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

<a NAME="E1:22_1_1"/>
<font color="Maroon" title="c2">S</font> <a href="filter_2.html#K3" title="FILTER_2:func.3">.:</a>  is  <a href="lattices.html#V21" title="LATTICES:attr.21">join-closed</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="22_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">p9</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a></span>)</span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="lattices.html#D25" onmouseover="rs('lattices/D25')" onmouseout="rh()">LATTICES:def 25</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive">b<sub>1</sub></font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a></span>)</span>  holds <br/>(  not <font color="Maroon" title="c3">p9</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">S</font> <a href="filter_2.html#K3" title="FILTER_2:func.3">.:</a>  or  not <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">S</font> <a href="filter_2.html#K3" title="FILTER_2:func.3">.:</a>  or <font color="Maroon" title="c3">p9</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">S</font> <a href="filter_2.html#K3" title="FILTER_2:func.3">.:</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">q9</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a></span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c3">p9</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">S</font> <a href="filter_2.html#K3" title="FILTER_2:func.3">.:</a>  or  not <font color="Maroon" title="c4">q9</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">S</font> <a href="filter_2.html#K3" title="FILTER_2:func.3">.:</a>  or <font color="Maroon" title="c3">p9</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c4">q9</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">S</font> <a href="filter_2.html#K3" title="FILTER_2:func.3">.:</a>  )</span><br/>



<a NAME="E1:22_1_1_1"/>
<font color="Maroon" title="c3">p9</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">q9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="filter_2.html#K2" title="FILTER_2:func.2">.:</a> <font color="Maroon" title="c3">p9</font></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><a href="filter_2.html#K2" title="FILTER_2:func.2">.:</a> <font color="Maroon" title="c4">q9</font></span>)</span>
 
;<br/>
<span class="kw">hence </span><a NAME="E2:22_1_1_1"/>
(  not <font color="Maroon" title="c3">p9</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">S</font> <a href="filter_2.html#K3" title="FILTER_2:func.3">.:</a>  or  not <font color="Maroon" title="c4">q9</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">S</font> <a href="filter_2.html#K3" title="FILTER_2:func.3">.:</a>  or <font color="Maroon" title="c3">p9</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c4">q9</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">S</font> <a href="filter_2.html#K3" title="FILTER_2:func.3">.:</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#D24" onmouseover="rs('lattices/D24')" onmouseout="rh()">LATTICES:def 24</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:22_1_1"/>
 for <font color="Olive">b<sub>1</sub></font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a></span>)</span>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">S</font> <a href="filter_2.html#K3" title="FILTER_2:func.3">.:</a>  holds <br/><font color="Olive">b<sub>1</sub></font> is  <a href="lattices.html#V21" title="LATTICES:attr.21">join-closed</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
