FinMeet</a> <font color="Maroon" title="c10">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c2">BL</font>
 ;<br/>

<a NAME="E3:40_2"/><i><font color="Green" title="E45">A13</font></i>: 
<font color="Maroon" title="c10">B</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">BL</font>
 
<b>by </b><i><a class="ref" href="finsub_1.html#T32" title="FINSUB_1:th.32">FINSUB_1:32</a></i>;<br/>
<a NAME="E4:40_2"/><i><font color="Green" title="E46">A14</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="lopclset.html#K9" title="LOPCLSET:func.9">UFilter</a> <font color="Maroon" title="c2">BL</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">BL</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E5:40_2"/><i><font color="Green" title="E47">A15</font></i>: 
<span class="p1">(<span class="default"><a href="lopclset.html#K9" title="LOPCLSET:func.9">UFilter</a> <font color="Maroon" title="c2">BL</font></span>)</span> <a href="setwiseo.html#K8" title="SETWISEO:func.8">.:</a> <font color="Maroon" title="c10">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="lopclset.html#K9" title="LOPCLSET:func.9">UFilter</a> <font color="Maroon" title="c2">BL</font></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T144" title="RELAT_1:th.144">RELAT_1:144</a></i>;<br/>
<a NAME="E6:40_2"/>
<span class="p1">(<span class="default"><a href="lopclset.html#K9" title="LOPCLSET:func.9">UFilter</a> <font color="Maroon" title="c2">BL</font></span>)</span> <a href="setwiseo.html#K8" title="SETWISEO:func.8">.:</a> <font color="Maroon" title="c10">B</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> <span class="p1">(<span class="default"><a href="lopclset.html#K7" title="LOPCLSET:func.7">ultraset</a> <font color="Maroon" title="c2">BL</font></span>)</span>
 
<b>by </b><i><a class="txt" href="lopclset.html#E5:40_2"><i><font color="Green" title="E47">A15</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c11">D</font> = <span class="p1">(<span class="default"><a href="lopclset.html#K9" title="LOPCLSET:func.9">UFilter</a> <font color="Maroon" title="c2">BL</font></span>)</span> <a href="setwiseo.html#K8" title="SETWISEO:func.8">.:</a> <font color="Maroon" title="c10">B</font> as  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 <span class="p1">(<span class="default"><a href="lopclset.html#K7" title="LOPCLSET:func.7">ultraset</a> <font color="Maroon" title="c2">BL</font></span>)</span> <b>by </b><i><a class="txt" href="lopclset.html#E3:40_2"><i><font color="Green" title="E45">A13</font></i></a>, <a class="txt" href="lopclset.html#E4:40_2"><i><font color="Green" title="E46">A14</font></i></a>, <a class="ref" href="relat_1.html#T152" title="RELAT_1:th.152">RELAT_1:152</a></i>;<br/>
<div><i><font color="Green" title="E48">A16</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2_1"><b>now </b></a><div class="add"><b>consider </b><font color="Maroon" title="c12">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="lopclset.html#K9" title="LOPCLSET:func.9">UFilter</a> <font color="Maroon" title="c2">BL</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c2">BL</font></span>)</span>;<br/><b>assume </b><a NAME="E2:40_2_1"/>
 <a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <span class="p1">(<span class="default"><a href="lopclset.html#K7" title="LOPCLSET:func.7">ultraset</a> <font color="Maroon" title="c2">BL</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="lopclset.html#K9" title="LOPCLSET:func.9">UFilter</a> <font color="Maroon" title="c2">BL</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c2">BL</font></span>)</span>
 ;<br/><a NAME="E3:40_2_1"/><b>then </b>
 ex <font color="Olive" title="b1">F</font> being   <a href="filter_0.html#M1" title="FILTER_0:mode.1">Filter</a> of <font color="Maroon" title="c2">BL</font> st <br/>( <font color="Olive" title="b1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">x</font> &amp; <font color="Olive" title="b1">F</font> <a href="filter_0.html#NR1" title="FILTER_0:NR.1">is_ultrafilter</a>  &amp;  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c2">BL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">F</font> )
 <b>by </b><i><a class="ref" href="lopclset.html#T20" target="_self" title="LOPCLSET:th.20">Th20</a></i>;<br/><b>hence </b><a NAME="E4:40_2_1"/>
contradiction
 <b>by </b><i><a class="ref" href="lopclset.html#T35" target="_self" title="LOPCLSET:th.35">Th35</a></i>;<br/></div><b>end;</b></div>
<a NAME="E9:40_2"/><i><font color="Green" title="E49">A17</font></i>: 
( the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <font color="Maroon" title="c2">BL</font> is <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a> &amp; the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <font color="Maroon" title="c2">BL</font> is <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a> &amp; the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <font color="Maroon" title="c2">BL</font> is <a href="binop_1.html#V3" title="BINOP_1:attr.3">idempotent</a> )
 
