<?xml version="1.0"?>
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<span class="kw">let </span><font color="Maroon" title="c1">B</font> be   <a href="lattices.html#V17" title="LATTICES:attr.17">Boolean</a>  <a href="lattices.html#NM1" title="LATTICES:NM.1">Lattice</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" title="b1">L</font> being   <a href="lattices.html#NM1" title="LATTICES:NM.1">Lattice</a><br/>  for <font color="Olive" title="b2">x1</font>, <font color="Olive" title="b3">x2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">B</font><br/>  for <font color="Olive" title="b4">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b1">L</font>  st <font color="Olive" title="b1">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">B</font> <a href="latstone.html#K10" title="LATSTONE:func.10">squared-latt</a>  &amp; <font color="Olive" title="b4">x</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="Olive" title="b2">x1</font>,<font color="Olive" title="b3">x2</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span> holds <br/><font color="Olive" title="b4">x</font> <a href="latstone.html#K1" title="LATSTONE:func.1">*</a>  <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="Olive" title="b3">x2</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b3">x2</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a></span>)</span></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">L</font> be   <a href="lattices.html#NM1" title="LATTICES:NM.1">Lattice</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" title="b1">x1</font>, <font color="Olive" title="b2">x2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">B</font><br/>  for <font color="Olive" title="b3">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L</font>  st <font color="Maroon" title="c2">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">B</font> <a href="latstone.html#K10" title="LATSTONE:func.10">squared-latt</a>  &amp; <font color="Olive" title="b3">x</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="Olive" title="b1">x1</font>,<font color="Olive" title="b2">x2</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span> holds <br/><font color="Olive" title="b3">x</font> <a href="latstone.html#K1" title="LATSTONE:func.1">*</a>  <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="Olive" title="b2">x2</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b2">x2</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a></span>)</span></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">x1</font>, <font color="Maroon" title="c4">x2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">B</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">  for <font color="Olive" title="b1">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L</font>  st <font color="Maroon" title="c2">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">B</font> <a href="latstone.html#K10" title="LATSTONE:func.10">squared-latt</a>  &amp; <font color="Olive" title="b1">x</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="c3">x1</font>,<font color="Maroon" title="c4">x2</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span> holds <br/><font color="Olive" title="b1">x</font> <a href="latstone.html#K1" title="LATSTONE:func.1">*</a>  <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="c4">x2</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">x2</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a></span>)</span></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c5">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">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">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">B</font> <a href="latstone.html#K10" title="LATSTONE:func.10">squared-latt</a>  &amp; <font color="Maroon" title="c5">x</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="c3">x1</font>,<font color="Maroon" title="c4">x2</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span> implies <font color="Maroon" title="c5">x</font> <a href="latstone.html#K1" title="LATSTONE:func.1">*</a>  <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="c4">x2</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">x2</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a></span>)</span></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span> )</span><br/>







<span class="kw">assume </span><a NAME="E1:84"/><span class="lab"><font color="Green" title="E34">A0</font></span>: 
( <font color="Maroon" title="c2">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">B</font> <a href="latstone.html#K10" title="LATSTONE:func.10">squared-latt</a>  &amp; <font color="Maroon" title="c5">x</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="c3">x1</font>,<font color="Maroon" title="c4">x2</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</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="c5">x</font> <a href="latstone.html#K1" title="LATSTONE:func.1">*</a>  <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="c4">x2</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">x2</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a></span>)</span></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span></span><br/>

<a NAME="E2:84"/>
<font color="Maroon" title="c5">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L</font>
 
