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<div class="add">

<span class="kw">set </span><font color="Maroon" title="c2">S</font> = <font color="Maroon" title="c1">B</font> <a href="latstone.html#K8" title="LATSTONE:func.8">squared</a> ;<br/>
<a NAME="E1:74_1_1"/><span class="lab"><font color="Green" title="E30">AA</font></span>: 
 for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">B</font>,<font color="Maroon" title="c1">B</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span>  st <font color="Olive" title="b1">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">B</font> <a href="latstone.html#K8" title="LATSTONE:func.8">squared</a>  &amp; <font color="Olive" title="b2">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">B</font> <a href="latstone.html#K8" title="LATSTONE:func.8">squared</a>  holds <br/><font color="Olive" title="b1">p</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Olive" title="b2">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">B</font> <a href="latstone.html#K8" title="LATSTONE:func.8">squared</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="74_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">p</font>, <font color="Maroon" title="c4">q</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">B</font>,<font color="Maroon" title="c1">B</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</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"> ( <font color="Maroon" title="c3">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">B</font> <a href="latstone.html#K8" title="LATSTONE:func.8">squared</a>  &amp; <font color="Maroon" title="c4">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">B</font> <a href="latstone.html#K8" title="LATSTONE:func.8">squared</a>  implies <font color="Maroon" title="c3">p</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">B</font> <a href="latstone.html#K8" title="LATSTONE:func.8">squared</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:74_1_1_1"/><span class="lab"><font color="Green" title="E31">A0</font></span>: 
( <font color="Maroon" title="c3">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">B</font> <a href="latstone.html#K8" title="LATSTONE:func.8">squared</a>  &amp; <font color="Maroon" title="c4">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">B</font> <a href="latstone.html#K8" title="LATSTONE:func.8">squared</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"> <font color="Maroon" title="c3">p</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">B</font> <a href="latstone.html#K8" title="LATSTONE:func.8">squared</a> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">a1</font>, <font color="Maroon" title="c6">b1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">B</font><span class="kw"> such that </span><br/><a NAME="E3:74_1_1_1"/><span class="lab"><font color="Green" title="E32">A1</font></span>: 
( <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><font color="Maroon" title="c5">a1</font>,<font color="Maroon" title="c6">b1</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span> &amp; <font color="Maroon" title="c5">a1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c6">b1</font> )
 ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">a2</font>, <font color="Maroon" title="c8">b2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">B</font><span class="kw"> such that </span><br/><a NAME="E5:74_1_1_1"/><span class="lab"><font color="Green" title="E33">A2</font></span>: 
( <font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><font color="Maroon" title="c7">a2</font>,<font color="Maroon" title="c8">b2</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span> &amp; <font color="Maroon" title="c7">a2</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c8">b2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:74_1_1_1"><span class="lab"><font color="Green" title="E31">A0</font></span></a></span>;<br/>
<a NAME="E6:74_1_1_1"/><span class="lab"><font color="Green" title="E34">A3</font></span>: 
<font color="Maroon" title="c3">p</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">a1</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c7">a2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">b1</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c8">b2</font></span>)</span></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_1.html#T35" onmouseover="rs('filter_1/T35')" onmouseout="rh()">FILTER_1:35</a>, <a class="txt" href="#E3:74_1_1_1"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="txt" href="#E5:74_1_1_1"><span class="lab"><font color="Green" title="E33">A2</font></span></a></span>;<br/>
<a NAME="E7:74_1_1_1"/>
<font color="Maroon" title="c5">a1</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c7">a2</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c6">b1</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c8">b2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:74_1_1_1"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="txt" href="#E5:74_1_1_1"><span class="lab"><font color="Green" title="E33">A2</font></span></a>, <a class="ref" href="filter_0.html#T4" onmouseover="rs('filter_0/T4')" onmouseout="rh()">FILTER_0:4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:74_1_1_1"/>
<font color="Maroon" title="c3">p</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">B</font> <a href="latstone.html#K8" title="LATSTONE:func.8">squared</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:74_1_1_1"><span class="lab"><font color="Green" title="E34">A3</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>
<a NAME="E2:74_1_1"/>