<b>by </b><i><a class="ref" href="lattice2.html#T30" title="LATTICE2:th.30">LATTICE2:30</a>, <a class="ref" href="lattice2.html#T31" title="LATTICE2:th.31">LATTICE2:31</a>, <a class="ref" href="lattice2.html#T32" title="LATTICE2:th.32">LATTICE2:32</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="lopclset.html#K7" title="LOPCLSET:func.7">ultraset</a> <font color="Maroon" title="c2">BL</font></span>)</span>,  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="lopclset.html#K7" title="LOPCLSET:func.7">ultraset</a> <font color="Maroon" title="c2">BL</font></span>)</span>) <b>-&gt; </b>   <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">bool</a> <span class="p1">(<span class="default"><a href="lopclset.html#K7" title="LOPCLSET:func.7">ultraset</a> <font color="Maroon" title="c2">BL</font></span>)</span> = $1 <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> $2;<br/>
<b>consider </b><font color="Maroon" title="c12">G</font> being   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1">(<span class="default"><a href="lopclset.html#K7" title="LOPCLSET:func.7">ultraset</a> <font color="Maroon" title="c2">BL</font></span>)</span><b> such that </b><br/><a NAME="E11:40_2"/><i><font color="Green" title="E50">A18</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="lopclset.html#K7" title="LOPCLSET:func.7">ultraset</a> <font color="Maroon" title="c2">BL</font></span>)</span> holds  <font color="Maroon" title="c12">G</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>)
 <b>from </b><i><a class="ref" href="binop_1.html#S4" title="BINOP_1:sch.4">BINOP_1:sch 4</a>();<br/></i>
<a NAME="E12:40_2"/><i><font color="Green" title="E51">A19</font></i>: 
<font color="Maroon" title="c12">G</font> is <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">x</font>, <font color="Maroon" title="c14">y</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="lopclset.html#K7" title="LOPCLSET:func.7">ultraset</a> <font color="Maroon" title="c2">BL</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="binop_1.html#D13" title="BINOP_1:def.13">BINOP_1:def 13</a><br/>

<font color="Maroon" title="c12">G</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c14">y</font> = 
<font color="Maroon" title="c14">y</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c13">x</font>
<b>by </b><i><a class="txt" href="lopclset.html#E11:40_2"><i><font color="Green" title="E50">A18</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c12">G</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c14">y</font>,<font color="Maroon" title="c13">x</font>
<b>by </b><i><a class="txt" href="lopclset.html#E11:40_2"><i><font color="Green" title="E50">A18</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E2:40_2_2"/>
<font color="Maroon" title="c12">G</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c14">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">G</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c14">y</font>,<font color="Maroon" title="c13">x</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E13:40_2"/><i><font color="Green" title="E52">A20</font></i>: 
<font color="Maroon" title="c12">G</font> is <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">x</font>, <font color="Maroon" title="c14">y</font>, <font color="Maroon" title="c15">z</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="lopclset.html#K7" title="LOPCLSET:func.7">ultraset</a> <font color="Maroon" title="c2">BL</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="binop_1.html#D14" title="BINOP_1:def.14">BINOP_1:def 14</a><br/>

<font color="Maroon" title="c12">G</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c13">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">G</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c14">y</font>,<font color="Maroon" title="c15">z</font></span>)</span> = 
<font color="Maroon" title="c12">G</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c13">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">y</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c15">z</font></span>)</span>
<b>by </b><i><a class="txt" href="lopclset.html#E11:40_2"><i><font color="Green" title="E50">A18</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c13">x</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">y</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c15">z</font></span>)</span>
<b>by </b><i><a class="txt" href="lopclset.html#E11:40_2"><i><font color="Green" title="E50">A18</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">x</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c14">y</font></span>)</span> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c15">z</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T16" title="XBOOLE_1:th.16">XBOOLE_1:16</a></i>
<br/>.= 
<font color="Maroon" title="c12">G</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">x</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c14">y</font></span>)</span>,<font color="Maroon" title="c15">z</font>
<b>by </b><i><a class="txt" href="lopclset.html#E11:40_2"><i><font color="Green" title="E50">A18</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c12">G</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">G</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c14">y</font></span>)</span>,<font color="Maroon" title="c15">z</font>
<b>by </b><i><a class="txt" href="lopclset.html#E11:40_2"><i><font color="Green" title="E50">A18</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E2:40_2_3"/>
<font color="Maroon" title="c12">G</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c13">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">G</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c14">y</font>,<font color="Maroon" title="c15">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">G</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">G</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c14">y</font></span>)</span>,<font color="Maroon" title="c15">z</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E14:40_2"/><i><font color="Green" title="E53">A21</font></i>: 
<font color="Maroon" title="c12">G</font> is <a href="binop_1.html#V3" title="BINOP_1:attr.3">idempotent</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">x</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="lopclset.html#K7" title="LOPCLSET:func.7">ultraset</a> <font color="Maroon" title="c2">BL</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="binop_1.html#D15" title="BINOP_1:def.15">BINOP_1:def 15</a><br/>

<font color="Maroon" title="c12">G</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c13">x</font> = 
<font color="Maroon" title="c13">x</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c13">x</font>
<b>by </b><i><a class="txt" href="lopclset.html#E11:40_2"><i><font color="Green" title="E50">A18</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c13">x</font>

;<br/>
<b>hence </b><a NAME="E2:40_2_4"/>
<font color="Maroon" title="c12">G</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">x</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E15:40_2"/><i><font color="Green" title="E54">A22</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">BL</font> holds  <span class="p1">(<span class="default"><a href="lopclset.html#K9" title="LOPCLSET:func.9">UFilter</a> <font color="Maroon" title="c2">BL</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default">the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <font color="Maroon" title="c2">BL</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">G</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopclset.html#K9" title="LOPCLSET:func.9">UFilter</a> <font color="Maroon" title="c2">BL</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopclset.html#K9" title="LOPCLSET:func.9">UFilter</a> <font color="Maroon" title="c2">BL</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">y</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">x</font>, <font color="Maroon" title="c14">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">BL</font>;<br/>

