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

<b>let </b><font color="Maroon" title="c1">T</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/>

<b>set </b><font color="Maroon" title="c2">OCL</font> =  <a href="lopclset.html#K6" title="LOPCLSET:func.6">OpenClosedSetLatt</a> <font color="Maroon" title="c1">T</font>;<br/>
<a NAME="E1:12"/><i><font color="Green" title="E11">A1</font></i>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lopclset.html#K6" title="LOPCLSET:func.6">OpenClosedSetLatt</a> <font color="Maroon" title="c1">T</font></span>)</span> holds  <font color="Olive" title="b1">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">b</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Olive" title="b3">c</font></span>)</span> <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="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <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="b3">c</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font>, <font color="Maroon" title="c5">c</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lopclset.html#K6" title="LOPCLSET:func.6">OpenClosedSetLatt</a> <font color="Maroon" title="c1">T</font></span>)</span>;<br/>

<b>set </b><font color="Maroon" title="c6">m</font> = <font color="Maroon" title="c3">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c5">c</font>;<br/>
<b>consider </b><font color="Maroon" title="c7">a'</font>, <font color="Maroon" title="c8">b'</font>, <font color="Maroon" title="c9">c'</font>, <font color="Maroon" title="c10">k'</font>, <font color="Maroon" title="c11">l'</font>, <font color="Maroon" title="c12">m'</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="lopclset.html#K1" title="LOPCLSET:func.1">OpenClosedSet</a> <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E2:12_1"/><i><font color="Green" title="E12">A2</font></i>: 
( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a'</font> &amp; <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b'</font> &amp; <font color="Maroon" title="c5">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">c'</font> &amp; <font color="Maroon" title="c4">b</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c5">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">k'</font> &amp; <font color="Maroon" title="c3">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">l'</font> &amp; <font color="Maroon" title="c3">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c5">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">m'</font> )
 ;<br/>
<a NAME="E3:12_1"/><i><font color="Green" title="E13">A3</font></i>: 
( <font color="Maroon" title="c7">a'</font> <a href="lopclset.html#K3" title="LOPCLSET:func.3">/\</a> <font color="Maroon" title="c9">c'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c5">c</font> &amp; <font color="Maroon" title="c3">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c5">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">m'</font> )
 