 for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">B</font>,<font color="Maroon" title="c1">B</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span>  st <font color="Olive" title="b1">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">B</font> <a href="latstone.html#K8" title="LATSTONE:func.8">squared</a>  &amp; <font color="Olive" title="b2">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">B</font> <a href="latstone.html#K8" title="LATSTONE:func.8">squared</a>  holds <br/><font color="Olive" title="b1">p</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b2">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">B</font> <a href="latstone.html#K8" title="LATSTONE:func.8">squared</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="74_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">p</font>, <font color="Maroon" title="c4">q</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">B</font>,<font color="Maroon" title="c1">B</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</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"> ( <font color="Maroon" title="c3">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">B</font> <a href="latstone.html#K8" title="LATSTONE:func.8">squared</a>  &amp; <font color="Maroon" title="c4">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">B</font> <a href="latstone.html#K8" title="LATSTONE:func.8">squared</a>  implies <font color="Maroon" title="c3">p</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c4">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">B</font> <a href="latstone.html#K8" title="LATSTONE:func.8">squared</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:74_1_1_2"/><span class="lab"><font color="Green" title="E31">A0</font></span>: 
( <font color="Maroon" title="c3">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">B</font> <a href="latstone.html#K8" title="LATSTONE:func.8">squared</a>  &amp; <font color="Maroon" title="c4">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">B</font> <a href="latstone.html#K8" title="LATSTONE:func.8">squared</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"> <font color="Maroon" title="c3">p</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c4">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">B</font> <a href="latstone.html#K8" title="LATSTONE:func.8">squared</a> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">a1</font>, <font color="Maroon" title="c6">b1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">B</font><span class="kw"> such that </span><br/><a NAME="E3:74_1_1_2"/><span class="lab"><font color="Green" title="E32">A1</font></span>: 
( <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><font color="Maroon" title="c5">a1</font>,<font color="Maroon" title="c6">b1</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span> &amp; <font color="Maroon" title="c5">a1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c6">b1</font> )
 ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">a2</font>, <font color="Maroon" title="c8">b2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">B</font><span class="kw"> such that </span><br/><a NAME="E5:74_1_1_2"/><span class="lab"><font color="Green" title="E33">A2</font></span>: 
( <font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><font color="Maroon" title="c7">a2</font>,<font color="Maroon" title="c8">b2</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span> &amp; <font color="Maroon" title="c7">a2</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c8">b2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:74_1_1_2"><span class="lab"><font color="Green" title="E31">A0</font></span></a></span>;<br/>
<a NAME="E6:74_1_1_2"/><span class="lab"><font color="Green" title="E34">A3</font></span>: 
<font color="Maroon" title="c3">p</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">a1</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c7">a2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">b1</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c8">b2</font></span>)</span></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_1.html#T35" onmouseover="rs('filter_1/T35')" onmouseout="rh()">FILTER_1:35</a>, <a class="txt" href="#E3:74_1_1_2"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="txt" href="#E5:74_1_1_2"><span class="lab"><font color="Green" title="E33">A2</font></span></a></span>;<br/>
<a NAME="E7:74_1_1_2"/>
<font color="Maroon" title="c5">a1</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c7">a2</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c6">b1</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c8">b2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:74_1_1_2"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="txt" href="#E5:74_1_1_2"><span class="lab"><font color="Green" title="E33">A2</font></span></a>, <a class="ref" href="filter_0.html#T5" onmouseover="rs('filter_0/T5')" onmouseout="rh()">FILTER_0:5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:74_1_1_2"/>
<font color="Maroon" title="c3">p</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c4">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">B</font> <a href="latstone.html#K8" title="LATSTONE:func.8">squared</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:74_1_1_2"><span class="lab"><font color="Green" title="E34">A3</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="E3:74_1_1"/>
( <font color="Maroon" title="c1">B</font> <a href="latstone.html#K8" title="LATSTONE:func.8">squared</a>  is  <a href="lattices.html#V21" title="LATTICES:attr.21">join-closed</a>  &amp; <font color="Maroon" title="c1">B</font> <a href="latstone.html#K8" title="LATSTONE:func.8">squared</a>  is  <a href="lattices.html#V20" title="LATTICES:attr.20">meet-closed</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>, <a class="ref" href="lattices.html#D25" onmouseover="rs('lattices/D25')" onmouseout="rh()">LATTICES:def 25</a>, <a class="txt" href="#E1:74_1_1"><span class="lab"><font color="Green" title="E30">AA</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>