<b>thus </b><span class="p1">(<span class="default"><a href="lopclset.html#K9" title="LOPCLSET:func.9">UFilter</a> <font color="Maroon" title="c2">BL</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default">the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <font color="Maroon" title="c2">BL</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c14">y</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="lopclset.html#K9" title="LOPCLSET:func.9">UFilter</a> <font color="Maroon" title="c2">BL</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">x</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c14">y</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopclset.html#K9" title="LOPCLSET:func.9">UFilter</a> <font color="Maroon" title="c2">BL</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">x</font></span>)</span> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopclset.html#K9" title="LOPCLSET:func.9">UFilter</a> <font color="Maroon" title="c2">BL</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">y</font></span>)</span>
<b>by </b><i><a class="ref" href="lopclset.html#T23" target="_self" title="LOPCLSET:th.23">Th23</a></i>
<br/>.= 
<font color="Maroon" title="c12">G</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopclset.html#K9" title="LOPCLSET:func.9">UFilter</a> <font color="Maroon" title="c2">BL</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">x</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopclset.html#K9" title="LOPCLSET:func.9">UFilter</a> <font color="Maroon" title="c2">BL</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">y</font></span>)</span>
<b>by </b><i><a class="txt" href="lopclset.html#E11:40_2"><i><font color="Green" title="E50">A18</font></i></a></i>
;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon" title="c13">DD</font> = <font color="Maroon" title="c11">D</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p2">(<span class="default"><a href="lopclset.html#K7" title="LOPCLSET:func.7">ultraset</a> <font color="Maroon" title="c2">BL</font></span>)</span></span>)</span> ;<br/>
<a NAME="E17:40_2"/>
 <a href="grcat_1.html#K7" title="GRCAT_1:func.7">id</a> <font color="Maroon" title="c2">BL</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">BL</font>
 
<b>by </b><i><a class="ref" href="grcat_1.html#D11" title="GRCAT_1:def.11">GRCAT_1:def 11</a></i>;<br/>
<b>then </b><i><font color="Green" title="E55">A23</font></i>: <span class="p1">(<span class="default"><a href="lopclset.html#K9" title="LOPCLSET:func.9">UFilter</a> <font color="Maroon" title="c2">BL</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="lattice4.html#K2" title="LATTICE4:func.2">FinMeet</a> <font color="Maroon" title="c10">B</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="lopclset.html#K9" title="LOPCLSET:func.9">UFilter</a> <font color="Maroon" title="c2">BL</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="lattice2.html#K3" title="LATTICE2:func.3">FinMeet</a> <font color="Maroon" title="c10">B</font>,<span class="p2">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">BL</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lattice4.html#D13" title="LATTICE4:def.13">LATTICE4:def 13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lopclset.html#K9" title="LOPCLSET:func.9">UFilter</a> <font color="Maroon" title="c2">BL</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default">the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <font color="Maroon" title="c2">BL</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c10">B</font>,<span class="p2">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">BL</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lattice2.html#D4" title="LATTICE2:def.4">LATTICE2:def 4</a></i>
<br/>.= 
<font color="Maroon" title="c12">G</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c10">B</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopclset.html#K9" title="LOPCLSET:func.9">UFilter</a> <font color="Maroon" title="c2">BL</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">BL</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="lopclset.html#E9:40_2"><i><font color="Green" title="E49">A17</font></i></a>, <a class="txt" href="lopclset.html#E12:40_2"><i><font color="Green" title="E51">A19</font></i></a>, <a class="txt" href="lopclset.html#E13:40_2"><i><font color="Green" title="E52">A20</font></i></a>, <a class="txt" href="lopclset.html#E14:40_2"><i><font color="Green" title="E53">A21</font></i></a>, <a class="txt" href="lopclset.html#E15:40_2"><i><font color="Green" title="E54">A22</font></i></a>, <a class="ref" href="setwiseo.html#T39" title="SETWISEO:th.39">SETWISEO:39</a></i>
<br/>.= 
<font color="Maroon" title="c12">G</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c10">B</font>,<span class="p1">(<span class="default"><a href="lopclset.html#K9" title="LOPCLSET:func.9">UFilter</a> <font color="Maroon" title="c2">BL</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T23" title="FUNCT_2:th.23">FUNCT_2:23</a></i>
<br/>.= 
<font color="Maroon" title="c12">G</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c10">B</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p3">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p4">(<span class="default"><a href="lopclset.html#K7" title="LOPCLSET:func.7">ultraset</a> <font color="Maroon" title="c2">BL</font></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="lopclset.html#K9" title="LOPCLSET:func.9">UFilter</a> <font color="Maroon" title="c2">BL</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T23" title="FUNCT_2:th.23">FUNCT_2:23</a></i>
<br/>.= 
<font color="Maroon" title="c12">G</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c13">DD</font>,<span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p2">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p3">(<span class="default"><a href="lopclset.html#K7" title="LOPCLSET:func.7">ultraset</a> <font color="Maroon" title="c2">BL</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="lopclset.html#E12:40_2"><i><font color="Green" title="E51">A19</font></i></a>, <a class="txt" href="lopclset.html#E13:40_2"><i><font color="Green" title="E52">A20</font></i></a>, <a class="txt" href="lopclset.html#E14:40_2"><i><font color="Green" title="E53">A21</font></i></a>, <a class="ref" href="setwiseo.html#T38" title="SETWISEO:th.38">SETWISEO:38</a></i>
;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p2">(<span class="default"><a href="lopclset.html#K7" title="LOPCLSET:func.7">ultraset</a> <font color="Maroon" title="c2">BL</font></span>)</span></span>)</span>] <b>means </b>for <font color="Olive" title="b1">D</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <span class="p1">(<span class="default"><a href="lopclset.html#K7" title="LOPCLSET:func.7">ultraset</a> <font color="Maroon" title="c2">BL</font></span>)</span>  st <font color="Olive" title="b1">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 holds <br/><font color="Maroon" title="c12">G</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> $1,<span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p2">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p3">(<span class="default"><a href="lopclset.html#K7" title="LOPCLSET:func.7">ultraset</a> <font color="Maroon" title="c2">BL</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Olive" title="b1">D</font>;<br/>
<a NAME="E19:40_2"/><i><font color="Green" title="E56">A24</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="setwiseo.html#K1" title="SETWISEO:func.1">{}.</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p2">(<span class="default"><a href="lopclset.html#K7" title="LOPCLSET:func.7">ultraset</a> <font color="Maroon" title="c2">BL</font></span>)</span></span>)</span>]
 