<b>by </b><i><a class="txt" href="lopclset.html#E2:12_1"><i><font color="Green" title="E12">A2</font></i></a>, <a class="ref" href="lopclset.html#D3" target="_self" title="LOPCLSET:def.3">Def3</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c3">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c5">c</font></span>)</span> = 
<font color="Maroon" title="c7">a'</font> <a href="lopclset.html#K3" title="LOPCLSET:func.3">/\</a> <font color="Maroon" title="c10">k'</font>
<b>by </b><i><a class="txt" href="lopclset.html#E2:12_1"><i><font color="Green" title="E12">A2</font></i></a>, <a class="ref" href="lopclset.html#D3" target="_self" title="LOPCLSET:def.3">Def3</a></i>
<br/>.= 
<font color="Maroon" title="c7">a'</font> <a href="lopclset.html#K3" title="LOPCLSET:func.3">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">b'</font> <a href="lopclset.html#K2" title="LOPCLSET:func.2">\/</a> <font color="Maroon" title="c9">c'</font></span>)</span>
<b>by </b><i><a class="txt" href="lopclset.html#E2:12_1"><i><font color="Green" title="E12">A2</font></i></a>, <a class="ref" href="lopclset.html#D2" target="_self" title="LOPCLSET:def.2">Def2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">a'</font> <a href="lopclset.html#K3" title="LOPCLSET:func.3">/\</a> <font color="Maroon" title="c8">b'</font></span>)</span> <a href="lopclset.html#K2" title="LOPCLSET:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">a'</font> <a href="lopclset.html#K3" title="LOPCLSET:func.3">/\</a> <font color="Maroon" title="c9">c'</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T23" title="XBOOLE_1:th.23">XBOOLE_1:23</a></i>
<br/>.= 
<font color="Maroon" title="c11">l'</font> <a href="lopclset.html#K2" title="LOPCLSET:func.2">\/</a> <font color="Maroon" title="c12">m'</font>
<b>by </b><i><a class="txt" href="lopclset.html#E2:12_1"><i><font color="Green" title="E12">A2</font></i></a>, <a class="txt" href="lopclset.html#E3:12_1"><i><font color="Green" title="E13">A3</font></i></a>, <a class="ref" href="lopclset.html#D3" target="_self" title="LOPCLSET:def.3">Def3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c5">c</font></span>)</span>
<b>by </b><i><a class="txt" href="lopclset.html#E2:12_1"><i><font color="Green" title="E12">A2</font></i></a>, <a class="ref" href="lopclset.html#D2" target="_self" title="LOPCLSET:def.2">Def2</a></i>
;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon" title="c3">bot</font> =  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c1">T</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lopclset.html#K6" title="LOPCLSET:func.6">OpenClosedSetLatt</a> <font color="Maroon" title="c1">T</font></span>)</span> <b>by </b><i><a class="ref" href="lopclset.html#T7" target="_self" title="LOPCLSET:th.7">Th7</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">top</font> =  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lopclset.html#K6" title="LOPCLSET:func.6">OpenClosedSetLatt</a> <font color="Maroon" title="c1">T</font></span>)</span> <b>by </b><i><a class="ref" href="lopclset.html#T8" target="_self" title="LOPCLSET:th.8">Th8</a></i>;<br/>
<a NAME="E4:12"/><i><font color="Green" title="E12">A4</font></i>: 
for <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lopclset.html#K6" title="LOPCLSET:func.6">OpenClosedSetLatt</a> <font color="Maroon" title="c1">T</font></span>)</span> holds <br/> ( <font color="Maroon" title="c4">top</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">top</font> &amp; <font color="Olive" title="b1">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">top</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">top</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lopclset.html#K6" title="LOPCLSET:func.6">OpenClosedSetLatt</a> <font color="Maroon" title="c1">T</font></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon" title="c6">a'</font> = <font color="Maroon" title="c5">a</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="lopclset.html#K1" title="LOPCLSET:func.1">OpenClosedSet</a> <font color="Maroon" title="c1">T</font> ;<br/>
<b>thus </b><font color="Maroon" title="c4">top</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c5">a</font> = 
<span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font></span>)</span> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <font color="Maroon" title="c6">a'</font>
<b>by </b><i><a class="ref" href="lopclset.html#D2" target="_self" title="LOPCLSET:def.2">Def2</a></i>
<br/>.= 
<font color="Maroon" title="c4">top</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T12" title="XBOOLE_1:th.12">XBOOLE_1:12</a></i>
;<br/>

<b>hence </b><a NAME="E3:12_2"/>
<font color="Maroon" title="c5">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">top</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">top</font>
 ;<br/>

<b>thus </b><a NAME="E4:12_2"/>
verum
 ;<br/>


</div><b>end;</b></div>
<a NAME="E5:12"/><i><font color="Green" title="E13">A5</font></i>: 
for <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lopclset.html#K6" title="LOPCLSET:func.6">OpenClosedSetLatt</a> <font color="Maroon" title="c1">T</font></span>)</span> holds <br/> ( <font color="Maroon" title="c3">bot</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">bot</font> &amp; <font color="Olive" title="b1">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c3">bot</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">bot</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lopclset.html#K6" title="LOPCLSET:func.6">OpenClosedSetLatt</a> <font color="Maroon" title="c1">T</font></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon" title="c6">a'</font> = <font color="Maroon" title="c5">a</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="lopclset.html#K1" title="LOPCLSET:func.1">OpenClosedSet</a> <font color="Maroon" title="c1">T</font> ;<br/>
<b>thus </b><font color="Maroon" title="c3">bot</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c5">a</font> = 
<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c6">a'</font>
<b>by </b><i><a class="ref" href="lopclset.html#D3" target="_self" title="LOPCLSET:def.3">Def3</a></i>
<br/>.= 
<font color="Maroon" title="c3">bot</font>

