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

<span class="kw">let </span><font color="Maroon" title="c1">L</font> be   <a href="lattices.html#V11" title="LATTICES:attr.11">distributive</a>   <a href="lattices.html#V15" title="LATTICES:attr.15">bounded</a>   <a href="latstone.html#V1" title="LATSTONE:attr.1">pseudocomplemented</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"> ( <font color="Maroon" title="c1">L</font> is  <a href="latstone.html#V2" title="LATSTONE:attr.2">satisfying_Stone_identity</a>  iff  for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b2">b</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="Olive" title="b1">a</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="Olive" title="b2">b</font> <a href="latstone.html#K1" title="LATSTONE:func.1">*</a></span>)</span> )</span><br/>

<div><span class="kw">hereby </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"> ( (  for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b2">b</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="Olive" title="b1">a</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="Olive" title="b2">b</font> <a href="latstone.html#K1" title="LATSTONE:func.1">*</a></span>)</span> ) implies <font color="Maroon" title="c1">L</font> is  <a href="latstone.html#V2" title="LATSTONE:attr.2">satisfying_Stone_identity</a>  )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:31_1"/><span class="lab"><font color="Green" title="E20">a11</font></span>: 
<font color="Maroon" title="c1">L</font> is  <a href="latstone.html#V2" title="LATSTONE:attr.2">satisfying_Stone_identity</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">a</font>, <font color="Olive" title="b2">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b2">b</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="Olive" title="b1">a</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="Olive" title="b2">b</font> <a href="latstone.html#K1" title="LATSTONE:func.1">*</a></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</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"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c3">b</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">a</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">b</font> <a href="latstone.html#K1" title="LATSTONE:func.1">*</a></span>)</span></span><br/><span class="kw">set </span><font color="Maroon" title="c4">g</font> = <font color="Maroon" title="c2">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c3">b</font>;<br/><span class="lab"><font color="Green" title="E21">a0</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</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="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="latstone.html#K1" title="LATSTONE:func.1">*</a></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="latstone.html#K1" title="LATSTONE:func.1">*</a></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="latstone.html#K1" title="LATSTONE:func.1">*</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#D11" onmouseover="rs('lattices/D11')" onmouseout="rh()">LATTICES:def 11</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="latstone.html#K1" title="LATSTONE:func.1">*</a></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="latstone.html#K1" title="LATSTONE:func.1">*</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#D7" onmouseover="rs('lattices/D7')" onmouseout="rh()">LATTICES:def 7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="latstone.html#K1" title="LATSTONE:func.1">*</a></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="latstone.html#K1" title="LATSTONE:func.1">*</a></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#D7" onmouseover="rs('lattices/D7')" onmouseout="rh()">LATTICES:def 7</a></span>
<br/>.= 
<span class="p1">(<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">L</font></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="latstone.html#K1" title="LATSTONE:func.1">*</a></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="latstone.html#T6" target="_self" onmouseover="rs('latstone/T6')" onmouseout="rh()">ThD</a></span>
<br/>.= 
<span class="p1">(<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">L</font></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p2">(<span class="default"><a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="latstone.html#T6" target="_self" onmouseover="rs('latstone/T6')" onmouseout="rh()">ThD</a></span>
<br/>.= 
 <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">L</font>

;<br/><a NAME="E3:31_1"/>
 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="c1">L</font>  st <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b1">x</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">L</font> holds <br/><font color="Olive" title="b1">x</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</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">b</font> <a href="latstone.html#K1" title="LATSTONE:func.1">*</a></span>)</span>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="31_1_2"><span class="kw">proof </span></a><div class="add">