;<br/>
<a NAME="E20:40_2"/><i><font color="Green" title="E57">A25</font></i>: 
for <font color="Olive" title="b1">DD</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p2">(<span class="default"><a href="lopclset.html#K7" title="LOPCLSET:func.7">ultraset</a> <font color="Maroon" title="c2">BL</font></span>)</span></span>)</span><br/> for <font color="Olive" title="b2">b</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="lopclset.html#K7" title="LOPCLSET:func.7">ultraset</a> <font color="Maroon" title="c2">BL</font></span>)</span>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">DD</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">DD</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p1"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Olive" title="b2">b</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">DD</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p2">(<span class="default"><a href="lopclset.html#K7" title="LOPCLSET:func.7">ultraset</a> <font color="Maroon" title="c2">BL</font></span>)</span></span>)</span>;<br/><b>let </b><font color="Maroon" title="c15">b</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="lopclset.html#K7" title="LOPCLSET:func.7">ultraset</a> <font color="Maroon" title="c2">BL</font></span>)</span>;<br/>



<b>assume </b><a NAME="E1:40_2_7"/><i><font color="Green" title="E58">A26</font></i>: 
for <font color="Olive" title="b1">D</font> being 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 <span class="p1">(<span class="default"><a href="lopclset.html#K7" title="LOPCLSET:func.7">ultraset</a> <font color="Maroon" title="c2">BL</font></span>)</span>  st <font color="Olive" title="b1">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">DD</font> holds <br/><font color="Maroon" title="c12">G</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c14">DD</font>,<span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p2">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p3">(<span class="default"><a href="lopclset.html#K7" title="LOPCLSET:func.7">ultraset</a> <font color="Maroon" title="c2">BL</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Olive" title="b1">D</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c16">D</font> be  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 <span class="p1">(<span class="default"><a href="lopclset.html#K7" title="LOPCLSET:func.7">ultraset</a> <font color="Maroon" title="c2">BL</font></span>)</span>;<br/>