;<br/>
<a NAME="E3:84"/><span class="kw">then </span>
<font color="Maroon" title="c5">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">B</font> <a href="latstone.html#K9" title="LATSTONE:func.9">squared</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="latstone.html#T34" target="_self" onmouseover="rs('latstone/T34')" onmouseout="rh()">SquaredCarrier</a>, <a class="txt" href="#E1:84"><span class="lab"><font color="Green" title="E34">A0</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">xx1</font>, <font color="Maroon" title="c7">xx2</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="E5:84"/><span class="lab"><font color="Green" title="E35">W1</font></span>: 
( <font color="Maroon" title="c5">x</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="c6">xx1</font>,<font color="Maroon" title="c7">xx2</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span> &amp; <font color="Maroon" title="c6">xx1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c7">xx2</font> )
 ;<br/>
<a NAME="E6:84"/><span class="lab"><font color="Green" title="E36">aa</font></span>: 
( <font color="Maroon" title="c6">xx1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x1</font> &amp; <font color="Maroon" title="c7">xx2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:84"><span class="lab"><font color="Green" title="E35">W1</font></span></a>, <a class="txt" href="#E1:84"><span class="lab"><font color="Green" title="E34">A0</font></span></a>, <a class="ref" href="xtuple_0.html#T1" onmouseover="rs('xtuple_0/T1')" onmouseout="rh()">XTUPLE_0:1</a></span>;<br/>
<a NAME="E7:84"/>
<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="c4">x2</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">x2</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a></span>)</span></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">B</font> <a href="latstone.html#K9" title="LATSTONE:func.9">squared</a> 
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">y</font> = <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="c4">x2</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">x2</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a></span>)</span></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:84"><span class="lab"><font color="Green" title="E34">A0</font></span></a>, <a class="ref" href="latstone.html#T34" target="_self" onmouseover="rs('latstone/T34')" onmouseout="rh()">SquaredCarrier</a></span>;<br/>
<span class="lab"><font color="Green" title="E37">Z1</font></span>: <font color="Maroon" title="c5">x</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c8">y</font> = 
<span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c4">x2</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</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="c4">x2</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">x2</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a></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="txt" href="#E1:84"><span class="lab"><font color="Green" title="E34">A0</font></span></a>, <a class="ref" href="msualg_7.html#T11" onmouseover="rs('msualg_7/T11')" onmouseout="rh()">MSUALG_7:11</a></span>
<br/>.= 
<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="c3">x1</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">x2</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">x2</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">x2</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a></span>)</span></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></span>
<br/>.= 
<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="c3">x1</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">x2</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">B</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="lattices.html#T20" onmouseover="rs('lattices/T20')" onmouseout="rh()">LATTICES:20</a></span>
;<br/>
<a NAME="E10:84"/>
<font color="Maroon" title="c4">x2</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a>  <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">x1</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:84"><span class="lab"><font color="Green" title="E36">aa</font></span></a>, <a class="txt" href="#E5:84"><span class="lab"><font color="Green" title="E35">W1</font></span></a>, <a class="ref" href="lattices.html#T26" onmouseover="rs('lattices/T26')" onmouseout="rh()">LATTICES:26</a></span>;<br/>
<a NAME="E11:84"/><span class="kw">then </span>
<font color="Maroon" title="c3">x1</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x2</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a></span>)</span> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">x1</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_0.html#T5" onmouseover="rs('filter_0/T5')" onmouseout="rh()">FILTER_0:5</a></span>;<br/>
<a NAME="E12:84"/><span class="kw">then </span>
<font color="Maroon" title="c3">x1</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x2</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a></span>)</span> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a>  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#T20" onmouseover="rs('lattices/T20')" onmouseout="rh()">LATTICES:20</a></span>;<br/>
<a NAME="E13:84"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">tt</font></span>: 
<font color="Maroon" title="c3">x1</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x2</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a></span>)</span> <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">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="boolealg.html#T9" onmouseover="rs('boolealg/T9')" onmouseout="rh()">BOOLEALG:9</a></span>;<br/>
<a NAME="E14:84"/>
 for <font color="Olive" title="b1">w</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L</font>  st <font color="Maroon" title="c5">x</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b1">w</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">L</font> holds <br/><font color="Olive" title="b1">w</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c8">y</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="84_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">w</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">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="c5">x</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c9">w</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">L</font> implies <font color="Maroon" title="c9">w</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c8">y</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:84_2"/><span class="lab"><font color="Green" title="E39">O1</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c9">w</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">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="c9">w</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c8">y</font></span><br/>

<a NAME="E2:84_2"/>
<font color="Maroon" title="c9">w</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L</font>
 