;<br/>

<b>hence </b><a NAME="E3:12_3"/>
<font color="Maroon" title="c5">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c3">bot</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">bot</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E6:12"/><b>then </b><i><font color="Green" title="E14">A6</font></i>: 
 <a href="lopclset.html#K6" title="LOPCLSET:func.6">OpenClosedSetLatt</a> <font color="Maroon" title="c1">T</font> is <a href="lattices.html#V13" title="LATTICES:attr.13">lower-bounded</a>
 
<b>by </b><i><a class="ref" href="lattices.html#D13" title="LATTICES:def.13">LATTICES:def 13</a></i>;<br/>
<a NAME="E7:12"/><i><font color="Green" title="E15">A7</font></i>: 
 <a href="lopclset.html#K6" title="LOPCLSET:func.6">OpenClosedSetLatt</a> <font color="Maroon" title="c1">T</font> is <a href="lattices.html#V14" title="LATTICES:attr.14">upper-bounded</a>
 
<b>by </b><i><a class="txt" href="lopclset.html#E4:12"><i><font color="Green" title="E12">A4</font></i></a>, <a class="ref" href="lattices.html#D14" title="LATTICES:def.14">LATTICES:def 14</a></i>;<br/>
<a NAME="E8:12"/><i><font color="Green" title="E16">A8</font></i>: 
 <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c1">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <span class="p1">(<span class="default"><a href="lopclset.html#K6" title="LOPCLSET:func.6">OpenClosedSetLatt</a> <font color="Maroon" title="c1">T</font></span>)</span>
 
<b>by </b><i><a class="txt" href="lopclset.html#E5:12"><i><font color="Green" title="E13">A5</font></i></a>, <a class="txt" href="lopclset.html#E6:12"><i><font color="Green" title="E14">A6</font></i></a>, <a class="ref" href="lattices.html#D16" title="LATTICES:def.16">LATTICES:def 16</a></i>;<br/>
<a NAME="E9:12"/><i><font color="Green" title="E17">A9</font></i>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <span class="p1">(<span class="default"><a href="lopclset.html#K6" title="LOPCLSET:func.6">OpenClosedSetLatt</a> <font color="Maroon" title="c1">T</font></span>)</span>
 
<b>by </b><i><a class="txt" href="lopclset.html#E4:12"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="lopclset.html#E7:12"><i><font color="Green" title="E15">A7</font></i></a>, <a class="ref" href="lattices.html#D17" title="LATTICES:def.17">LATTICES:def 17</a></i>;<br/>
<b>thus </b><a NAME="E10:12"/>
 <a href="lopclset.html#K6" title="LOPCLSET:func.6">OpenClosedSetLatt</a> <font color="Maroon" title="c1">T</font> is <a href="lattices.html#V15" title="LATTICES:attr.15">bounded</a>
 <b>by </b><i><a class="txt" href="lopclset.html#E6:12"><i><font color="Green" title="E14">A6</font></i></a>, <a class="txt" href="lopclset.html#E7:12"><i><font color="Green" title="E15">A7</font></i></a>, <a class="ref" href="lattices.html#D15" title="LATTICES:def.15">LATTICES:def 15</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D20" title="LATTICES:def.20">LATTICES:def 20</a><br/>

<b>reconsider </b><font color="Maroon" title="c5">OCL</font> =  <a href="lopclset.html#K6" title="LOPCLSET:func.6">OpenClosedSetLatt</a> <font color="Maroon" title="c1">T</font> as   <a href="lattices.html#NM6" title="LATTICES:NM.6">01_Lattice</a> <b>by </b><i><a class="txt" href="lopclset.html#E6:12"><i><font color="Green" title="E14">A6</font></i></a>, <a class="txt" href="lopclset.html#E7:12"><i><font color="Green" title="E15">A7</font></i></a>, <a class="ref" href="lattices.html#D15" title="LATTICES:def.15">LATTICES:def 15</a></i>;<br/>
<a NAME="E12:12"/>
for <font color="Olive" title="b1">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">OCL</font>  ex <font color="Olive" title="b2">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">OCL</font> st <font color="Olive" title="b2">a</font> <a href="lattices.html#R2" title="LATTICES:pred.2">is_a_complement_of</a> <font color="Olive" title="b1">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">OCL</font>;<br/>