<b>assume </b><a NAME="E2:40_2_7"/><i><font color="Green" title="E59">A27</font></i>: 
<font color="Maroon" title="c16">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">DD</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c15">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2_7_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_2_7_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:40_2_7_1_1"/>
( <font color="Maroon" title="c14">DD</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> or  not <font color="Maroon" title="c14">DD</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_2_7_1_1_1"><b>suppose </b></a><a NAME="E1:40_2_7_1_1_1"/><i><font color="Green" title="E60">A28</font></i>: 
<font color="Maroon" title="c14">DD</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><div class="add"><b>hence </b><font color="Maroon" title="c12">G</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">DD</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p2"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c15">b</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p2">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p3">(<span class="default"><a href="lopclset.html#K7" title="LOPCLSET:func.7">ultraset</a> <font color="Maroon" title="c2">BL</font></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p2">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p3">(<span class="default"><a href="lopclset.html#K7" title="LOPCLSET:func.7">ultraset</a> <font color="Maroon" title="c2">BL</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">b</font>
<b>by </b><i><a class="txt" href="lopclset.html#E12:40_2"><i><font color="Green" title="E51">A19</font></i></a>, <a class="txt" href="lopclset.html#E13:40_2"><i><font color="Green" title="E52">A20</font></i></a>, <a class="ref" href="setwiseo.html#T26" title="SETWISEO:th.26">SETWISEO:26</a></i>
<br/>.= 
<font color="Maroon" title="c15">b</font>
<b>by </b><i><a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></i>
<br/>.= 
 <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c16">D</font>
<b>by </b><i><a class="txt" href="lopclset.html#E2:40_2_7"><i><font color="Green" title="E59">A27</font></i></a>, <a class="txt" href="lopclset.html#E1:40_2_7_1_1_1"><i><font color="Green" title="E60">A28</font></i></a>, <a class="ref" href="setfam_1.html#T11" title="SETFAM_1:th.11">SETFAM_1:11</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_2_7_1_1_2"><b>suppose </b></a><a NAME="E1:40_2_7_1_1_2"/><i><font color="Green" title="E60">A29</font></i>: 
not <font color="Maroon" title="c14">DD</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><div class="add"><a NAME="E2:40_2_7_1_1_2"/>
<font color="Maroon" title="c14">DD</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c16">D</font>
 <b>by </b><i><a class="txt" href="lopclset.html#E2:40_2_7"><i><font color="Green" title="E59">A27</font></i></a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c17">D1</font> = <font color="Maroon" title="c14">DD</font> as  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 <span class="p1">(<span class="default"><a href="lopclset.html#K7" title="LOPCLSET:func.7">ultraset</a> <font color="Maroon" title="c2">BL</font></span>)</span> <b>by </b><i><a class="txt" href="lopclset.html#E1:40_2_7_1_1_2"><i><font color="Green" title="E60">A29</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c18">D1</font> = <font color="Maroon" title="c17">D1</font> as  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 <span class="p1">(<span class="default"><a href="lopclset.html#K7" title="LOPCLSET:func.7">ultraset</a> <font color="Maroon" title="c2">BL</font></span>)</span> ;<br/><b>thus </b><font color="Maroon" title="c12">G</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">DD</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p2"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c15">b</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p2">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p3">(<span class="default"><a href="lopclset.html#K7" title="LOPCLSET:func.7">ultraset</a> <font color="Maroon" title="c2">BL</font></span>)</span></span>)</span></span>)</span> = 
<font color="Maroon" title="c12">G</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">G</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c14">DD</font>,<span class="p2">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p3">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p4">(<span class="default"><a href="lopclset.html#K7" title="LOPCLSET:func.7">ultraset</a> <font color="Maroon" title="c2">BL</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p3">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p4">(<span class="default"><a href="lopclset.html#K7" title="LOPCLSET:func.7">ultraset</a> <font color="Maroon" title="c2">BL</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">b</font></span>)</span>
<b>by </b><i><a class="txt" href="lopclset.html#E12:40_2"><i><font color="Green" title="E51">A19</font></i></a>, <a class="txt" href="lopclset.html#E13:40_2"><i><font color="Green" title="E52">A20</font></i></a>, <a class="txt" href="lopclset.html#E14:40_2"><i><font color="Green" title="E53">A21</font></i></a>, <a class="txt" href="lopclset.html#E1:40_2_7_1_1_2"><i><font color="Green" title="E60">A29</font></i></a>, <a class="ref" href="setwiseo.html#T29" title="SETWISEO:th.29">SETWISEO:29</a></i>
<br/>.= 
<font color="Maroon" title="c12">G</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">G</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c14">DD</font>,<span class="p2">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p3">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p4">(<span class="default"><a href="lopclset.html#K7" title="LOPCLSET:func.7">ultraset</a> <font color="Maroon" title="c2">BL</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c15">b</font>
<b>by </b><i><a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">G</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c14">DD</font>,<span class="p2">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p3">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p4">(<span class="default"><a href="lopclset.html#K7" title="LOPCLSET:func.7">ultraset</a> <font color="Maroon" title="c2">BL</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c15">b</font>
<b>by </b><i><a class="txt" href="lopclset.html#E11:40_2"><i><font color="Green" title="E50">A18</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c18">D1</font></span>)</span> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c15">b</font>
<b>by </b><i><a class="txt" href="lopclset.html#E1:40_2_7"><i><font color="Green" title="E58">A26</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c18">D1</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c15">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
<b>by </b><i><a class="ref" href="setfam_1.html#T11" title="SETFAM_1:th.11">SETFAM_1:11</a></i>
<br/>.= 
 <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c16">D</font>
<b>by </b><i><a class="txt" href="lopclset.html#E2:40_2_7"><i><font color="Green" title="E59">A27</font></i></a>, <a class="ref" href="setfam_1.html#T10" title="SETFAM_1:th.10">SETFAM_1:10</a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:40_2_7"/>
<font color="Maroon" title="c12">G</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">DD</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p2"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c15">b</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p2">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p3">(<span class="default"><a href="lopclset.html#K7" title="LOPCLSET:func.7">ultraset</a> <font color="Maroon" title="c2">BL</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c16">D</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E21:40_2"/>
for <font color="Olive" title="b1">DD</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p2">(<span class="default"><a href="lopclset.html#K7" title="LOPCLSET:func.7">ultraset</a> <font color="Maroon" title="c2">BL</font></span>)</span></span>)</span> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">DD</font>]
 
<b>from </b><i><a class="ref" href="setwiseo.html#S4" title="SETWISEO:sch.4">SETWISEO:sch 4</a>(<a class="txt" href="lopclset.html#E19:40_2"><i><font color="Green" title="E56">A24</font></i></a>, <a class="txt" href="lopclset.html#E20:40_2"><i><font color="Green" title="E57">A25</font></i></a>);<br/></i>
<a NAME="E22:40_2"/><b>then </b>
 <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c11">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <span class="p1">(<span class="default"><a href="lopclset.html#K7" title="LOPCLSET:func.7">ultraset</a> <font color="Maroon" title="c2">BL</font></span>)</span>
 
<b>by </b><i><a class="txt" href="lopclset.html#E2:40_2"><i><font color="Green" title="E44">A12</font></i></a>, <a class="txt" href="lopclset.html#E8:40_2"><i><font color="Green" title="E48">A16</font></i></a>, <a class="txt" href="lopclset.html#E18:40_2"><i><font color="Green" title="E55">A23</font></i></a></i>;<br/>
<b>then </b><i><font color="Green" title="E58">A30</font></i>:  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K7" title="SETFAM_1:func.7">COMPLEMENT</a> <font color="Maroon" title="c11">D</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="subset_1.html#K2" title="SUBSET_1:func.2">[#]</a> <span class="p2">(<span class="default"><a href="lopclset.html#K7" title="LOPCLSET:func.7">ultraset</a> <font color="Maroon" title="c2">BL</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
<b>by </b><i><a class="ref" href="setfam_1.html#T48" title="SETFAM_1:th.48">SETFAM_1:48</a></i>
<br/>.= 
 <a href="lopclset.html#K7" title="LOPCLSET:func.7">ultraset</a> <font color="Maroon" title="c2">BL</font>

