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<a NAME="E1:51_1_1"/><span class="lab"><font color="Green" title="E24">A1</font></span>: 
 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="c1">L</font>  st <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="latstone.html#K5" title="LATSTONE:func.5">DenseElements</a> <font color="Maroon" title="c1">L</font> &amp; <font color="Olive" title="b2">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="latstone.html#K5" title="LATSTONE:func.5">DenseElements</a> <font color="Maroon" title="c1">L</font> holds <br/><font color="Olive" title="b1">x</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Olive" title="b2">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="latstone.html#K5" title="LATSTONE:func.5">DenseElements</a> <font color="Maroon" title="c1">L</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="51_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <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"> ( <font color="Maroon" title="c2">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="latstone.html#K5" title="LATSTONE:func.5">DenseElements</a> <font color="Maroon" title="c1">L</font> &amp; <font color="Maroon" title="c3">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="latstone.html#K5" title="LATSTONE:func.5">DenseElements</a> <font color="Maroon" title="c1">L</font> implies <font color="Maroon" title="c2">x</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c3">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="latstone.html#K5" title="LATSTONE:func.5">DenseElements</a> <font color="Maroon" title="c1">L</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:51_1_1_1"/>
( <font color="Maroon" title="c2">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="latstone.html#K5" title="LATSTONE:func.5">DenseElements</a> <font color="Maroon" title="c1">L</font> &amp; <font color="Maroon" title="c3">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="latstone.html#K5" title="LATSTONE:func.5">DenseElements</a> <font color="Maroon" title="c1">L</font> )
 ; <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"> <font color="Maroon" title="c2">x</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c3">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="latstone.html#K5" title="LATSTONE:func.5">DenseElements</a> <font color="Maroon" title="c1">L</font></span><br/>

<a NAME="E2:51_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A2</font></span>: 
( <font color="Maroon" title="c2">x</font> <a href="latstone.html#K1" title="LATSTONE:func.1">*</a>  <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="c1">L</font> &amp; <font color="Maroon" title="c3">y</font> <a href="latstone.html#K1" title="LATSTONE:func.1">*</a>  <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="c1">L</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="latstone.html#T24" target="_self" onmouseover="rs('latstone/T24')" onmouseout="rh()">DenseIsBot</a></span>;<br/>
<a NAME="E3:51_1_1_1"/><span class="lab"><font color="Green" title="E26">F1</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="latstone.html#K1" title="LATSTONE:func.1">*</a>  <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">x</font> <a href="latstone.html#K1" title="LATSTONE:func.1">*</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="latstone.html#T8" target="_self" onmouseover="rs('latstone/T8')" onmouseout="rh()">Th6</a>, <a class="ref" href="lattices.html#T5" onmouseover="rs('lattices/T5')" onmouseout="rh()">LATTICES:5</a></span>;<br/>
<a NAME="E4:51_1_1_1"/>
 <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">L</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="latstone.html#K1" title="LATSTONE:func.1">*</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#T16" onmouseover="rs('lattices/T16')" onmouseout="rh()">LATTICES:16</a></span>;<br/>
<a NAME="E5:51_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="latstone.html#K1" title="LATSTONE:func.1">*</a>  <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="c1">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#T8" onmouseover="rs('lattices/T8')" onmouseout="rh()">LATTICES:8</a>, <a class="txt" href="#E2:51_1_1_1"><span class="lab"><font color="Green" title="E25">A2</font></span></a>, <a class="txt" href="#E3:51_1_1_1"><span class="lab"><font color="Green" title="E26">F1</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:51_1_1_1"/>
<font color="Maroon" title="c2">x</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c3">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="latstone.html#K5" title="LATSTONE:func.5">DenseElements</a> <font color="Maroon" title="c1">L</font>
 ; <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>
<a NAME="E2:51_1_1"/>
 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="c1">L</font>  st <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="latstone.html#K5" title="LATSTONE:func.5">DenseElements</a> <font color="Maroon" title="c1">L</font> &amp; <font color="Olive" title="b2">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="latstone.html#K5" title="LATSTONE:func.5">DenseElements</a> <font color="Maroon" title="c1">L</font> holds <br/><font color="Olive" title="b1">x</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b2">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="latstone.html#K5" title="LATSTONE:func.5">DenseElements</a> <font color="Maroon" title="c1">L</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="51_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <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"> ( <font color="Maroon" title="c2">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="latstone.html#K5" title="LATSTONE:func.5">DenseElements</a> <font color="Maroon" title="c1">L</font> &amp; <font color="Maroon" title="c3">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="latstone.html#K5" title="LATSTONE:func.5">DenseElements</a> <font color="Maroon" title="c1">L</font> implies <font color="Maroon" title="c2">x</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c3">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="latstone.html#K5" title="LATSTONE:func.5">DenseElements</a> <font color="Maroon" title="c1">L</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:51_1_1_2"/>
( <font color="Maroon" title="c2">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="latstone.html#K5" title="LATSTONE:func.5">DenseElements</a> <font color="Maroon" title="c1">L</font> &amp; <font color="Maroon" title="c3">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="latstone.html#K5" title="LATSTONE:func.5">DenseElements</a> <font color="Maroon" title="c1">L</font> )
 ; <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"> <font color="Maroon" title="c2">x</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c3">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="latstone.html#K5" title="LATSTONE:func.5">DenseElements</a> <font color="Maroon" title="c1">L</font></span><br/>

<a NAME="E2:51_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A2</font></span>: 
( <font color="Maroon" title="c2">x</font> <a href="latstone.html#K1" title="LATSTONE:func.1">*</a>  <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="c1">L</font> &amp; <font color="Maroon" title="c3">y</font> <a href="latstone.html#K1" title="LATSTONE:func.1">*</a>  <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="c1">L</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="latstone.html#T24" target="_self" onmouseover="rs('latstone/T24')" onmouseout="rh()">DenseIsBot</a></span>;<br/>
<a NAME="E3:51_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="latstone.html#K1" title="LATSTONE:func.1">*</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="latstone.html#K1" title="LATSTONE:func.1">*</a></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="latstone.html#K1" title="LATSTONE:func.1">*</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="latstone.html#T19" target="_self" onmouseover="rs('latstone/T19')" onmouseout="rh()">Th12</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:51_1_1_2"/>
<font color="Maroon" title="c2">x</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c3">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="latstone.html#K5" title="LATSTONE:func.5">DenseElements</a> <font color="Maroon" title="c1">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51_1_1_2"><span class="lab"><font color="Green" title="E25">A2</font></span></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="E3:51_1_1"/>
(  <a href="latstone.html#K5" title="LATSTONE:func.5">DenseElements</a> <font color="Maroon" title="c1">L</font> is  <a href="lattices.html#V21" title="LATTICES:attr.21">join-closed</a>  &amp;  <a href="latstone.html#K5" title="LATSTONE:func.5">DenseElements</a> <font color="Maroon" title="c1">L</font> is  <a href="lattices.html#V20" title="LATTICES:attr.20">meet-closed</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_1_1"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="ref" href="lattices.html#D25" onmouseover="rs('lattices/D25')" onmouseout="rh()">LATTICES:def 25</a>, <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>