<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="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"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c5">x</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">L</font> implies <font color="Maroon" title="c5">x</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</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">b</font> <a href="latstone.html#K1" title="LATSTONE:func.1">*</a></span>)</span> )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c6">z</font> = <font color="Maroon" title="c3">b</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c5">x</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">w</font> = <font color="Maroon" title="c5">x</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="latstone.html#K1" title="LATSTONE:func.1">*</a></span>)</span> <a href="latstone.html#K1" title="LATSTONE:func.1">*</a></span>)</span>;<br/>
<span class="kw">assume </span><a NAME="E1:31_1_2"/><span class="lab"><font color="Green" title="E22">a1</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c5">x</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">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#R3" title="LATTICES:pred.3">[=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</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">b</font> <a href="latstone.html#K1" title="LATSTONE:func.1">*</a></span>)</span></span><br/>

<a NAME="E2:31_1_2"/><span class="lab"><font color="Green" title="E23">a0</font></span>: 
 <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="c3">b</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="latstone.html#K1" title="LATSTONE:func.1">*</a></span>)</span> <a href="latstone.html#K1" title="LATSTONE:func.1">*</a></span>)</span>
 
<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="E3:31_1_2"/><span class="lab"><font color="Green" title="E24">a2</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c2">a</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">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31_1_2"><span class="lab"><font color="Green" title="E22">a1</font></span></a>, <a class="ref" href="lattices.html#D7" onmouseover="rs('lattices/D7')" onmouseout="rh()">LATTICES:def 7</a></span>;<br/>
<a NAME="E4:31_1_2"/>
<font color="Maroon" title="c2">a</font> <a href="latstone.html#K1" title="LATSTONE:func.1">*</a>  <a href="latstone.html#R1" title="LATSTONE:pred.1">is_a_pseudocomplement_of</a> <font color="Maroon" title="c2">a</font>
 
<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></span>;<br/>
<a NAME="E5:31_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="latstone.html#K1" title="LATSTONE:func.1">*</a></span>)</span> <a href="latstone.html#K1" title="LATSTONE:func.1">*</a></span>)</span> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="latstone.html#K1" title="LATSTONE:func.1">*</a></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="latstone.html#K1" title="LATSTONE:func.1">*</a></span>)</span> <a href="latstone.html#K1" title="LATSTONE:func.1">*</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#T9" onmouseover="rs('lattices/T9')" onmouseout="rh()">LATTICES:9</a>, <a class="txt" href="#E3:31_1_2"><span class="lab"><font color="Green" title="E24">a2</font></span></a></span>;<br/>
<a NAME="E6:31_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="latstone.html#K1" title="LATSTONE:func.1">*</a></span>)</span> <a href="latstone.html#K1" title="LATSTONE:func.1">*</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">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="latstone.html#T6" target="_self" onmouseover="rs('latstone/T6')" onmouseout="rh()">ThD</a></span>;<br/>
<a NAME="E7:31_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="latstone.html#K1" title="LATSTONE:func.1">*</a></span>)</span> <a href="latstone.html#K1" title="LATSTONE:func.1">*</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">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="#E2:31_1_2"><span class="lab"><font color="Green" title="E23">a0</font></span></a></span>;<br/>
<a NAME="E8:31_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">a4</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="latstone.html#K1" title="LATSTONE:func.1">*</a></span>)</span> <a href="latstone.html#K1" title="LATSTONE:func.1">*</a></span>)</span></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c3">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="c1">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#D7" onmouseover="rs('lattices/D7')" onmouseout="rh()">LATTICES:def 7</a></span>;<br/>
<a NAME="E9:31_1_2"/>
<font color="Maroon" title="c3">b</font> <a href="latstone.html#K1" title="LATSTONE:func.1">*</a>  <a href="latstone.html#R1" title="LATSTONE:pred.1">is_a_pseudocomplement_of</a> <font color="Maroon" title="c3">b</font>
 