;<br/>
<a NAME="E24:40_2"/>
 <a href="setfam_1.html#K7" title="SETFAM_1:func.7">COMPLEMENT</a> <font color="Maroon" title="c11">D</font> is   <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of <font color="Maroon" title="c1">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2_9"><b>proof </b></a><div class="add">

<a NAME="E1:40_2_9"/><i><font color="Green" title="E59">A31</font></i>: 
 <a href="setfam_1.html#K7" title="SETFAM_1:func.7">COMPLEMENT</a> <font color="Maroon" title="c11">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2_9_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">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="E1:40_2_9_1"/><i><font color="Green" title="E60">A32</font></i>: 
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K7" title="SETFAM_1:func.7">COMPLEMENT</a> <font color="Maroon" title="c11">D</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c15">c</font> = <font color="Maroon" title="c14">x</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="lopclset.html#K7" title="LOPCLSET:func.7">ultraset</a> <font color="Maroon" title="c2">BL</font></span>)</span> ;<br/>
<a NAME="E3:40_2_9_1"/>
<font color="Maroon" title="c15">c</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">D</font>
 
<b>by </b><i><a class="txt" href="lopclset.html#E1:40_2_9_1"><i><font color="Green" title="E60">A32</font></i></a>, <a class="ref" href="setfam_1.html#D8" title="SETFAM_1:def.8">SETFAM_1:def 8</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c16">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:40_2_9_1"/><i><font color="Green" title="E61">A33</font></i>: 
( <font color="Maroon" title="c16">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="lopclset.html#K9" title="LOPCLSET:func.9">UFilter</a> <font color="Maroon" title="c2">BL</font></span>)</span> &amp; <font color="Maroon" title="c16">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">B</font> &amp; <font color="Maroon" title="c15">c</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="lopclset.html#K9" title="LOPCLSET:func.9">UFilter</a> <font color="Maroon" title="c2">BL</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">a</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/>
<b>reconsider </b><font color="Maroon" title="c17">a</font> = <font color="Maroon" title="c16">a</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">BL</font> <b>by </b><i><a class="txt" href="lopclset.html#E5:40_2_9_1"><i><font color="Green" title="E61">A33</font></i></a></i>;<br/>
<a NAME="E7:40_2_9_1"/>
<font color="Maroon" title="c17">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">CY</font>
 
<b>by </b><i><a class="txt" href="lopclset.html#E1:40_2"><i><font color="Green" title="E43">A11</font></i></a>, <a class="txt" href="lopclset.html#E5:40_2_9_1"><i><font color="Green" title="E61">A33</font></i></a></i>;<br/>
<a NAME="E8:40_2_9_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c17">a</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a> </span>)</span> <a href="lattices.html#K7" title="LATTICES:func.7">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">CY</font>
 
<b>by </b><i><a class="ref" href="lattices.html#T49" title="LATTICES:th.49">LATTICES:49</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c18">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">BL</font><b> such that </b><br/><a NAME="E10:40_2_9_1"/><i><font color="Green" title="E62">A34</font></i>: 
( <font color="Maroon" title="c18">b</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">a</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a> </span>)</span> <a href="lattices.html#K7" title="LATTICES:func.7">`</a>  &amp; <span class="p1">(<span class="default"><a href="lopclset.html#K9" title="LOPCLSET:func.9">UFilter</a> <font color="Maroon" title="c2">BL</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> )
 ;<br/>
<font color="Maroon" title="c14">x</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopclset.html#K9" title="LOPCLSET:func.9">UFilter</a> <font color="Maroon" title="c2">BL</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">a</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
<b>by </b><i><a class="txt" href="lopclset.html#E5:40_2_9_1"><i><font color="Green" title="E61">A33</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lopclset.html#K9" title="LOPCLSET:func.9">UFilter</a> <font color="Maroon" title="c2">BL</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">a</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a> </span>)</span>
<b>by </b><i><a class="ref" href="lopclset.html#T28" target="_self" title="LOPCLSET:th.28">Th28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lopclset.html#K9" title="LOPCLSET:func.9">UFilter</a> <font color="Maroon" title="c2">BL</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">b</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a> </span>)</span> <a href="lattices.html#K7" title="LATTICES:func.7">`</a> </span>)</span>
<b>by </b><i><a class="txt" href="lopclset.html#E10:40_2_9_1"><i><font color="Green" title="E62">A34</font></i></a>, <a class="ref" href="lattices.html#T49" title="LATTICES:th.49">LATTICES:49</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lopclset.html#K9" title="LOPCLSET:func.9">UFilter</a> <font color="Maroon" title="c2">BL</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">b</font>
<b>by </b><i><a class="ref" href="lattices.html#T49" title="LATTICES:th.49">LATTICES:49</a></i>
;<br/>
<b>hence </b><a NAME="E12:40_2_9_1"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="lopclset.html#E10:40_2_9_1"><i><font color="Green" title="E62">A34</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:40_2_9"/>
 <a href="setfam_1.html#K7" title="SETFAM_1:func.7">COMPLEMENT</a> <font color="Maroon" title="c11">D</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2_9_2"><b>proof </b></a><div class="add">