;<br/>
<a NAME="E3:84_2"/><span class="kw">then </span>
<font color="Maroon" title="c9">w</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">B</font> <a href="latstone.html#K9" title="LATSTONE:func.9">squared</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:84"><span class="lab"><font color="Green" title="E34">A0</font></span></a>, <a class="ref" href="latstone.html#T34" target="_self" onmouseover="rs('latstone/T34')" onmouseout="rh()">SquaredCarrier</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">w1</font>, <font color="Maroon" title="c11">w2</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="E5:84_2"/><span class="lab"><font color="Green" title="E40">Y1</font></span>: 
( <font color="Maroon" title="c9">w</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="c10">w1</font>,<font color="Maroon" title="c11">w2</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span> &amp; <font color="Maroon" title="c10">w1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c11">w2</font> )
 ;<br/>
<a NAME="E6:84_2"/>
<span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c4">x2</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><font color="Maroon" title="c10">w1</font>,<font color="Maroon" title="c11">w2</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span> <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">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:84_2"><span class="lab"><font color="Green" title="E39">O1</font></span></a>, <a class="txt" href="#E5:84_2"><span class="lab"><font color="Green" title="E40">Y1</font></span></a>, <a class="txt" href="#E1:84"><span class="lab"><font color="Green" title="E34">A0</font></span></a>, <a class="ref" href="msualg_7.html#T11" onmouseover="rs('msualg_7/T11')" onmouseout="rh()">MSUALG_7:11</a></span>;<br/>
<a NAME="E7:84_2"/><span class="kw">then </span>
<span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c4">x2</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><font color="Maroon" title="c10">w1</font>,<font color="Maroon" title="c11">w2</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span> <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"><a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">B</font></span>)</span>,<span class="p2">(<span class="default"><a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">B</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="txt" href="#E1:84"><span class="lab"><font color="Green" title="E34">A0</font></span></a>, <a class="ref" href="latstone.html#T37" target="_self" onmouseover="rs('latstone/T37')" onmouseout="rh()">SquaredBottom</a></span>;<br/>
<a NAME="E8:84_2"/><span class="kw">then </span>
<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="c3">x1</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c10">w1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">x2</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c11">w2</font></span>)</span></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span> <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"><a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">B</font></span>)</span>,<span class="p2">(<span class="default"><a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">B</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></span>;<br/>
<a NAME="E9:84_2"/><span class="kw">then </span>
( <font color="Maroon" title="c3">x1</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c10">w1</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="c1">B</font> &amp; <font color="Maroon" title="c4">x2</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c11">w2</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="c1">B</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xtuple_0.html#T1" onmouseover="rs('xtuple_0/T1')" onmouseout="rh()">XTUPLE_0:1</a></span>;<br/>
<a NAME="E10:84_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">Y2</font></span>: 
<font color="Maroon" title="c11">w2</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">x2</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#T25" onmouseover="rs('lattices/T25')" onmouseout="rh()">LATTICES:25</a></span>;<br/>
<a NAME="E11:84_2"/><span class="kw">then </span>
<font color="Maroon" title="c10">w1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">x2</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:84_2"><span class="lab"><font color="Green" title="E40">Y1</font></span></a>, <a class="ref" href="lattices.html#T7" onmouseover="rs('lattices/T7')" onmouseout="rh()">LATTICES:7</a></span>;<br/>
<a NAME="E12:84_2"/><span class="kw">then </span>
<span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><font color="Maroon" title="c10">w1</font>,<font color="Maroon" title="c11">w2</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</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="c4">x2</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">x2</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a></span>)</span></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span> <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="c10">w1</font>,<font color="Maroon" title="c11">w2</font></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="lattices.html#T4" onmouseover="rs('lattices/T4')" onmouseout="rh()">LATTICES:4</a>, <a class="txt" href="#E10:84_2"><span class="lab"><font color="Green" title="E41">Y2</font></span></a>, <a class="ref" href="filter_1.html#T36" onmouseover="rs('filter_1/T36')" onmouseout="rh()">FILTER_1:36</a></span>;<br/>
<a NAME="E13:84_2"/><span class="kw">then </span>
<font color="Maroon" title="c9">w</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">w</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:84_2"><span class="lab"><font color="Green" title="E40">Y1</font></span></a>, <a class="ref" href="msualg_7.html#T11" onmouseover="rs('msualg_7/T11')" onmouseout="rh()">MSUALG_7:11</a>, <a class="txt" href="#E1:84"><span class="lab"><font color="Green" title="E34">A0</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:84_2"/>
<font color="Maroon" title="c9">w</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c8">y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#T4" onmouseover="rs('lattices/T4')" onmouseout="rh()">LATTICES:4</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="E15:84"/><span class="kw">then </span>
<font color="Maroon" title="c8">y</font> <a href="latstone.html#R1" title="LATSTONE:pred.1">is_a_pseudocomplement_of</a> <font color="Maroon" title="c5">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:84"><span class="lab"><font color="Green" title="E38">tt</font></span></a>, <a class="txt" href="#E9:84"><span class="lab"><font color="Green" title="E37">Z1</font></span></a>, <a class="txt" href="#E1:84"><span class="lab"><font color="Green" title="E34">A0</font></span></a>, <a class="ref" href="latstone.html#T37" target="_self" onmouseover="rs('latstone/T37')" onmouseout="rh()">SquaredBottom</a></span>;<br/>
<span class="kw">hence </span><a NAME="E16:84"/>
<font color="Maroon" title="c5">x</font> <a href="latstone.html#K1" title="LATSTONE:func.1">*</a>  <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="c4">x2</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">x2</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a></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="latstone.html#D3" target="_self" onmouseover="rs('latstone/D3')" onmouseout="rh()">def3</a>, <a class="txt" href="#E1:84"><span class="lab"><font color="Green" title="E34">A0</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>