<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></span>;<br/>
<a NAME="E10:31_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">a5</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="latstone.html#K1" title="LATSTONE:func.1">*</a></span>)</span> <a href="latstone.html#K1" title="LATSTONE:func.1">*</a></span>)</span> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">b</font> <a href="latstone.html#K1" title="LATSTONE:func.1">*</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:31_1_2"><span class="lab"><font color="Green" title="E25">a4</font></span></a></span>;<br/>
<a NAME="E11:31_1_2"/><span class="lab"><font color="Green" title="E27">a6</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="latstone.html#K1" title="LATSTONE:func.1">*</a></span>)</span> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">a</font> <a href="latstone.html#K1" title="LATSTONE:func.1">*</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#T6" onmouseover="rs('lattices/T6')" onmouseout="rh()">LATTICES:6</a></span>;<br/>
<font color="Maroon" title="c5">x</font> = 
<font color="Maroon" title="c5">x</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c1">L</font></span>)</span>

<br/>.= 
<font color="Maroon" title="c5">x</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</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="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="latstone.html#K1" title="LATSTONE:func.1">*</a></span>)</span> <a href="latstone.html#K1" title="LATSTONE:func.1">*</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31_1"><span class="lab"><font color="Green" title="E20">a11</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="latstone.html#K1" title="LATSTONE:func.1">*</a></span>)</span></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="latstone.html#K1" title="LATSTONE:func.1">*</a></span>)</span> <a href="latstone.html#K1" title="LATSTONE:func.1">*</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#D11" onmouseover="rs('lattices/D11')" onmouseout="rh()">LATTICES:def 11</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E13:31_1_2"/>
<font color="Maroon" title="c5">x</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</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">b</font> <a href="latstone.html#K1" title="LATSTONE:func.1">*</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:31_1_2"><span class="lab"><font color="Green" title="E26">a5</font></span></a>, <a class="txt" href="#E11:31_1_2"><span class="lab"><font color="Green" title="E27">a6</font></span></a>, <a class="ref" href="filter_0.html#T4" onmouseover="rs('filter_0/T4')" onmouseout="rh()">FILTER_0:4</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="E4:31_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</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">b</font> <a href="latstone.html#K1" title="LATSTONE:func.1">*</a></span>)</span> <a href="latstone.html#R1" title="LATSTONE:pred.1">is_a_pseudocomplement_of</a> <font color="Maroon" title="c2">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c3">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:31_1"><span class="lab"><font color="Green" title="E21">a0</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E5:31_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c3">b</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">a</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">b</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#D3" target="_self" onmouseover="rs('latstone/D3')" onmouseout="rh()">def3</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">assume </span><a NAME="E2:31"/><span class="lab"><font color="Green" title="E20">a1</font></span>: 
 for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b2">b</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="Olive" title="b1">a</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="Olive" title="b2">b</font> <a href="latstone.html#K1" title="LATSTONE: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"> <font color="Maroon" title="c1">L</font> is  <a href="latstone.html#V2" title="LATSTONE:attr.2">satisfying_Stone_identity</a> </span><br/>

<span class="kw">let </span><font color="Maroon" title="c2">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="latstone.html#D5" target="_self" onmouseover="rs('latstone/D5')" onmouseout="rh()">LATSTONE:def 5</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"> <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#K1" title="LATTICES:func.1">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="latstone.html#K1" title="LATSTONE:func.1">*</a></span>)</span> <a href="latstone.html#K1" title="LATSTONE:func.1">*</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c1">L</font></span><br/>

<a NAME="E3:31"/>
<font color="Maroon" title="c2">x</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</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="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#T6" target="_self" onmouseover="rs('latstone/T6')" onmouseout="rh()">ThD</a></span>;<br/>
<a NAME="E4:31"/><span class="kw">then </span>
<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"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="latstone.html#K1" title="LATSTONE:func.1">*</a></span>)</span> <a href="latstone.html#K1" title="LATSTONE:func.1">*</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="latstone.html#K1" title="LATSTONE:func.1">*</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E20">a1</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:31"/>
<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#K1" title="LATTICES:func.1">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="latstone.html#K1" title="LATSTONE:func.1">*</a></span>)</span> <a href="latstone.html#K1" title="LATSTONE:func.1">*</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c1">L</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="latstone.html#T10" target="_self" onmouseover="rs('latstone/T10')" onmouseout="rh()">Th11</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>