<a NAME="E1:40_2_9_2"/><i><font color="Green" title="E60">A35</font></i>: 
<font color="Maroon" title="c11">D</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 
;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="lopclset.html#K7" title="LOPCLSET:func.7">ultraset</a> <font color="Maroon" title="c2">BL</font></span>)</span>) <b>-&gt; </b>   <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">bool</a> <span class="p1">(<span class="default"><a href="lopclset.html#K7" title="LOPCLSET:func.7">ultraset</a> <font color="Maroon" title="c2">BL</font></span>)</span> = $1 <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> ;<br/>
<a NAME="E2:40_2_9_2"/><i><font color="Green" title="E61">A36</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">w</font>) where <font color="Olive" title="b1">w</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="lopclset.html#K7" title="LOPCLSET:func.7">ultraset</a> <font color="Maroon" title="c2">BL</font></span>)</span> : <font color="Olive" title="b1">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">D</font> </span>}</span>  is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 
<b>from </b><i><a class="ref" href="fraenkel.html#S21" title="FRAENKEL:sch.21">FRAENKEL:sch 21</a>(<a class="txt" href="lopclset.html#E1:40_2_9_2"><i><font color="Green" title="E60">A35</font></i></a>);<br/></i>
<a NAME="E3:40_2_9_2"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">w</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> where <font color="Olive" title="b1">w</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="lopclset.html#K7" title="LOPCLSET:func.7">ultraset</a> <font color="Maroon" title="c2">BL</font></span>)</span> : <font color="Olive" title="b1">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">D</font> </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K7" title="SETFAM_1:func.7">COMPLEMENT</a> <font color="Maroon" title="c11">D</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2_9_2_1"><b>proof </b></a><div class="add">

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a>,<a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a>
<div class="add"><b>let </b><font color="Maroon" title="c14">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:40_2_9_2_1_1"/>
<font color="Maroon" title="c14">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">w</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> where <font color="Olive" title="b1">w</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="lopclset.html#K7" title="LOPCLSET:func.7">ultraset</a> <font color="Maroon" title="c2">BL</font></span>)</span> : <font color="Olive" title="b1">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">D</font> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon" title="c15">w</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="lopclset.html#K7" title="LOPCLSET:func.7">ultraset</a> <font color="Maroon" title="c2">BL</font></span>)</span><b> such that </b><br/><a NAME="E3:40_2_9_2_1_1"/><i><font color="Green" title="E62">A37</font></i>: 
( <font color="Maroon" title="c15">w</font> <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="c14">x</font> &amp; <font color="Maroon" title="c15">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">D</font> )
 ;<br/><a NAME="E4:40_2_9_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c15">w</font> <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#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">D</font>
 <b>by </b><i><a class="txt" href="lopclset.html#E3:40_2_9_2_1_1"><i><font color="Green" title="E62">A37</font></i></a></i>;<br/><b>hence </b><a NAME="E5:40_2_9_2_1_1"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K7" title="SETFAM_1:func.7">COMPLEMENT</a> <font color="Maroon" title="c11">D</font>
 <b>by </b><i><a class="txt" href="lopclset.html#E3:40_2_9_2_1_1"><i><font color="Green" title="E62">A37</font></i></a>, <a class="ref" href="setfam_1.html#D8" title="SETFAM_1:def.8">SETFAM_1:def 8</a></i>;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon" title="c14">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="E2:40_2_9_2_1"/><i><font color="Green" title="E62">A38</font></i>: 
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K7" title="SETFAM_1:func.7">COMPLEMENT</a> <font color="Maroon" title="c11">D</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c15">x'</font> = <font color="Maroon" title="c14">x</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="lopclset.html#K7" title="LOPCLSET:func.7">ultraset</a> <font color="Maroon" title="c2">BL</font></span>)</span> ;<br/>
<a NAME="E4:40_2_9_2_1"/><i><font color="Green" title="E63">A39</font></i>: 
<font color="Maroon" title="c15">x'</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">D</font>
 
<b>by </b><i><a class="txt" href="lopclset.html#E2:40_2_9_2_1"><i><font color="Green" title="E62">A38</font></i></a>, <a class="ref" href="setfam_1.html#D8" title="SETFAM_1:def.8">SETFAM_1:def 8</a></i>;<br/>
<a NAME="E5:40_2_9_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c15">x'</font> <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> <font color="Maroon" title="c15">x'</font>
 
;<br/>
<b>hence </b><a NAME="E6:40_2_9_2_1"/>
<font color="Maroon" title="c14">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">w</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> where <font color="Olive" title="b1">w</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="lopclset.html#K7" title="LOPCLSET:func.7">ultraset</a> <font color="Maroon" title="c2">BL</font></span>)</span> : <font color="Olive" title="b1">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">D</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="lopclset.html#E4:40_2_9_2_1"><i><font color="Green" title="E63">A39</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:40_2_9_2"/>
 <a href="setfam_1.html#K7" title="SETFAM_1:func.7">COMPLEMENT</a> <font color="Maroon" title="c11">D</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>by </b><i><a class="txt" href="lopclset.html#E2:40_2_9_2"><i><font color="Green" title="E61">A36</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:40_2_9"/>
 <a href="setfam_1.html#K7" title="SETFAM_1:func.7">COMPLEMENT</a> <font color="Maroon" title="c11">D</font> is   <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="lopclset.html#E1:40_2_9"><i><font color="Green" title="E59">A31</font></i></a>, <a class="ref" href="finsub_1.html#D5" title="FINSUB_1:def.5">FINSUB_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E25:40_2"/>
contradiction
 <b>by </b><i><a class="txt" href="lopclset.html#E2:40"><i><font color="Green" title="E35">A2</font></i></a>, <a class="txt" href="lopclset.html#E23:40_2"><i><font color="Green" title="E58">A30</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E18:40"/><b>then </b>
<span class="p1"><a href="filter_0.html#K3" title="FILTER_0:func.3">&lt;.</a><span class="default"><font color="Maroon" title="c8">CY</font></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">BL</font>
 