<b>reconsider </b><font color="Maroon" title="c7">c</font> = <font color="Maroon" title="c6">b</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="lopclset.html#K1" title="LOPCLSET:func.1">OpenClosedSet</a> <font color="Maroon" title="c1">T</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c8">a</font> = <font color="Maroon" title="c7">c</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">OCL</font> <b>by </b><i><a class="ref" href="lopclset.html#T9" target="_self" title="LOPCLSET:th.9">Th9</a></i>;<br/>
<a NAME="E3:12_4"/><i><font color="Green" title="E18">A10</font></i>: 
<font color="Maroon" title="c8">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c6">b</font> <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="c5">OCL</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_4_1"><b>proof </b></a><div class="add">

<b>thus </b><font color="Maroon" title="c8">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c6">b</font> = 
<font color="Maroon" title="c7">c</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">c</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span>
<b>by </b><i><a class="ref" href="lopclset.html#D2" target="_self" title="LOPCLSET:def.2">Def2</a></i>
<br/>.= 
 <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c5">OCL</font>
<b>by </b><i><a class="txt" href="lopclset.html#E9:12"><i><font color="Green" title="E17">A9</font></i></a>, <a class="ref" href="pre_topc.html#T18" title="PRE_TOPC:th.18">PRE_TOPC:18</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E4:12_4"/><i><font color="Green" title="E19">A11</font></i>: 
<font color="Maroon" title="c7">c</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c7">c</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<b>by </b><i><a class="ref" href="xboole_1.html#T79" title="XBOOLE_1:th.79">XBOOLE_1:79</a></i>;<br/>
<i><font color="Green" title="E20">A12</font></i>: <font color="Maroon" title="c8">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c6">b</font> = 
<font color="Maroon" title="c7">c</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">c</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span>
<b>by </b><i><a class="ref" href="lopclset.html#D3" target="_self" title="LOPCLSET:def.3">Def3</a></i>
<br/>.= 
 <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c5">OCL</font>
<b>by </b><i><a class="txt" href="lopclset.html#E8:12"><i><font color="Green" title="E16">A8</font></i></a>, <a class="txt" href="lopclset.html#E4:12_4"><i><font color="Green" title="E19">A11</font></i></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>
;<br/>
<b>take </b>
<font color="Maroon" title="c8">a</font>
;<br/>

<b>thus </b><a NAME="E6:12_4"/>
<font color="Maroon" title="c8">a</font> <a href="lattices.html#R2" title="LATTICES:pred.2">is_a_complement_of</a> <font color="Maroon" title="c6">b</font>
 <b>by </b><i><a class="txt" href="lopclset.html#E3:12_4"><i><font color="Green" title="E18">A10</font></i></a>, <a class="txt" href="lopclset.html#E5:12_4"><i><font color="Green" title="E20">A12</font></i></a>, <a class="ref" href="lattices.html#D18" title="LATTICES:def.18">LATTICES:def 18</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E13:12"/>
(  <a href="lopclset.html#K6" title="LOPCLSET:func.6">OpenClosedSetLatt</a> <font color="Maroon" title="c1">T</font> is <a href="lattices.html#V16" title="LATTICES:attr.16">complemented</a> &amp;  <a href="lopclset.html#K6" title="LOPCLSET:func.6">OpenClosedSetLatt</a> <font color="Maroon" title="c1">T</font> is <a href="lattices.html#V11" title="LATTICES:attr.11">distributive</a> )
 <b>by </b><i><a class="txt" href="lopclset.html#E1:12"><i><font color="Green" title="E11">A1</font></i></a>, <a class="ref" href="lattices.html#D11" title="LATTICES:def.11">LATTICES:def 11</a>, <a class="ref" href="lattices.html#D19" title="LATTICES:def.19">LATTICES:def 19</a></i>;<br/>


</div>