<b>by </b><i><a class="ref" href="lopclset.html#T34" target="_self" title="LOPCLSET:th.34">Th34</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c10">F</font> being    <a href="filter_0.html#M1" title="FILTER_0:mode.1">Filter</a> of <font color="Maroon" title="c2">BL</font><b> such that </b><br/><a NAME="E20:40"/><i><font color="Green" title="E43">A40</font></i>: 
( <span class="p1"><a href="filter_0.html#K3" title="FILTER_0:func.3">&lt;.</a><span class="default"><font color="Maroon" title="c8">CY</font></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">F</font> &amp; <font color="Maroon" title="c10">F</font> <a href="filter_0.html#NR1" title="FILTER_0:NR.1">is_ultrafilter</a>  )
 <b>by </b><i><a class="ref" href="filter_0.html#T22" title="FILTER_0:th.22">FILTER_0:22</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c11">F1</font> being    <a href="filter_0.html#M1" title="FILTER_0:mode.1">Filter</a> of <font color="Maroon" title="c2">BL</font><b> such that </b><br/><a NAME="E22:40"/><i><font color="Green" title="E44">A41</font></i>: 
( <font color="Maroon" title="c10">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">F1</font> &amp; <font color="Maroon" title="c11">F1</font> <a href="filter_0.html#NR1" title="FILTER_0:NR.1">is_ultrafilter</a>  )
 <b>by </b><i><a class="txt" href="lopclset.html#E20:40"><i><font color="Green" title="E43">A40</font></i></a></i>;<br/>
<a NAME="E23:40"/><i><font color="Green" title="E45">A42</font></i>: 
<font color="Maroon" title="c8">CY</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="filter_0.html#K3" title="FILTER_0:func.3">&lt;.</a><span class="default"><font color="Maroon" title="c8">CY</font></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span>
 
<b>by </b><i><a class="ref" href="filter_0.html#D5" title="FILTER_0:def.5">FILTER_0:def 5</a></i>;<br/>
<a NAME="E24:40"/>
not <font color="Maroon" title="c10">F</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="c1">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:40_3"/>
<font color="Maroon" title="c10">F</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="c1">X</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c12">Y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:40_3"/><i><font color="Green" title="E46">A43</font></i>: 
( <font color="Maroon" title="c10">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">Y</font> &amp; <font color="Maroon" title="c12">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<a NAME="E4:40_3"/>
<font color="Maroon" title="c12">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lopclset.html#K10" title="LOPCLSET:func.10">StoneR</a> <font color="Maroon" title="c2">BL</font>
 
<b>by </b><i><a class="txt" href="lopclset.html#E1:40"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="lopclset.html#E3:40_3"><i><font color="Green" title="E46">A43</font></i></a></i>;<br/>
<a NAME="E5:40_3"/><b>then </b>
<font color="Maroon" title="c12">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="lopclset.html#K9" title="LOPCLSET:func.9">UFilter</a> <font color="Maroon" title="c2">BL</font></span>)</span>
 
;<br/>
<b>then consider </b><font color="Maroon" title="c13">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E7:40_3"/><i><font color="Green" title="E47">A44</font></i>: 
( <font color="Maroon" title="c13">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="lopclset.html#K9" title="LOPCLSET:func.9">UFilter</a> <font color="Maroon" title="c2">BL</font></span>)</span> &amp; <font color="Maroon" title="c12">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="lopclset.html#K9" title="LOPCLSET:func.9">UFilter</a> <font color="Maroon" title="c2">BL</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">a</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">a</font> = <font color="Maroon" title="c13">a</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">BL</font> <b>by </b><i><a class="txt" href="lopclset.html#E7:40_3"><i><font color="Green" title="E47">A44</font></i></a></i>;<br/>
<a NAME="E9:40_3"/>
<font color="Maroon" title="c14">a</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">CY</font>
 
<b>by </b><i><a class="txt" href="lopclset.html#E3:40_3"><i><font color="Green" title="E46">A43</font></i></a>, <a class="txt" href="lopclset.html#E7:40_3"><i><font color="Green" title="E47">A44</font></i></a></i>;<br/>
<a NAME="E10:40_3"/><b>then </b>
<font color="Maroon" title="c14">a</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="filter_0.html#K3" title="FILTER_0:func.3">&lt;.</a><span class="default"><font color="Maroon" title="c8">CY</font></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span>
 
<b>by </b><i><a class="txt" href="lopclset.html#E23:40"><i><font color="Green" title="E45">A42</font></i></a></i>;<br/>
<a NAME="E11:40_3"/><b>then </b>
( <font color="Maroon" title="c14">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">F</font> &amp; <font color="Maroon" title="c14">a</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">F</font> )
 
<b>by </b><i><a class="txt" href="lopclset.html#E20:40"><i><font color="Green" title="E43">A40</font></i></a>, <a class="txt" href="lopclset.html#E3:40_3"><i><font color="Green" title="E46">A43</font></i></a>, <a class="txt" href="lopclset.html#E7:40_3"><i><font color="Green" title="E47">A44</font></i></a>, <a class="ref" href="lopclset.html#T21" target="_self" title="LOPCLSET:th.21">Th21</a></i>;<br/>
<b>hence </b><a NAME="E12:40_3"/>
contradiction
 <b>by </b><i><a class="txt" href="lopclset.html#E20:40"><i><font color="Green" title="E43">A40</font></i></a>, <a class="ref" href="filter_0.html#T59" title="FILTER_0:th.59">FILTER_0:59</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E25:40"/>
contradiction
 <b>by </b><i><a class="txt" href="lopclset.html#E1:40"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="lopclset.html#E22:40"><i><font color="Green" title="E44">A41</font></i></a></i>;<br/>


</div>
